 This article is a continuation of Introductory mathematics
It has been known since the time of Euclid^{w} that all of geometry can be derived from a handful of objects (points, lines...), a few actions on those objects, and a small number of axoims^{w}. Every field of science likewise can be reduced to a small set of objects, actions, and rules. Math itself is not a single field but rather a constellation of related fields. One way in which new fields are created is by the process of generalization.
A generalization is the formulation of general concepts from specific instances by abstracting common properties. Generalization is the process of identifying the parts of a whole, as belonging to the whole.^{[1]}
Foreword:
Mathematical notation^{w} can be extremely intimidating. Wikipedia is full of articles with page after page of indecipherable text. At first glance this article might appear to be the same. I want to assure the reader that every effort has been made to simplify everything as much as possible while also providing links to articles with more indepth information.
The following has been assembled from countless small pieces gathered from throughout the world wide web. I cant guarantee that there are no errors in it. Please report any errors or omissions on this articles talk page.
Numbers
Scalars
 See also: Peano axioms^{w} and Hyperoperation^{*}
The basis of all of mathematics is the "Next"^{*} function. See Graph theory^{w}. Next(0)=1, Next(1)=2, Next(2)=3, Next(3)=4. (We might express this by saying that One differs from nothing as two differs from one.) This defines the Natural numbers^{w} (denoted ). Natural numbers are those used for counting.
 These have the convenient property of being transitive^{w}. That means that if a<b and b<c then it follows that a<c. In fact they are totally ordered^{w}. See Order theory^{*}.
Addition^{w} (See Tutorial:arithmetic) is defined as repeatedly calling the Next function, and its inverse is subtraction^{w}. But this leads to the ability to write equations like for which there is no answer among natural numbers. To provide an answer mathematicians generalize to the set of all integers^{w} (denoted because zahlen means count in german) which includes negative integers.
 The Additive identity^{w} is zero because x + 0 = x.
 The absolute value or modulus of x is defined as
 Integers form a ring^{*} (denoted ) over the field of rational numbers. Ring^{w} is defined below.
 Z_{n} is used to denote the set of integers modulo n ^{*}.
 Modular arithmetic^{*} is essentially arithmetic in the quotient ring^{w} Z/nZ (which has n elements).
 An ideal^{*} is a special subset of a ring. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3.
 The study of integers is called Number theory^{w}.
 means a divides b.
 means a does not divide b.
 means p^{a} exactly divides n (i.e. p^{a} divides n but p^{a+1} does not).
 A prime number is a number that can only be divided by itself and one.
 If a, b, c, and d are primes and x=abc and y=c^{2}d then:
Multiplication^{w} (See Tutorial:multiplication) is defined as repeated addition, and its inverse is division^{w}. But this leads to equations like for which there is no answer. The solution is to generalize to the set of rational numbers^{w} (denoted ) which include fractions (See Tutorial:fractions). Any number which isnt rational is irrational^{w}. See also padic number^{*}
 Rational numbers form a division algebra^{*} because every nonzero element has an inverse.
 The set of all rational numbers minus zero forms a multiplicative group^{*}.
 The Multiplicative identity^{w} is one because x * 1 = x.
 Division by zero is undefined and undefinable^{w}. 1/0 exists nowhere on the complex plane^{w}. It does, however, exist on the Riemann sphere^{w} (often called the extended complex plane) where it is surprisingly well behaved. See also Wheel theory^{*} and L'Hôpital's rule^{w}.
 (Addition and multiplication are fast but division is slow even for computers^{*}.)
Exponentiation^{w} (See Tutorial:exponents) is defined as repeated multiplication, and its inverses are roots^{w} and logarithms^{w}. But this leads to multiple equations with no solutions:
 Equations like The solution is to generalize to the set of algebraic numbers^{w} (denoted ). See also algebraic integer^{*}. To see a proof that the square root of two is irrational see Square root of 2^{w}.
 Equations like The solution (because x is transcendental^{w}) is to generalize to the set of Real numbers^{w} (denoted ).
 Equations like and The solution is to generalize to the set of complex numbers^{w} (denoted ) by defining i = sqrt(1). A single complex number consists of a real part a and an imaginary part bi (See Tutorial:complex numbers). Imaginary numbers^{w} (denoted ) often occur in equations involving change with respect to time. If friction is resistance to motion then imaginary friction would be resistance to change of motion wrt time. (In other words, imaginary friction would be mass.) In fact, in the equation for the Spacetime interval^{w} (given below), time itself is an imaginary quantity^{*}.
 Complex numbers can be used to represent and perform rotations^{w} but only in 2 dimensions. Hypercomplex numbers^{w} like quaternions^{w} (denoted ), octonions^{w} (denoted ), and sedenions^{*} (denoted ) are one way to generalize complex numbers to some (but not all) higher dimensions. A quaternion can be thought of as a complex number whose coefficients are themselves complex numbers.
 Where
 and
 and
 Splitcomplex numbers^{*} (hyperbolic complex numbers) are similar to complex numbers except that i^{2} = +1.
 The Complex conjugate^{w} of the complex number is (Not to be confused with the dual^{w} of a vector.)
 The complex numbers are not ordered^{w}. However the absolute value^{w} or modulus^{*} of a complex number is:
 There are n solutions of
 0^0 = 1. See Empty product^{w}.
Tetration^{w} is defined as repeated exponentiation and its inverses are called superroot and superlogarithm.
When a quantity, like the charge of a single electron, becomes so small that it is insignificant we, quite justifiably, treat it as though it were zero. A quantity that can be treated as though it were zero, even though it very definitely is not, is called infinitesimal. If is a finite amount of charge then using Leibniz's notation^{w} would be an infinitesimal amount of charge. See Differential^{w}
Likewise when a quantity becomes so large that a regular finite quantity becomes insignificant then we call it infinite. We would say that the mass of the ocean is infinite . But compared to the mass of the Milky Way galaxy our ocean is insignificant. So we would say the mass of the Galaxy is doubly infinite .
Infinity and the infinitesimal are called Hyperreal numbers^{w} (denoted ). Hyperreals behave, in every way, exactly like real numbers. For example, is exactly twice as big as In reality, the mass of the ocean is a real number so it is hardly surprising that it behaves like one. See Epsilon numbers^{*} and Big O notation^{*}
Binary
Decimal pattern  Binary number 

0  0 
1  1 
2  10 
3  11 
4  100 
5  101 
6  110 
7  111 
8  1000 
9  1001 
10  1010 
11  1011 
12  1100 
13  1101 
14  1110 
15  1111 
The binary numbers 1011 and 1010 are multiplied as follows:
1 0 1 1 (A) (11 in decimal) × 1 0 1 0 (B) (10 in decimal)  0 0 0 0 + 1 0 1 1 + 0 0 0 0 + 1 0 1 1  = 1 1 0 1 1 1 0
Binary numbers can also be multiplied with bits after a binary point:
1 0 1 . 1 0 1 A (5.625 in decimal) × 1 1 0 . 0 1 B (6.25 in decimal)  1 . 0 1 1 0 1 + 0 0 . 0 0 0 0 + 0 0 0 . 0 0 0 + 1 0 1 1 . 0 1 + 1 0 1 1 0 . 1  = 1 0 0 0 1 1 . 0 0 1 0 1 (35.15625 in decimal)
 2^{1} = 2
 2^{2} = 4
 2^{4} = 16
 2^{8} = 256
 2^{16} = 65,536
 2^{32} = 4,294,967,296
 2^{64} = 18,446,744,073,709,551,616 (20 digits)
 2^{128} = 340,282,366,920,938,463,463,374,607,431,768,211,456 (39 digits)
 2^{256} = 115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,
639,936 (78 digits)
Our universe is tiny. Starting with only 2 people and doubling the population every 100 years will in only 27,000 years result in enough people to completely fill the observable universe.
Intervals
 [2,5[ or [2,5) denotes the interval^{w} from 2 to 5, including 2 but excluding 5.
 [3..7] denotes all integers from 3 to 7.
 The set of all reals is unbounded at both ends.
 An open interval does not include its endpoints.
 Compactness^{*} is a property that generalizes the notion of a subset being closed and bounded.
 The unit interval^{*} is the closed interval [0,1]. It is often denoted I.
 The unit square^{*} is a square whose sides have length 1.
 Often, "the" unit square refers specifically to the square in the Cartesian plane^{w} with corners at the four points (0, 0), (1, 0), (0, 1), and (1, 1).
 The unit disk^{*} in the complex plane is the set of all complex numbers of absolute value less than one and is often denoted
Vectors
 See also: Algebraic geometry^{*}, Algebraic variety^{*}, Scheme^{*}, Algebraic manifold^{*}, and Linear algebra^{w}
The one dimensional number line can be generalized to a multidimensional Cartesian coordinate system^{w} thereby creating multidimensional math (i.e. geometry^{w}). See also Curvilinear coordinates^{*}
For sets A and B, the Cartesian product A × B is the set of all ordered pairs^{w} (a, b) where a ∈ A and b ∈ B.^{[2]} The direct product^{*} generalizes the Cartesian product. (See also Direct sum^{*})
 is the Cartesian product^{w}
 is the Cartesian product^{w} (See Complexification^{*})
A vector space^{w} is a coordinate space^{w} with vector addition^{w} and scalar multiplication^{w} (multiplication of a vector and a scalar^{w} belonging to a field^{w}.
 If are orthogonal^{w} unit^{w} basis vectors^{*}
 and are arbitrary vectors then we can (and usually do) write:
 See also: Linear independence^{w}
 A module^{*} generalizes a vector space by allowing multiplication of a vector and a scalar belonging to a ring^{w}.
Coordinate systems define the length of vectors parallel to one of the axes but leave all other lengths undefined. This concept of "length" which only works for certain vectors is generalized as the "norm^{w}" which works for all vectors. The norm of vector is denoted The double bars are used to avoid confusion with the absolute value of the function.
 Taxicab metric^{w} (called L^{1} norm. See L^{p} space^{*}. Sometimes called Lebesgue spaces. See also Lebesgue measure^{w}.)
 In Euclidean space^{w} the norm (called L^{2} norm) doesnt depend on the choice of coordinate system. As a result, rigid objects can rotate in Euclidean space. See proof of the Pythagorean theorem^{w} to the right. L^{2} is the only Hilbert space^{*} among L^{p} spaces.
 In Minkowski space^{w} (See PseudoEuclidean space^{*}) the Spacetime interval^{w} is
 In complex space^{*} the most common norm of an n dimensional vector is obtained by treating it as though it were a regular real valued 2n dimensional vector in Euclidean space
 A Banach space^{*} is a normed vector space^{*} that is also a complete metric space^{w} (there are no points missing from it).
Manifolds 

A manifold^{w} is a type of topological space^{w} in which each point has an infinitely small neighbourhood^{w} that is homeomorphic^{w} to Euclidean space^{w}. A manifold is locally, but not globally, Euclidean. A Riemannian metric^{*} on a manifold allows distances and angles to be measured.
A Lie group^{*} is a group that is also a finitedimensional real smooth manifold, in which the group operation is multiplication rather than addition.^{[4]} n×n invertible matrices^{*} (See below) are a Lie group.

Multiplication of vectors
Multiplication can be generalized to allow for multiplication of vectors in 3 different ways:
Dot product
Dot product^{w} (a Scalar^{w}):
 Strangely, only parallel components multiply.
 The dot product can be generalized to the bilinear form^{w} where A is an (0,2) tensor. (For the dot product in Euclidean space A is the identity tensor. But in Minkowski space A is the Minkowski metric^{*}).
 Two vectors are orthogonal if
 A bilinear form is symmetric if
 Its associated quadratic form^{*} is
 In Euclidean space
 The inner product^{w} is a generalization of the dot product to complex vector space.
 The 2 vectors are called "bra" and "ket"^{*}.
 A Hilbert space^{*} is an inner product space^{w} that is also a Complete metric space^{w}.
 The inner product can be generalized to (a sesquilinear form^{w})
 A complex Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form h : V × V → C such that^{[5]}
 A is a Hermitian operator^{*} iff^{w} Often written as
 The curl operator, is Hermitian.
Outer product
Outer product^{w} (a tensor^{w} called a dyadic^{w}):
 As one would expect, every component of one vector multipies with every component of the other vector.

 Taking the dot product of u⊗v and any vector x (See Visualization of Tensor multiplication^{w}) causes the components of x not pointing in the direction of v to become zero. What remains is then rotated from v to u.
 A rotation matrix can be constructed by summing three outer products. The first two sum to form a bivector. The third one rotates the axis of rotation zero degrees.
 The Tensor product^{w} generalizes the outer product^{w}.
Wedge product
Wedge product^{w} (a simple bivector^{w}):
 The wedge product is also called the exterior product^{w} (sometimes mistakenly called the outer product).
 The term "exterior" comes from the exterior product of two vectors not being a vector.
 Just as a vector has length and direction so a bivector has an area and an orientation.
 In three dimensions is a pseudovector^{w} and its dual^{w} is the cross product^{w}.
 The triple product^{w} a∧b∧c is a trivector which is a 3rd degree tensor.
 In 3 dimensions a trivector is a pseudoscalar so in 3 dimensions every trivector can be represented as a scalar times the unit trivector. See LeviCivita symbol^{w}
 The Matrix commutator^{w} generalizes the wedge product.
 The dual^{w} of vector a is bivector ā:
Tensors
Multiplying a tensor and a vector results in a new vector that can not only have a different magnitude but can even point in a completely different direction:
Some special cases:
One can also multiply a tensor with another tensor. Each column of the second tensor is transformed exactly as a vector would be.
And we can also switch things around:
This is called a Permutation matrix^{*}. See also Permutation group^{*}.
Complex numbers can be used to represent and perform rotations^{w} but only in 2 dimensions.
Tensors^{w}, on the other hand, can be used in any number of dimensions to represent and perform rotations and other linear transformations^{w}. See the image to the right.
 Any affine transformation^{w} is equivalent to a linear transformation followed by a translation^{w} of the origin. (The origin^{w} is always a fixed point for any linear transformation.) "Translation" is just a fancy word for "move".
Just as a vector is a sum of unit vectors multiplied by constants so a tensor is a sum of unit dyadics () multiplied by constants. Each dyadic is associated with a certain plane segment having a certain orientation and magnitude.
The order or degree of the tensor is the dimension of the tensor which is the total number of indices required to identify each component uniquely.^{[6]} A vector is a 1storder tensor.
A simple tensor is a tensor that can be written as a product of tensors of the form (See Outer Product above.) The rank of a tensor T is the minimum number of simple tensors that sum to T.^{[7]} A bivector^{w} is a tensor of rank 2.
The Determinant^{w} of a matrix is the area or volume spanned by its column vectors and is frequently useful.
From Wikipedia:Matrix similarity
In linear algebra, two nbyn matrices A and B are called similar if
for some invertible nbyn matrix P. Similar matrices represent the same linear operator^{*} under two (possibly) different bases^{*}, with P being the change of basis^{*} matrix.^{[8]}^{[9]}
A transformation A ↦ P^{−1}AP is called a similarity transformation or conjugation of the matrix A. In the general linear group^{*}, similarity is therefore the same as conjugacy^{*}, and similar matrices are also called conjugate; however in a given subgroup H of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that P be chosen to lie in H.
Decomposition of tensors 

Every tensor of degree 2 can be decomposed into a symmetric and an antisymmetric tensor The Outer product (tensor product) of a vector with itself is a symmetric tensor: The wedge product of 2 vectors is antisymmetric: Any matrix X with complex entries can be expressed as where
This is the Jordan–Chevalley decomposition^{*}. 
Block matrix 

The matrix can be partitioned into 4 2×2 blocks The partitioned matrix can then be written as the matrix product can be formed blockwise, yielding as an matrix with row partitions and column partitions. The matrices in the resulting matrix are calculated by multiplying: Or, using the Einstein notation^{*} that implicitly sums over repeated indices: 
Linear groups
A square matrix^{w} of order n is an nbyn matrix. Any two square matrices of the same order can be added and multiplied. A matrix is invertible if and only if its determinant is nonzero.
GL_{n}(F) or GL(n, F), or simply GL(n) is the Lie group^{*} of n×n invertible matrices with entries from the field F. The group GL(n, F) and its subgroups are often called linear groups or matrix groups.
 SL(n, F) or SL_{n}(F), is the subgroup^{*} of GL(n, F) consisting of matrices with a determinant^{w} of 1.
 U(n), the Unitary group of degree n is the group^{w} of n × n unitary matrices^{w}. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication^{w}.^{[10]}
 SU(n), the special unitary group of degree n, is the Lie group^{*} of n×n unitary matrices^{w} with determinant^{w} 1.
Symmetry groups
Affine group^{*}
 Poincaré group^{*}: boosts, rotations, translations
 Lorentz group^{*}: boosts, rotations
 The set of all boosts, however, does not form a subgroup, since composing two boosts does not, in general, result in another boost. (Rather, a pair of noncolinear boosts is equivalent to a boost and a rotation, and this relates to Thomas rotation.)
Aff(n,K): the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.
 E(n): rotations, reflections, and translations.
 O(n): rotations, reflections
 SO(n): rotations
 so(3) is the Lie algebra of SO(3) and consists of all skewsymmetric^{w} 3 × 3 matrices.
Clifford group: The set of invertible elements x such that for all v in V The spinor norm^{*} Q is defined on the Clifford group by
 Pin_{V}(K): The subgroup of elements of spinor norm 1. Maps 2to1 to the orthogonal group
 Spin_{V}(K): The subgroup of elements of Dickson invariant 0 in Pin_{V}(K). When the characteristic is not 2, these are the elements of determinant 1. Maps 2to1 to the special orthogonal group. Elements of the spin group act as linear transformations on the space of spinors
Rotations
In 4 spatial dimensions a rigid object can rotate in 2 different ways simultaneously^{*}.
 See also: Hypersphere of rotations^{*}, Rotation group SO(3)^{*}, Special unitary group^{*}, Plate trick^{*}, Spin representation^{*}, Spin group^{*}, Pin group^{*}, Spinor^{*}, Clifford algebra^{w}, Indefinite orthogonal group^{*}, Root system^{*}, Bivectors^{w}, Curl^{w}
Consider the solid ball in R^{3} of radius π. For every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The two rotations through π and through −π are the same. So we identify^{*} (or "glue together") antipodal points^{*} on the surface of the ball.
The ball with antipodal surface points identified is a smooth manifold^{*}, and this manifold is diffeomorphic^{*} to the rotation group. It is also diffeomorphic to the real 3dimensional projective space^{*} RP^{3}, so the latter can also serve as a topological model for the rotation group.
These identifications illustrate that SO(3) is connected^{*} but not simply connected^{*}. As to the latter, consider the path running from the "north pole" straight through the interior down to the south pole. This is a closed loop, since the north pole and the south pole are identified. This loop cannot be shrunk to a point, since no matter how you deform the loop, the start and end point have to remain antipodal, or else the loop will "break open".
Surprisingly, if you run through the path twice, i.e., run from north pole down to south pole, jump back to the north pole (using the fact that north and south poles are identified), and then again run from north pole down to south pole, so that φ runs from 0 to 4π, you get a closed loop which can be shrunk to a single point: first move the paths continuously to the ball's surface, still connecting north pole to south pole twice. The second half of the path can then be mirrored over to the antipodal side without changing the path at all. Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle. This circle can be shrunk to the north pole without problems. The Balinese plate trick^{*} and similar tricks demonstrate this practically.
The same argument can be performed in general, and it shows that the fundamental group^{*} of SO(3) is cyclic group^{w} of order 2. In physics applications, the nontriviality of the fundamental group allows for the existence of objects known as spinors^{*}, and is an important tool in the development of the spinstatistics theorem^{*}.
Spin group 

The universal cover^{*} of SO(3) is a Lie group^{*} called Spin(3)^{*}. The group Spin(3) is isomorphic to the special unitary group^{*} SU(2); it is also diffeomorphic to the unit 3sphere^{*} S^{3} and can be understood as the group of versors^{*} (quaternions^{w} with absolute value^{w} 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotation^{*}. The map from S^{3} onto SO(3) that identifies antipodal points of S^{3} is a surjective^{*} homomorphism^{*} of Lie groups, with kernel^{*} {±1}. Topologically, this map is a twotoone covering map^{*}. (See the plate trick^{*}.)
The spin group Spin(n)^{[11]}^{[12]} is the double cover^{*} of the special orthogonal group^{*} SO(n) = SO(n, R), such that there exists a short exact sequence^{*} of Lie groups^{*} (with n ≠ 2) As a Lie group, Spin(n) therefore shares its dimension^{*}, n(n − 1)/2, and its Lie algebra^{*} with the special orthogonal group. For n > 2, Spin(n) is simply connected^{*} and so coincides with the universal cover^{*} of SO(n)^{*}. The nontrivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin^{*}, generally denoted −I . Spin(n) can be constructed as a subgroup^{*} of the invertible elements in the Clifford algebra^{w} Cl(n). A distinct article discusses the spin representations^{*}. 
Matrix representations
 See also: Group representation^{*}
Real numbers
If a vector is multiplied with the the identity matrix^{*} then the vector is completely unchanged:
And if then
Therefore can be thought of as the matrix form of the scalar a.
 .
 .
(Note: Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.)
Complex numbers
Complex numbers can also be written in matrix form^{w} in such a way that complex multiplication corresponds perfectly to matrix multiplication:
Quaternions
There are at least two ways of representing quaternions as matrices^{w} in such a way that quaternion addition and multiplication correspond to matrix addition and matrix multiplication^{w}.
Using 2 × 2 complex matrices, the quaternion a + bi + cj + dk can be represented as
Using 4 × 4 real matrices, that same quaternion can be written as
Therefore:
The obvious way of representing quaternions with 3 × 3 real matrices does not work because:
Vectors
 See also: Splitcomplex numbers^{*}
Unfortunately the matrix representation of a vector is not so obvious. First we must decide what properties the matrix should have. To see consider the square (quadratic form^{*}) of a single vector:
From the Pythagorean theorem we know that:
So we know that
The set of 3 matrices in 3 dimensions that have these properties are called Pauli matrices.
The Pauli matrices are a set of three 2 × 2 complex^{w} matrices^{w} which are Hermitian^{w} and unitary^{w}.^{[13]} They are
Squaring a Pauli matrix results in a "scalar":
Multiplication is anticommutative^{*}:
And
commutation^{w} relations:
anticommutation^{*} relations:
Exponential of a Pauli vector:
Adding the commutator to the anticommutator gives:
If is identified with the pseudoscalar then the right hand side becomes which is also the definition for the geometric product^{w} of two vectors in geometric algebra^{w} (Clifford algebra^{w}). The geometric product of two vectors is a multivector^{w}.
Isomorphism to quaternions 

Multiplying any 2 Pauli matrices results in a quaternion: Quaternions form a division algebra^{*}—every nonzero element has an inverse—whereas Pauli matrices do not. And multiplying a Pauli matrix and a quaternion results in a Pauli matrix: It would appear therefore that quaternions are to the matrix representations of vectors what bivectors are to ordinary vectors. Note: The (real) spinors^{*} in threedimensions are quaternions, and the action of an evengraded element on a spinor is given by ordinary quaternionic multiplication.^{[14]} 
Further reading: Generalizations of Pauli matrices^{*}, GellMann matrices^{*} and Pauli equation^{*}
Multivectors
 See also: Dirac algebra^{*}
External links:
 A brief introduction to geometric algebra
 A brief introduction to Clifford algebra
 The Construction of Spinors in Geometric Algebra
 Functions of Multivector Variables
 From Wikipedia:Multivector:
The wedge product^{w} operation (See Exterior algebra^{w}) used to construct multivectors is linear, associative and alternating, which reflect the properties of the determinant. This means for vectors u, v and w in a vector space V and for scalars α, β, the wedge product has the properties,
 Linear:
 Associative:
 Alternating:
However the wedge product is not invertible because many different pairs of vectors can have the same wedge product.
The product of p vectors, , is called a grade p multivector, or a pvector. The maximum grade of a multivector is the dimension of the vector space V.
The set of all possible products of orthogonal basis vectors with indices in increasing order, including as the empty product, forms a basis for the entire geometric algebra (an analogue of the PBW theorem^{*}).
Canonical basis 

For example, the following is a basis for the geometric algebra : A basis formed this way is called a canonical basis for the geometric algebra, and any other orthogonal basis for will produce another canonical basis. Each canonical basis consists of elements. Every multivector of the geometric algebra can be expressed as a linear combination of the canonical basis elements. The general element of the Clifford algebra Cℓ_{0,3}(R) is given by The linear combination of the even degree elements of Cℓ_{0,3}(R) defines the even subalgebra Cℓ[0] The basis elements can be identified with the quaternion basis elements i, j, k as The linearity of the wedge product allows a multivector to be defined as the linear combination of basis multivectors. There are (n 
W. K. Clifford^{*} combined multivectors with the inner product^{w} defined on the vector space, in order to obtain a general construction for hypercomplex numbers that includes the usual complex numbers and Hamilton's quaternions^{w}.^{[16]}^{[17]}
The Clifford product between two vectors is linear and associative like the wedge product. But unlike the wedge product the Clifford product is invertible.
Clifford's relation preserves the alternating property for the product of vectors that are perpendicular. But in contrast to the wedge product, the Clifford product of a vector with itself is no longer zero.
We know that velocity is a vector and that velocity^2 = energy. We also know that energy is a scalar.
Therefore the rules of Clifford algebra require:
Now would be a good time to point out that and are in reality gamma matrices not vectors. Gamma matrices are constructed in such a way as to cause the mathematical relationships shown above to be true. See below
And futher that:
And i, as we already know, has the effect of rotating complex numbers.
For any 2 arbitrary vectors:
Applying the rules of Clifford algebra we get:
For comparison here is the outer product of the same 2 vectors:
 (See divergence, curl, & gradient below)
This particular Clifford algebra is known as Cl_{2,0}. The subscript 2 indicates that the 2 basis vectors are square roots of +1. See Metric signature^{*}. If we had used then the result would have been Cl_{0,2}.
Every nondegenerate quadratic form on a finitedimensional real vector space is equivalent to the standard diagonal form:
where n = p + q is the dimension of the vector space. The pair of integers (p, q) is called the signature^{*} of the quadratic form. The real vector space with this quadratic form is often denoted R^{p,q}. The Clifford algebra on R^{p,q} is denoted Cℓ_{p,q}(R). The symbol Cℓ_{n}(R) means either Cℓ_{n,0}(R) or Cℓ_{0,n}(R) depending on whether the author prefers positivedefinite or negativedefinite spaces.
A standard basis^{w} {e_{i}} for R^{p,q} consists of n = p + q mutually orthogonal vectors, p of which square to +1 and q of which square to −1. The algebra Cℓ_{p,q}(R) will therefore have p vectors that square to +1 and q vectors that square to −1.
Spacetime algebra^{*} (STA) is a name for the Clifford algebra^{w} Cl_{1,3}(R), or equivalently the geometric algebra^{w} G(M^{4}), which can be particularly closely associated with the geometry of special relativity^{w} and relativistic spacetime^{w}. See also Algebra of physical space^{*}.
The spacetime algebra may be built up from an orthogonal basis of one timelike vector and three spacelike vectors, , with the multiplication rule
where is the Minkowski metric^{w} with signature (+ − − −).
Thus, , , otherwise .
The basis vectors share these properties with the Gamma matrices^{*}, but no explicit matrix representation need be used in STA.
Associated with the orthogonal basis is the reciprocal basis for , satisfying the relation
 (See Kronecker delta^{*})
These reciprocal frame vectors differ only by a sign, with , and for .
A vector may be represented in either upper or lower index coordinates with summation over , according to the Einstein notation^{*}, where the coordinates may be extracted by taking dot products with the basis vectors or their reciprocals.
Gamma matrices
 See also: Electron magnetic moment^{*}
Gamma matrices^{*}, , also known as the Dirac matrices, are a set of 4 × 4 conventional matrices with specific anticommutation^{*} relations that ensure they generate^{*} a matrix representation of the Clifford algebra^{w} Cℓ_{1,3}(R). One gamma matrix squares to 1 times the identity matrix^{*} and three gamma matrices square to 1 times the identity matrix.
The defining property for the gamma matrices to generate a Clifford algebra^{w} is the anticommutation relation
where is the anticommutator^{*}, is the Minkowski metric^{*} with signature (+ − − −) and is the 4 × 4 identity matrix.
Minkowski metric 

From Wikipedia:Minkowski_space#Minkowski_metric The simplest example of a Lorentzian manifold is flat spacetime^{*}, which can be given as R^{4} with coordinates and the metric Note that these coordinates actually cover all of R^{4}. The flat space metric (or Minkowski metric^{*}) is often denoted by the symbol η and is the metric used in special relativity^{*}. A standard basis for Minkowski space is a set of four mutually orthogonal vectors { e_{0}, e_{1}, e_{2}, e_{3} } such that These conditions can be written compactly in the form Relative to a standard basis, the components of a vector v are written (v^{0}, v^{1}, v^{2}, v^{3}) where the Einstein summation convention^{*} is used to write v = v^{μ}e_{μ}. The component v^{0} is called the timelike component of v while the other three components are called the spatial components. The spatial components of a 4vector v may be identified with a 3vector v = (v_{1}, v_{2}, v_{3}). In terms of components, the Minkowski inner product between two vectors v and w is given by and Here lowering of an index with the metric was used. The Minkowski metric^{[18]} η is the metric tensor of Minkowski space. It is a pseudoEuclidean metric, or more generally a constant pseudoRiemannian metric in Cartesian coordinates. As such it is a nondegenerate symmetric bilinear form, a type (0,2) tensor. It accepts two arguments u, v. The definition yields an inner productlike structure on M, previously and also henceforth, called the Minkowski inner product, similar to the Euclidean inner product, but it describes a different geometry. It is also called the relativistic dot product. If the two arguments are the same, the resulting quantity will be called the Minkowski norm squared. This bilinear form can in turn be written as where [η] is a 4×4 matrix associated with η. Possibly confusingly, denote [η] with just η as is common practice. The matrix is read off from the explicit bilinear form as and the bilinear form with which this section started by assuming its existence, is now identified. 
When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant^{*} vectors in Minkowski space^{w}, the column vectors on which the matrices act become a space of spinors^{*}, on which the Clifford algebra of spacetime^{*} acts. This in turn makes it possible to represent infinitesimal spatial rotations^{*} and Lorentz boosts^{w}. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation^{*} for relativistic spin½ particles.
In Dirac representation^{w}, the four contravariant^{*} gamma matrices are
is the timelike matrix and the other three are spacelike matrices.
The matrices are also sometimes written using the 2×2 [[[Wikipedia:[identity matrix[identity matrix]]^{[[[identity matrix*]]}, , and the Pauli matrices^{*}.
The gamma matrices we have written so far are appropriate for acting on Dirac spinors^{*} written in the Dirac basis; in fact, the Dirac basis is defined by these matrices. To summarize, in the Dirac basis:
Another common choice is the Weyl or chiral basis,^{[19]} in which remains the same but is different, and so is also different, and diagonal,
Original Dirac matrices 

where are the Pauli matrices and is the Kronecker product^{*} (not the tensor product) The 16 original Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214): Any of the 15 original Dirac matrices (excluding the identity matrix ) anticommute with eight other original Dirac matrices and commute with the remaining eight, including itself and the identity matrix. Any of the 16 original Dirac matrices multiplied times itself equals 
Higherdimensional gamma matrices 

Analogous sets of gamma matrices can be defined in any dimension^{*} and for any signature of the metric. For example, the Pauli matrices are a set of "gamma" matrices in dimension 3 with metric of Euclidean signature (3,0). In 5 spacetime dimensions, the 4 gammas above together with the fifth gamma matrix to be presented below generate the Clifford algebra. It is useful to define the product of the four gamma matrices as follows:
Although uses the letter gamma, it is not one of the gamma matrices of Cℓ_{1,3}(R). The number 5 is a relic of old notation in which was called "". From Wikipedia:Higherdimensional gamma matrices Consider a spacetime of dimension d with the flat Minkowski metric^{*}, where a,b = 0,1, ..., d−1. Set N= 2^{⌊d/2⌋}. The standard Dirac matrices correspond to taking d = N = 4. The higher gamma matrices are a dlong sequence of complex N×N matrices which satisfy the anticommutator^{*} relation from the Clifford algebra^{*} Cℓ_{1,d−1}(R) (generating a representation for it), where I_{N} is the identity matrix^{*} in N dimensions. (The spinors acted on by these matrices have N components in d dimensions.) Such a sequence exists for all values of d and can be constructed explicitly, as provided below. The gamma matrices have the following property under hermitian conjugation, 
Further reading: Quantum Mechanics for Engineers and How (not) to teach Lorentz covariance of the Dirac equation
Rotors
 See also: Rotor (mathematics)^{*}
The inverse of a vector is:
The projection of onto (or the parallel part) is
and the rejection of from (or the orthogonal part) is
The reflection of a vector along a vector , or equivalently across the hyperplane orthogonal to , is the same as negating the component of a vector parallel to . The result of the reflection will be

If a is a unit vector then and therefore
is called the sandwich product which is called a doublesided product.
If we have a product of vectors then we denote the reverse as
Any rotation is equivalent to 2 reflections.
R is called a Rotor
If a and b are unit vectors then the rotor is automatically normalised:
2 rotations becomes:
R_{2}R_{1} represents Rotor R_{1} rotated by Rotor R_{2}. This would be called a singlesided transformation. (R_{2}R_{1}R_{2} would be doublesided.) Therefore rotors do not transform doublesided the same way that other objects do. They transform singlesided.
Quaternions
The square root of the product of a quaternion with its conjugate is called its norm^{*}:
A unit quaternion is a quaternion of norm one. Unit quaternions, also known as versors^{*}, provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions.
Every nonzero quaternion has a multiplicative inverse
Thus quaternions form a division algebra^{*}.
The inverse of a unit quaternion is obtained simply by changing the sign of its imaginary components.
A 3D Euclidean vector^{*} such as (2, 3, 4) or (a_{x}, a_{y}, a_{z}) can be rewritten as 0 + 2 i + 3 j + 4 k or 0 + a_{x} i + a_{y} j + a_{z} k, where i, j, k are unit vectors representing the three Cartesian axes^{*}. A rotation through an angle of θ around the axis defined by a unit vector
can be represented by a quaternion. This can be done using an extension^{*} of Euler's formula^{w}:
It can be shown that the desired rotation can be applied to an ordinary vector in 3dimensional space, considered as a quaternion with a real coordinate equal to zero, by evaluating the conjugation of p by q:
using the Hamilton product^{*}
The conjugate of a product of two quaternions is the product of the conjugates in the reverse order.
Conjugation by the product of two quaternions is the composition of conjugations by these quaternions: If p and q are unit quaternions, then rotation (conjugation) by pq is
 ,
which is the same as rotating (conjugating) by q and then by p. The scalar component of the result is necessarily zero.
The imaginary part of a quaternion behaves like a vector in three dimension vector space, and the real part a behaves like a scalar^{*} in R. When quaternions are used in geometry, it is more convenient to define them as a scalar plus a vector^{*}:
When multiplying the vector/imaginary parts, in place of the rules i^{2} = j^{2} = k^{2} = ijk = −1 we have the quaternion multiplication rule:
From these rules it follows immediately that (see details^{*}):
It is important to note, however, that the vector part of a quaternion is, in truth, an "axial" vector or "pseudovector", not an ordinary or "polar" vector.
 From Wikipedia:Quaternion:
the reflection of a vector r in a plane perpendicular to a unit vector w can be written:
Two reflections make a rotation by an angle twice the angle between the two reflection planes, so
corresponds to a rotation of 180° in the plane containing σ_{1} and σ_{2}.
This is very similar to the corresponding quaternion formula,
In fact, the two are identical, if we make the identification
and it is straightforward to confirm that this preserves the Hamilton relations
In this picture, quaternions correspond not to vectors but to bivectors^{w} – quantities with magnitude and orientations associated with particular 2D planes rather than 1D directions. The relation to complex numbers^{w} becomes clearer, too: in 2D, with two vector directions σ_{1} and σ_{2}, there is only one bivector basis element σ_{1}σ_{2}, so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements σ_{1}σ_{2}, σ_{2}σ_{3}, σ_{3}σ_{1}, so three imaginaries.
The usefulness of quaternions for geometrical computations can be generalised to other dimensions, by identifying the quaternions as the even part Cℓ^{+}_{3,0}(R) of the Clifford algebra^{w} Cℓ_{3,0}(R).
Spinors
 See also: Bispinor^{*}
External link:An introduction to spinors
Spinors may be regarded as nonnormalised rotors which transform singlesided.^{[20]}
Note: The (real) spinors^{*} in threedimensions are quaternions, and the action of an evengraded element on a spinor is given by ordinary quaternionic multiplication.^{[21]}
A spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360°. This property characterizes spinors.^{[22]}
In three dimensions...the Lie group^{*} SO(3)^{*} is not simply connected^{*}. Mathematically, one can tackle this problem by exhibiting the special unitary group^{*} SU(2), which is also the spin group^{*} in three Euclidean^{*} dimensions, as a double cover^{*} of SO(3).
SU(2) is the following group,^{[23]}
where the overline denotes complex conjugation^{*}.
For comparison: Using 2 × 2 complex matrices, the quaternion a + bi + cj + dk can be represented as
If X = (x_{1},x_{2},x_{3}) is a vector in R^{3}, then we identify X with the 2 × 2 matrix with complex entries
Note that −det(X) gives the square of the Euclidean length of X regarded as a vector, and that X is a tracefree^{*}, or better, tracezero Hermitian matrix^{*}.
The unitary group acts on X via
where M ∈ SU(2). Note that, since M is unitary,
 , and
 is tracezero Hermitian.
Hence SU(2) acts via rotation on the vectors X. Conversely, since any change of basis^{*} which sends tracezero Hermitian matrices to tracezero Hermitian matrices must be unitary, it follows that every rotation also lifts to SU(2). However, each rotation is obtained from a pair of elements M and −M of SU(2). Hence SU(2) is a doublecover of SO(3). Furthermore, SU(2) is easily seen to be itself simply connected by realizing it as the group of unit quaternions^{*}, a space homeomorphic^{*} to the 3sphere^{*}.
A unit quaternion has the cosine of half the rotation angle as its scalar part and the sine of half the rotation angle multiplying a unit vector along some rotation axis (here assumed fixed) as its pseudovector (or axial vector) part. If the initial orientation of a rigid body (with unentangled connections to its fixed surroundings) is identified with a unit quaternion having a zero pseudovector part and +1 for the scalar part, then after one complete rotation (2pi rad) the pseudovector part returns to zero and the scalar part has become 1 (entangled). After two complete rotations (4pi rad) the pseudovector part again returns to zero and the scalar part returns to +1 (unentangled), completing the cycle.
The association of a spinor with a 2×2 complex Hermitian matrix^{*} was formulated by Élie Cartan.^{[24]}
In detail, given a vector x = (x_{1}, x_{2}, x_{3}) of real (or complex) numbers, one can associate the complex matrix
Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3space:
 det X = – (length x)^{2}.
 X ^{2} = (length x)^{2}I, where I is the identity matrix.
 ^{[24]}
 where Z is the matrix associated to the cross product z = x × y.
 If u is a unit vector, then −UXU is the matrix associated to the vector obtained from x by reflection in the plane orthogonal to u.
 It is an elementary fact from linear algebra^{*} that any rotation in 3space factors as a composition of two reflections. (Similarly, any orientation reversing orthogonal transformation is either a reflection or the product of three reflections.) Thus if R is a rotation, decomposing as the reflection in the plane perpendicular to a unit vector u_{1} followed by the plane perpendicular to u_{2}, then the matrix U_{2}U_{1}XU_{1}U_{2} represents the rotation of the vector x through R.
Having effectively encoded all of the rotational linear geometry of 3space into a set of complex 2×2 matrices, it is natural to ask what role, if any, the 2×1 matrices (i.e., the column vectors^{*}) play. Provisionally, a spinor is a column vector
 with complex entries ξ_{1} and ξ_{2}.
The space of spinors is evidently acted upon by complex 2×2 matrices. Furthermore, the product of two reflections in a given pair of unit vectors defines a 2×2 matrix whose action on euclidean vectors is a rotation, so there is an action of rotations on spinors.
Often, the first example of spinors that a student of physics encounters are the 2×1 spinors used in Pauli's theory of electron spin. The Pauli matrices^{*} are a vector of three 2×2 matrices^{*} that are used as spin^{*} operators^{*}.
Given a unit vector^{*} in 3 dimensions, for example (a, b, c), one takes a dot product^{*} with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector.
The eigenvectors^{*} of that spin matrix are the spinors for spin1/2 oriented in the direction given by the vector.
Example: u = (0.8, 0.6, 0) is a unit vector. Dotting this with the Pauli spin matrices gives the matrix:
The eigenvectors may be found by the usual methods of linear algebra^{*}, but a convenient trick is to note that a Pauli spin matrix is an involutory matrix^{*}, that is, the squareof the above matrix is the identity matrix.
Thus a (matrix) solution to the eigenvector problem with eigenvalues of ±1 is simply 1 ± S_{u}. That is,
One can then choose either of the columns of the eigenvector matrix as the vector solution, provided that the column chosen is not zero. Taking the first column of the above, eigenvector solutions for the two eigenvalues are:
The trick used to find the eigenvectors is related to the concept of ideals^{*}, that is, the matrix eigenvectors (1 ± S_{u})/2 are projection operators^{*} or idempotents^{*} and therefore each generates an ideal in the Pauli algebra. The same trick works in any Clifford algebra^{*}, in particular the Dirac algebra^{*} that are discussed below. These projection operators are also seen in density matrix^{*} theory where they are examples of pure density matrices.
More generally, the projection operator for spin in the (a, b, c) direction is given by
and any non zero column can be taken as the projection operator. While the two columns appear different, one can use a^{2} + b^{2} + c^{2} = 1 to show that they are multiples (possibly zero) of the same spinor.
 From Wikipedia:Tensor#Spinors:
When changing from one orthonormal basis^{*} (called a frame) to another by a rotation, the components of a tensor transform by that same rotation. This transformation does not depend on the path taken through the space of frames. However, the space of frames is not simply connected^{*} (see orientation entanglement^{*} and plate trick^{*}): there are continuous paths in the space of frames with the same beginning and ending configurations that are not deformable one into the other. It is possible to attach an additional discrete invariant to each frame that incorporates this path dependence, and which turns out (locally) to have values of ±1.^{[25]} A spinor^{*} is an object that transforms like a tensor under rotations in the frame, apart from a possible sign that is determined by the value of this discrete invariant.^{[26]}^{[27]}
Succinctly, spinors are elements of the spin representation^{*} of the rotation group, while tensors are elements of its tensor representations^{*}. Other classical groups^{*} have tensor representations, and so also tensors that are compatible with the group, but all noncompact classical groups have infinitedimensional unitary representations as well.
 From Wikipedia:Spinor:
Quote from Elie Cartan: The Theory of Spinors, Hermann, Paris, 1966: "Spinors...provide a linear representation of the group of rotations in a space with any number of dimensions, each spinor having components where or ." The star (*) refers to Cartan 1913.
(Note: is the number of simultaneous independent rotations^{*} an object can have in n dimensions.)
Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest to work with. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anticommutation relations. The spinors are the column vectors on which these matrices act. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, and the twocomponent complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. As a representation of the spin group, this realization of spinors as (complex) column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of socalled "halfspin" or Weyl representations if the dimension is even.
In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to (angular momenta about) the three coordinate axes. These are 2×2 matrices with complex entries, and the twocomponent complex column vectors on which these matrices act by matrix multiplication are the spinors. In this case, the spin group is isomorphic to the group of 2×2 unitary matrices with determinant one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves, realizing it as a group of rotations among them, but it also acts on the column vectors (that is, the spinors).
 From Wikipedia:Spinor:
In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles. More precisely, it is the fermions of spin1/2 that are described by spinors, which is true both in the relativistic and nonrelativistic theory. The wavefunction of the nonrelativistic electron has values in 2 component spinors transforming under threedimensional infinitesimal rotations. The relativistic Dirac equation^{*} for the electron is an equation for 4 component spinors transforming under infinitesimal Lorentz transformations for which a substantially similar theory of spinors exists.
Maxwell's equations
Analogous to the tensor formulation, two objects, one for the field and one for the current, are introduced. In geometric algebra^{w} (GA) these are multivectors^{w}. The field multivector, known as the Riemann–Silberstein vector^{*}, is
and the current multivector is
where, in the algebra of physical space^{*} (APS) with the vector basis . The unit pseudoscalar^{*} is (assuming an orthonormal basis^{*}). Orthonormal basis vectors share the algebra of the Pauli matrices^{*}, but are usually not equated with them. After defining the derivative
Maxwell's equations are reduced to the single equation^{[28]}
In three dimensions, the derivative has a special structure allowing the introduction of a cross product:
from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation. After expanding and rearranging, this can be written as
We can identify APS as a subalgebra of the spacetime algebra^{*} (STA) , defining and . The s have the same algebraic properties of the gamma matrices^{*} but their matrix representation is not needed. The derivative is now
The Riemann–Silberstein becomes a bivector
and the charge and current density become a vector
Owing to the identity
Maxwell's equations reduce to the single equation
Functions
From Wikipedia:Function (mathematics)
In mathematics, a function is a relation^{*} between a set^{*} of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number^{*} x to its square x^{2}. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. See Tutorial:Evaluate by Substitution. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s)^{*} are sometimes referred to as the argument(s) of the function.
Euclids "common notions"
Things that do not differ from one another are equal to one another
a=a 
Things that are equal to the same thing are also equal to one another
If 
 then a=c 
If equals are added to equals, then the wholes are equal
If 
 then a+c=b+d 
If equals are subtracted from equals, then the remainders are equal
If 
 then ac=bd 
The whole is greater than the part.
If  b≠0  then a+b>a 
Elementary algebra
Elementary algebra builds on and extends arithmetic by introducing letters called variables^{*} to represent general (nonspecified) numbers.
Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition^{w}, subtraction^{w}, multiplication^{w}, division^{w} and exponentiation^{w}). For example,
 Added terms are simplified using coefficients. For example, can be simplified as (where 3 is a numerical coefficient).
 Multiplied terms are simplified using exponents. For example, is represented as
 Like terms are added together,^{[29]} for example, is written as , because the terms containing are added together, and, the terms containing are added together.
 Brackets can be "multiplied out", using the distributive property^{w}. For example, can be written as which can be written as
 Expressions can be factored. For example, , by dividing both terms by can be written as
For any function , if then:
One must be careful though when squaring both sides of an equation since this can result is solutions that dont satisfy the original equation.
 yet
A function is an even function^{w} if f(x) = f(x)
A function is an odd function^{w} if f(x) = f(x)
Trigonometry
The law of cosines^{w} reduces to the Pythagorean theorem^{w} when gamma=90 degrees
The law of sines^{w} (also known as the "sine rule") for an arbitrary triangle states:
where is the area of the triangle
The law of tangents^{w}:
Right triangles
A right triangle is a triangle with gamma=90 degrees.
For small values of x, sin x ≈ x. (If x is in radians).
SOH → sin = "opposite" / "hypotenuse" CAH → cos = "adjacent" / "hypotenuse" TOA → tan = "opposite" / "adjacent" 
= sin A = a/c = cos A = b/c = tan A = a/b 
(Note: the expression of tan(x) has i in the numerator, not in the denominator, because the order of the terms (and thus the sign) of the numerator is changed w.r.t. the expression of sin(x).)
Hyperbolic functions
 See also: Hyperbolic angle^{*}
Hyperbolic functions^{w} are analogs of the ordinary trigonometric, or circular, functions.
 Hyperbolic sine:
 Hyperbolic cosine:
 Hyperbolic tangent:
 Hyperbolic cotangent:
 Hyperbolic secant:
 Hyperbolic cosecant:
Areas and volumes
The length of the circumference C of a circle is related to the radius r and diameter d by:
 where
 = 3.141592654
 = 2 * π
The area of a circle is:
The surface area of a sphere is
 The surface area of a sphere 1 unit in radius is:
 The surface area of a sphere 128 units in radius is:
The volume inside a sphere is
 The volume of a sphere 1 unit in radius is:
The area of a hexagon is:
 where a is the length of any side.
Polynomials
 See also: Runge's phenomenon^{*}, Polynomial ring^{*}, System of polynomial equations^{*}, Rational root theorem^{*}, Descartes' rule of signs^{*}, and Complex conjugate root theorem^{*}
 From Wikipedia:Polynomial:
A polynomial^{w} can always be written in the form
where are constants called coefficients and n is the degree^{w} of the polynomial.
 A linear polynomial^{*} is a polynomial of degree one.
Each individual term^{*} is the product of the coefficient^{*} and a variable raised to a nonnegative integer power.
 A monomial^{*} has only one term.
 A binomial^{*} has 2 terms.
Fundamental theorem of algebra^{*}:
 Every singlevariable, degree n polynomial with complex coefficients has exactly n complex roots^{w}.
 However, some or even all of the roots might be the same number.
 A root (or zero) of a function is a value of x for which Z(x)=0.
 If then z_{2} is a root of multiplicity^{*} k.^{[30]} z_{2} is a root of multiplicity k1 of the derivative (Derivative is defined below) of Z(x).
 If k=1 then z_{2} is a simple root.
 The graph is tangent to the x axis at the multiple roots of f and not tangent at the simple roots.
 The graph crosses the xaxis at roots of odd multiplicity and bounces off (not goes through) the xaxis at roots of even multiplicity.
 Near x=z_{2} the graph has the same general shape as
 The roots of the formula are given by the Quadratic formula^{w}:
See Completing the square^{w}
 This is a parabola shifted to the right h units, stretched by a factor of a, and moved upward k units.
 k is the value at x=h and is either the maximum or the minimum value.
 Where See Binomial coefficient^{w}
The polynomial remainder theorem^{w} states that the remainder of the division of a polynomial Z(x) by the linear polynomial xa is equal to Z(a). See Ruffini's rule^{*}.
Determining the value at Z(a) is sometimes easier if we use Horner's method^{*} (synthetic division^{*}) by writing the polynomial in the form
A monic polynomial^{*} is a one variable polynomial in which the leading coefficient is equal to 1.
Rational functions
A rational function^{*} is a function of the form
It has n zeros^{w} and m poles^{w}. A pole is a value of x for which f(x) = infinity.
 The vertical asymptotes^{w} are the poles of the rational function.
 If n<m then f(x) has a horizontal asymptote at the x axis
 If n=m then f(x) has a horizontal asymptote at k.
 If n>m then f(x) has no horizontal asymptote.
 See also Wikipedia:Asymptote#Oblique_asymptotes^{*}
 Given two polynomials and , where the p_{i} are distinct constants and deg Z < m, partial fractions^{w} are generally obtained by supposing that
 and solving for the c_{i} constants, by substitution, by equating the coefficients^{*} of terms involving the powers of x, or otherwise.
 (This is a variant of the method of undetermined coefficients^{*}.)^{[31]}
 If the degree of Z is not less than m then use long division to divide P into Z. The remainder then replaces Z in the equation above and one proceeds as before.
 If then
A Generalized hypergeometric series^{*} is given by
 where c_{0}=1 and
The function f(x) has n zeros and m poles.
 Basic hypergeometric series^{*}, or hypergeometric qseries, are qanalogue^{*} generalizations of generalized hypergeometric series.^{[32]}
 Roughly speaking a qanalog^{*} of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1^{[33]}
 We define the qanalog of n, also known as the qbracket or qnumber of n, to be
 one may define the qanalog of the factorial^{w}, known as the qfactorial^{*}, by
 Elliptic hypergeometric series^{*} are generalizations of basic hypergeometric series.
 An elliptic function is a meromorphic function that is periodic in two directions.
A generalized hypergeometric function^{*} is given by
So for e^{x} (see below) we have:
Integration and differentiation
 See also: Hyperreal number^{w} and Implicit differentiation^{w}
The integral^{w} is a generalization of multiplication.
 For example: a unit mass dropped from point x_{2} to point x_{1} will release energy.
 The usual equation is is a simple multiplication:
 But that equation cant be used if the strength of gravity is itself a function of x.
 The strength of gravity at x_{1} would be different than it is at x_{2}.
 And in reality gravity really does depend on x (x is the distance from the center of the earth):
 (See inversesquare law^{w}.)
 However, the corresponding Definite integral^{w} is easily solved:
The surprisingly simple rules for solving definite integrals F(x) is called the indefinite integral^{w}. (antiderivative^{w})
k and y are arbitrary constants:
(Units (feet, mm...) behave exactly like constants.)
And most conveniently :
 The integral of a function is equal to the area under the curve.
 When the "curve" is a constant (in other words, k•x^{0}) then the integral reduces to ordinary multiplication.
The derivative^{w} is a generalization of division.
The derivative of the integral of f(x) is just f(x).
The derivative of a function at any point is equal to the slope of the function at that point.
The equation of the line tangent to a function at point a is
The Lipschitz constant^{w} of a function is a real number for which the absolute value of the slope of the function at every point is not greater than this real number.
The derivative of f(x) where f(x) = k•x^{y} is
 The derivative of a is
 The integral of is ln(x)^{[34]}. See natural log^{w}
Chain rule^{w} for the derivative of a function of a function:
The Chain rule for a function of 2 functions:
 (See "partial derivatives" below)
The Product rule^{w} can be considered a special case of the chain rule^{w} for several variables^{[35]}
Product rule^{w}:
 (because is negligible)
General Leibniz rule^{*}:
By the chain rule:
Therefore the Quotient rule^{w}:
There is a chain rule for integration but the inner function must have the form so that its derivative and therefore
Actually the inner function can have the form so that its derivative and therefore provided that all factors involving x cancel out.
The product rule for integration is called Integration by parts^{w}
One can use partial fractions^{w} or even the Taylor series^{w} to convert difficult integrals into a more manageable form.
The fundamental theorem of Calculus is:
The fundamental theorem of calculus is just the particular case of the Leibniz integral rule^{*}:
In calculus, a function f defined on a subset of the real numbers with real values is called monotonic^{*} if and only if it is either entirely nonincreasing, or entirely nondecreasing.^{[36]}
A differential form^{w} is a generalisation of the notion of a differential^{w} that is independent of the choice of coordinate system^{*}. f(x,y) dx ∧ dy is a 2form in 2 dimensions (an area element). The derivative^{w} operation on an nform is an n+1form; this operation is known as the exterior derivative^{w}. By the generalized Stokes' theorem^{w}, the integral of a function over the boundary of a manifold^{w} is equal to the integral of its exterior derivative on the manifold itself.
Taylor & Maclaurin series
If we know the value of a smooth function^{w} at x=0 (smooth means all its derivatives are continuous^{w}) and we also know the value of all of its derivatives at x=0 then we can determine the value at any other point x by using the Maclaurin series^{w}. ("!" means factorial^{w})
The proof of this is actually quite simple. Plugging in a value of x=0 causes all terms but the first to become zero. So, assuming that such a function exists, a_{0} must be the value of the function at x=0. Simply differentiate both sides of the equation and repeat for the next term. And so on.
 The Taylor series^{w} generalizes this formula.
 An analytic function^{w} is a function whose Taylor series converges for every z_{0} in its domain^{w}; analytic functions are infinitely differentiable^{w}.
 Any vector g = (z_{0}, α_{0}, α_{1}, ...) is a germ^{*} if it represents a power series of an analytic function^{w} around z_{0} with some radius of convergence r > 0.
 The set of germs is a Riemann surface^{w}.
 Riemann surfaces are the objects on which multivalued functions become singlevalued.
 A connected component^{*} of (i.e., an equivalence class) is called a sheaf^{*}.
We can easily determine the Maclaurin series expansion of the exponential function^{w} (because it is equal to its own derivative).^{[34]}
 The above holds true even if x is a matrix. See Matrix exponential^{*}
And cos(x)^{w} and sin(x)^{w} (because cosine is the derivative of sine which is the derivative of cosine)
It then follows that and therefore See Euler's formula^{w}
 x is the angle in radians^{*}.
 This makes the equation for a circle in the complex plane, and by extension sine and cosine, extremely simple and easy to work with especially with regard to differentiation and integration.
 Differentiation and integration are replaced with multiplication and division. Calculus is replaced with algebra. Therefore any expression that can be represented as a sum of sine waves can be easily differentiated or integrated.
Fourier Series
The Maclaurin series cant be used for a discontinuous function like a square wave because it is not differentiable. (Distributions^{*} make it possible to differentiate functions whose derivatives do not exist in the classical sense. See Generalized function^{*}.)
But remarkably we can use the Fourier series^{w} to expand it or any other periodic function^{w} into an infinite sum of sine waves each of which is fully differentiable^{w}!
 The reason this works is because sine and cosine are orthogonal functions^{*}.
 That means that multiplying any 2 sine waves of frequency n and frequency m and integrating over one period will always equal zero unless n=m.
 See the graph of sin^{2}(x) to the right.
 See Amplitude_modulation^{*}
 And of course ∫ f_{n}*(f_{1}+f_{2}+f_{3}+...) = ∫ (f_{n}*f_{1}) + ∫ (f_{n}*f_{2}) + ∫ (f_{n}*f_{3}) +...
 The complex form of the Fourier series uses complex exponentials instead of sine and cosine and uses both positive and negative frequencies (clockwise and counter clockwise) whose imaginary parts cancel.
 The complex coefficients encode both amplitude and phase and are complex conjugates of each other.
 where the dot between x and ν indicates the inner product^{w} of R^{n}.
 A 2 dimensional Fourier series is used in video compression.
 A discrete Fourier transform^{*} can be computed very efficiently by a fast Fourier transform^{*}.
 In mathematical analysis, many generalizations of Fourier series have proven to be useful.
 They are all special cases of decompositions over an orthonormal basis of an inner product space.^{[37]}
 Spherical harmonics^{*} are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via Fourier series.^{[38]}
 Spherical harmonics are basis functions^{*} for SO(3). See Laplace series^{w}.
 Every continuous function in the function space can be represented as a linear combination^{*} of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.
 Every quadratic polynomial can be written as a1+bt+ct^{2}, that is, as a linear combination of the basis functions 1, t, and t^{2}.
Transforms
Fourier transforms^{w} generalize Fourier series to nonperiodic functions like a single pulse of a square wave.
The more localized in the time domain (the shorter the pulse) the more the Fourier transform is spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle^{w}.
The Fourier transform of the Dirac delta function^{w} gives G(f)=1
 Laplace transforms^{w} generalize Fourier transforms to complex frequency .
 Complex frequency includes a term corresponding to the amount of damping.
 , (assuming a > 0)
 The inverse Laplace transform^{w} is given by
 where the integration is done along the vertical line Re(s) = γ in the complex plane^{w} such that γ is greater than the real part of all singularities^{*} of F(s) and F(s) is bounded on the line, for example if contour path is in the region of convergence^{*}.
 If all singularities are in the left halfplane, or F(s) is an entire function^{*} , then γ can be set to zero and the above inverse integral formula becomes identical to the inverse Fourier transform^{*}.^{[39]}
 Integral transforms^{w} generalize Fourier transforms to other kernals^{w} (besides sine^{w} and cosine^{w})
 Cauchy kernel =
 Hilbert kernel =
 Poisson Kernel:
 For the ball of radius r, , in R^{n}, the Poisson kernel takes the form:
 where , (the surface of ), and is the surface area of the unit nsphere^{*}.
 unit disk (r=1) in the complex plane:^{[40]}
 Dirichlet kernel
The convolution^{*} theorem states that^{[41]}
where denotes pointwise multiplication. It also works the other way around:
By applying the inverse Fourier transform , we can write:
and:
This theorem also holds for the Laplace transform^{w}.
The Hilbert transform^{*} is a multiplier operator^{*}. The multiplier of H is σ_{H}(ω) = −i sgn(ω) where sgn is the signum function^{*}. Therefore:
where denotes the Fourier transform^{w}.
Since sgn(x) = sgn(2πx), it follows that this result applies to the three common definitions of .
By Euler's formula^{w},
Therefore, H(u)(t) has the effect of shifting the phase of the negative frequency^{*} components of u(t) by +90° (π/2 radians) and the phase of the positive frequency components by −90°.
And i·H(u)(t) has the effect of restoring the positive frequency components while shifting the negative frequency ones an additional +90°, resulting in their negation.
In electrical engineering, the convolution of one function (the input signal) with a second function (the impulse response) gives the output of a linear timeinvariant system (LTI).
At any given moment, the output is an accumulated effect of all the prior values of the input function
Differential equations
 See also: Variation of parameters^{*}
Simple harmonic motion^{*} of a mass on a spring is a secondorder linear ordinary differential equation^{w}.
where m is the inertial mass, x is its displacement from the equilibrium, and k is the spring constant.
Solving for x produces
A is the amplitude (maximum displacement from the equilibrium position), is the angular frequency^{w}, and φ is the phase.
Energy passes back and forth between the potential energy in the spring and the kinetic energy of the mass.
The important thing to note here is that the frequency of the oscillation depends only on the mass and the stiffness of the spring and is totally independent of the amplitude.
That is the defining characteristic of resonance.
Kirchhoff's voltage law^{*} states that the sum of the emfs in any closed loop of any electronic circuit is equal to the sum of the voltage drops^{*} in that loop.^{[42]}
V is the voltage, R is the resistance, L is the inductance, C is the capacitance.
I = dQ/dt is the current.
It makes no difference whether the current is a small number of charges moving very fast or a large number of charges moving slowly.
In reality the latter is the case^{*}.
If V(t)=0 then the only solution to the equation is the transient response which is a rapidly decaying sine wave with the same frequency as the resonant frequency of the circuit.
 Like a mass (inductance) on a spring (capacitance) the circuit will resonate at one frequency.
 Energy passes back and forth between the capacitor and the inductor with some loss as it passes through the resistor.
If V(t)=sin(t) from ∞ to +∞ then the only solution is a sine wave with the same frequency as V(t) but with a different amplitude and phase.
If V(t) is zero until t=0 and then equals sin(t) then I(t) will be zero until t=0 after which it will consist of the steady state response plus a transient response.
From Wikipedia:Characteristic equation (calculus):
Starting with a linear homogeneous differential equation with constant coefficients ,
it can be seen that if , each term would be a constant multiple of . This results from the fact that the derivative of the exponential function^{w} is a multiple of itself. Therefore, , , and are all multiples. This suggests that certain values of will allow multiples of to sum to zero, thus solving the homogeneous differential equation.^{[43]} In order to solve for , one can substitute and its derivatives into the differential equation to get
Since can never equate to zero, it can be divided out, giving the characteristic equation
By solving for the roots, , in this characteristic equation, one can find the general solution to the differential equation.^{[44]}^{[45]} For example, if is found to equal to 3, then the general solution will be , where is an arbitrary constant^{w}.
Partial derivatives
Partial derivatives^{w} and multiple integrals^{w} generalize derivatives and integrals to multiple dimensions.
The partial derivative with respect to one variable is found by simply treating all other variables as though they were constants.
Multiple integrals are found the same way.
Let f(x, y, z) be a scalar function^{w} (for example electric potential energy or temperature).
 A 2 dimensional example of a scalar function would be an elevation map.
 (Contour lines of an elevation map are an example of a level set^{*}.)
The total derivative^{w} of f(x(t), y(t)) with respect to t is^{[46]}
And the differential^{w} is
Gradient of scalar field
The Gradient^{w} of f(x, y, z) is a vector field whose value at each point is a vector (technically its a covector^{w} because it has units of distance^{−1}) that points "downhill" with a magnitude equal to the slope^{w} of the function at that point.
You can think of it as how much the function changes per unit distance.
For static (unchanging) fields the Gradient of the electric potential is the electric field^{w} itself.
The gradient of temperature gives heat flow.
Divergence
The Divergence^{w} of a vector field is a scalar.
The divergence of the electric field is nonzero wherever there is electric charge^{w} and zero everywhere else.
Field lines^{w} begin and end at charges because the charges create the electric field.
The Laplacian^{w} is the divergence of the gradient of a function:
 elliptic operators^{*} generalize the Laplacian.
Curl
The Curl^{w} of a vector field describes how much the vector field is twisted.
(The field may even go in circles.)
The curl at a certain point of a magnetic field^{w} is the current^{w} vector at that point because current creates the magnetic field^{w}.
In 3 dimensions the dual of the current vector is a bivector.
In 2 dimensions this reduces to a single scalar
The curl of the gradient of any scalar field is always zero.
The curl of a vector field in 4 dimensions would no longer be a vector. It would be a bivector. However the curl of a bivector field in 4 dimensions would still be a vector.
See also: differential forms^{*}.
Gradient of vector field
The Gradient^{w} of a vector field is a tensor field. Each row is the gradient of the corresponding scalar function:
 Remember that because rotation from y to x is the negative of rotation from x to y.
Partial differential equations can be classified as parabolic^{*}, hyperbolic^{*} and elliptic^{*}.
Green's theorem
The line integral^{w} along a 2D vector field is:

Green's theorem^{w} states that if you want to know how many field lines cross (or run parallel to) the boundary of a given region then you can either perform a line integral or you can simply count the number of charges (or the amount of current) within that region. See Divergence theorem^{w}
In 2 dimensions this is
Green's theorem is perfectly obvious when dealing with vector fields but is much less obvious when applied to complex valued functions in the complex plane.
The complex plane
 Highly recomend: Fundamentals of complex analysis with applications to engineering and science by Saff and Snider
 External link: http://www.solitaryroad.com/c606.html
The formula for the derivative of a complex function f at a point z_{0} is the same as for a real function:
Every complex function can be written in the form
Because the complex plane is two dimensional, z can approach z_{0} from an infinite number of different directions.
However, if within a certain region, the function f is holomorphic^{w} (that is, complex differentiable^{w}) then, within that region, it will only have a single derivative whose value does not depend on the direction in which z approaches z_{0} despite the fact that f_{x} and f_{y} each have 2 partial derivatives. One in the x and one in the y direction..

This is only possible if the Cauchy–Riemann conditions^{w} are true.
An entire function^{*}, also called an integral function, is a complexvalued function that is holomorphic at all finite points over the whole complex plane.
As with real valued functions, a line integral of a holomorphic function depends only on the starting point and the end point and is totally independant of the path taken.
The starting point and the end point for any loop are the same. This, of course, implies Cauchy's integral theorem^{w} for any holomorphic function f:
Therefore curl and divergence must both be zero for a function to be holomorphic.
Green's theorem^{w} for functions (not necessarily holomorphic) in the complex plane:
Computing the residue^{w} of a monomial^{[47]}
 where is the circle with radius therefore and
The last term in the equation above equals zero when r=0. Since its value is independent of r it must therefore equal zero for all values of r.
Cauchy's integral formula^{w} states that the value of a holomorphic function within a disc is determined entirely by the values on the boundary of the disc.
Divergence can be nonzero outside the disc.
Cauchy's integral formula can be generalized to more than two dimensions.
Which gives:
 Note that n does not have to be an integer. See Fractional calculus^{*}.
The Taylor series becomes:
The Laurent series^{*} for a complex function f(z) about a point z_{0} is given by:
The positive subscripts correspond to a line integral around the outer part of the annulus and the negative subscripts correspond to a line integral around the inner part of the annulus. In reality it makes no difference where the line integral is so both line integrals can be moved until they correspond to the same contour gamma. See also: Ztransform^{*}
The function has poles at z=1 and z=2. It therefore has 3 different Laurent series centered on the origin (z_{0} = 0):
 For 0 < z < 1 the Laurent series has only positive subscripts and is the Taylor series.
 For 1 < z < 2 the Laurent series has positive and negative subscripts.
 For 2 < z the Laurent series has only negative subscripts.
Cauchy formula for repeated integration^{*}:
For every holomorphic function^{w} both f_{x} and f_{y} are harmonic functions^{w}.
Any twodimensional harmonic function is the real part of a complex analytic function^{w}.
See also: complex analysis^{w}.^{[48]}
 f_{y} is the harmonic conjugate^{*} of f_{x}.
 Geometrically f_{x} and f_{y} are related as having orthogonal trajectories, away from the zeroes of the underlying holomorphic function; the contours on which f_{x} and f_{y} are constant (equipotentials^{*} and streamlines^{*}) cross at right angles.
 In this regard, f_{x}+if_{y} would be the complex potential, where f_{x} is the potential function^{*} and f_{y} is the stream function^{*}.^{[49]}
 f_{x} and f_{y} are both solutions of Laplace's equation^{w} so divergence of the gradient is zero
 Legendre function^{*} are solutions to Legendre's differential equation.
 This ordinary differential equation is frequently encountered when solving Laplace's equation (and related partial differential equations) in spherical coordinates.
 A harmonic function^{w} is a scalar potential function therefore the curl of the gradient will also be zero.
 See Potential theory^{*}
 Harmonic functions are real analogues to holomorphic functions.
 All harmonic functions are analytic, i.e. they can be locally expressed as power series.
 This is a general fact about elliptic operators^{*}, of which the Laplacian is a major example.
 The value of a harmonic function at any point inside a disk is a weighted average^{*} of the value of the function on the boundary of the disk.
 The Poisson kernel^{*} gives different weight to different points on the boundary except when x=0.
 The value at the center of the disk (x=0) equals the average of the equally weighted values on the boundary.
 All locally integrable functions satisfying the meanvalue property are both infinitely differentiable and harmonic.
 The kernel itself appears to simply be 1/r^n shifted to the point x and multiplied by different constants.
 For a circle (K = Poisson Kernel):

Geometric calculus
 See also: Geometric_algebra#Geometric_calculus^{*}
From Wikipedia:Geometric calculus^{w}:
Geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including differential geometry and differential forms.
With a geometric algebra given, let a and b be vectors^{*} and let F(a) be a multivector^{w}valued function. The directional derivative^{w} of F(a) along b is defined as
provided that the limit exists, where the limit is taken for scalar ε. This is similar to the usual definition of a directional derivative but extends it to functions that are not necessarily scalarvalued.
Next, choose a set of basis vector^{w}s and consider the operators, noted , that perform directional derivatives in the directions of :
Then, using the Einstein summation notation^{*}, consider the operator :
which means:
or, more verbosely:
It can be shown that this operator is independent of the choice of frame, and can thus be used to define the geometric derivative:
This is similar to the usual definition of the gradient^{w}, but it, too, extends to functions that are not necessarily scalarvalued.
It can be shown that the directional derivative is linear regarding its direction, that is:
From this follows that the directional derivative is the inner product of its direction by the geometric derivative. All needs to be observed is that the direction can be written , so that:
For this reason, is often noted .
The standard order of operations^{w} for the geometric derivative is that it acts only on the function closest to its immediate right. Given two functions F and G, then for example we have
Although the partial derivative exhibits a product rule^{w}, the geometric derivative only partially inherits this property. Consider two functions F and G:
Since the geometric product is not commutative^{w} with in general, we cannot proceed further without new notation. A solution is to adopt the overdot^{*} notation, in which the scope of a geometric derivative with an overdot is the multivectorvalued function sharing the same overdot. In this case, if we define
then the product rule for the geometric derivative is
Let F be an rgrade multivector. Then we can define an additional pair of operators, the interior and exterior derivatives,
In particular, if F is grade 1 (vectorvalued function), then we can write
and identify the divergence^{w} and curl^{w} as
Note, however, that these two operators are considerably weaker than the geometric derivative counterpart for several reasons. Neither the interior derivative operator nor the exterior derivative operator is invertible^{*}.
The reason for defining the geometric derivative and integral as above is that they allow a strong generalization of Stokes' theorem^{w}. Let be a multivectorvalued function of rgrade input A and general position x, linear in its first argument. Then the fundamental theorem of geometric calculus relates the integral of a derivative over the volume V to the integral over its boundary:
As an example, let for a vectorvalued function F(x) and a (n1)grade multivector A. We find that
and likewise
Thus we recover the divergence theorem^{w},
Calculus of variations
 Calculus of variations^{*}, Functional^{*}, Functional analysis^{*}, Higherorder function^{*}
Whereas calculus is concerned with infinitesimal changes of variables, calculus of variations is concerned with infinitesimal changes of the underlying function itself.
Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals.
A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is obviously a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, where the optical length depends upon the material of the medium. One corresponding concept in mechanics is the principle of least action.^{[50]}
Discrete mathematics
Groups and rings
 Main articles: Algebraic structure^{w}, Abstract algebra^{w}, and group theory^{*}
Addition and multiplication can be generalized in so many ways that mathematicians have created a whole system just to categorize them.
A magma^{*} is a set with a single closed^{*} binary operation (usually, but not always^{*}, addition).
 a + b = c
A semigroup^{*} is a magma where the addition is associative. See also Semigroupoid^{*}
 a + (b + c) = (a + b) + c
A monoid^{*} is a semigroup with an additive identity element.
 a + 0 = a
A group^{*} is a monoid with additive inverse elements.
 a + (a) = 0
An abelian group^{*} is a group where the addition is commutative.
 a + b = b + a
A pseudoring^{*} is an abelian group that also has a second closed, associative, binary operation (usually, but not always, multiplication).
 a * (b * c) = (a * b) * c
 And these two operations satisfy a distribution law.
 a(b + c) = ab + ac
A ring^{*} is a pseudoring that has a multiplicative identity
 a * 1 = a
A commutative ring^{*} is a ring where multiplication commutes, (e.g. integers^{*})
 a * b = b * a
A field^{*} is a commutative ring where every element has a multiplicative inverse (and thus there is a multiplicative identity),
 a * (1/a) = 1
 The existence of a multiplicative inverse for every nonzero element automatically implies that there are no zero divisors^{*} in a field
 if ab=0 for some a≠0, then we must have b=0 (we call this having no zerodivisors).
The characteristic^{*} of ring R, denoted char(R), is the number of times one must add the multiplicative identity^{*} to get the additive identity^{*}.
The center^{*} of a noncommutative ring^{*} is the subring of elements c such that cx = xc for every x. See also: Centralizer and normalizer^{*}.
All nonzero nilpotent^{*} elements are zero divisors^{*}.
 The square matrix^{w} is nilpotent
Set theory
 See also: Naive set theory^{*}, Zermelo–Fraenkel set theory^{*}, Set theory^{w}, Set notation^{*}, Setbuilder notation^{*}, Set^{w}, Algebra of sets^{*}, Field of sets^{*}, and Sigmaalgebra^{*}
is the empty set (the additive identity)
is the universe of all elements (the multiplicative identity)
means that a is a element^{w} (or member) of set A. In other words a is in A.
 means the set of all x's that are members of the set A such that x is not a member of the real numbers^{w}. Could also be written
A set^{w} does not allow multiple instances of an element.
 A multiset^{w} does allow multiple instances of an element.
A set can contain other sets.
means that A is a proper subset^{w} of B
 means that a is a subset^{w} of itself. But a set is not a proper subset^{w} of itself.
is the Union^{w} of the sets A and B. In other words,
is the Intersection^{w} of the sets A and B. In other words, All a's in B.
 Associative:
 Distributive:
 Commutative:
is the Set difference^{w} of A and B. In other words,
 or is the complement^{w} of A.
or is the Antiintersection^{w} of sets A and B which is the set of all objects that are a members of either A or B but not in both.
is the Cartesian product^{w} of A and B which is the set whose members are all possible ordered pairs^{w} (a, b) where a is a member of A and b is a member of B.
The Power set^{w} of a set A is the set whose members are all of the possible subsets of A.
A cover^{*} of a set X is a collection of sets whose union contains X as a subset.^{[51]}
A subset A of a topological space X is called dense^{*} (in X) if every point x in X either belongs to A or is arbitrarily "close" to a member of A.
 A subset A of X is meagre^{*} if it can be expressed as the union of countably many nowhere dense subsets of X.
Disjoint union^{*} of sets = {1, 2, 3} and = {1, 2, 3} can be computed by finding:
so
Let H be the subgroup of the integers (mZ, +) = ({..., −2m, −m, 0, m, 2m, ...}, +) where m is a positive integer.
 Then the cosets^{*} of H are the mZ + a = {..., −2m+a, −m+a, a, m+a, 2m+a, ...}.
 There are no more than m cosets, because mZ + m = m(Z + 1) = mZ.
 The coset (mZ + a, +) is the congruence class^{w} of a modulo m.^{[52]}
 Cosets are not usually themselves subgroups of G, only subsets.
means "there exists at least one"
means "there exists one and only one"
means "for all"
means "and" (not to be confused with wedge product^{w})
means "or" (not to be confused with antiwedge product^{w})
Probability
is the cardinality^{w} of A which is the number of elements in A. See measure^{w}.
is the unconditional probability^{w} that A will happen.
is the conditional probability^{w} that A will happen given that B has happened.
means that the probability that A or B will happen is the probability of A plus the probability of B minus the probability that both A and B will happen.
means that the probability that A and B will happen is the probability of "A and B given B" times the probability of B.
is Bayes' theorem^{*}
If you dont know the certainty then you can still know the probability. If you dont know the probability then you can always know the Bayesian probability. The Bayesian probability is the degree to which you expect something.
Even if you dont know anything about the system you can still know the A priori^{*} Bayesian probability. As new information comes in the Prior probability^{*} is updated and replaced with the Posterior probability^{*} by using Bayes' theorem^{*}.
From Wikipedia:Base rate fallacy:
In a city of 1 million inhabitants let there be 100 terrorists and 999,900 nonterrorists. In an attempt to catch the terrorists, the city installs an alarm system with a surveillance camera and automatic facial recognition software. 99% of the time it behaves correctly. 1% of the time it behaves incorrectly, ringing when it should not and failing to ring when it should. Suppose now that an inhabitant triggers the alarm. What is the chance that the person is a terrorist? In other words, what is P(T  B), the probability that a terrorist has been detected given the ringing of the bell? Someone making the 'base rate fallacy' would infer that there is a 99% chance that the detected person is a terrorist. But that is not even close. For every 1 million faces scanned it will see 100 terrorists and will correctly ring 99 times. But it will also ring falsely 9,999 times. So the true probability is only 99/(9,999+99) or about 1%.
permutation^{w} relates to the act of arranging all the members of a set^{w} into some sequence^{w} or order^{*}.
The number of permutations of n distinct objects is n!^{w}.^{[53]}
 A derangement is a permutation of the elements of a set, such that no element appears in its original position.
In other words, derangement is a permutation that has no fixed points^{*}.
The number of derangements^{*} of a set of size n, usually written !n^{*}, is called the "derangement number" or "de Montmort number".^{[54]}
 The rencontres numbers^{*} are a triangular array of integers that enumerate permutations of the set { 1, ..., n } with specified numbers of fixed points: in other words, partial derangements.^{[55]}
a combination^{w} is a selection of items from a collection, such that the order of selection does not matter.
For example, given three numbers, say 1, 2, and 3, there are three ways to choose two from this set of three: 12, 13, and 23.
More formally, a kcombination of a set^{w} S is a subset of k distinct elements of S.
If the set has n elements, the number of kcombinations is equal to the binomial coefficient^{w}
 Pronounced n choose k. The set of all kcombinations of a set S is often denoted by .
The central limit theorem (CLT) establishes that, in most situations, when independent random variables^{*} are added, their properly normalized sum tends toward a normal distribution^{w} (informally a "bell curve") even if the original variables themselves are not normally distributed.^{[56]}
In statistics^{w}, the standard deviation (SD, also represented by the Greek letter sigma σ^{w} or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion^{*} of a set of data values.^{[57]}
A low standard deviation indicates that the data points tend to be close to the mean^{w} (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.^{[58]}
The hypergeometric distribution^{*} is a discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, without replacement, from a finite population of size N that contains exactly K objects with that feature, wherein each draw is either a success or a failure.
 In contrast, the binomial distribution^{*} describes the probability of k successes in n draws with replacement.^{[59]}
See also Dirichlet distribution^{*} and Rice distribution^{*}
Morphisms
 See also: Higher category theory^{w} and Multivalued function (misnomer)^{*}
Every function^{w} has exactly one output for every input.
If the function f(x) is invertible^{*} then its inverse function^{w} f^{−1}(x) has exactly one output for every input.
If it isn't invertible then it doesn't have an inverse function.
 f(x)=x/(x1) is an involution^{*} which is a function that is its own inverse function. f(f(x))=x
Injection^{w}  Invertible function Injection+Surjection Bijection^{w}  Surjection^{w} 
A morphism^{w} is exactly the same as a function but in Category theory^{w} every morphism has an inverse which is allowed to have more than one value or no value at all.
Categories^{*} consist of:
 Objects (usually Sets^{w})
 one source object (domain)
 one target object (codomain)
a morphism is represented by an arrow:
 is written where x is in X and y is in Y.
 is written where y is in Y and z is in Z.
The image^{*} of y is z.
The preimage^{*} (or fiber^{*}) of z is the set of all y whose image is z and is denoted
A picture is worth 1000 words 
A space Y is a covering space^{*} (a fiber bundle) of space Z if the map is locally homeomorphic^{w}.
 A covering space is a universal covering space^{*} if it is simply connected^{*}.
 The concept of a universal cover was first developed to define a natural domain for the analytic continuation^{*} of an analytic function^{w}.
 The general theory of analytic continuation and its generalizations are known as sheaf theory^{*}.
 The set of germs^{*} can be considered to be the analytic continuation of an analytic function.
A topological space is (path)connected^{*} if no part of it is disconnected.
A space is simply connected^{*} if there are no holes passing all the way through it (therefore any loop can be shrunk to a point)
 See Homology^{*}
Composition of morphisms:
 is written
 f is the pullback^{*} of g
 f is the lift^{*} of
 ? is the pushforward^{*} of ?
A homomorphism^{*} is a map from one set to another of the same type which preserves the operations of the algebraic structure:
 See Cauchy's functional equation^{*}
 A Functor^{*} is a homomorphism with a domain in one category and a codomain in another.
 A group homomorphism^{*} from (G, ∗) to (H, ·) is a function^{*} h : G → H such that
 for all u*v = c in G.
 For example
 Since log is a homomorphism that has an inverse that is also a homomorphism, log is an isomorphism^{*} of groups.
 See also group action^{*} and group orbit^{*}
A Multicategory^{*} has morphisms with more than one source object.
A Multilinear map^{*} :
has a corresponding Linear map^{w}::
Numerical methods
 See also: Explicit and implicit methods^{*}
One of the simplest problems is the evaluation of a function at a given point.
The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient.
For polynomials, a better approach is using the Horner scheme^{*}, since it reduces the necessary number of multiplications and additions.
Generally, it is important to estimate and control roundoff errors^{*} arising from the use of floating point^{*} arithmetic.
Interpolation^{*} solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points?
Extrapolation^{*} is very similar to interpolation, except that now we want to find the value of the unknown function at a point which is outside the given points.
Regression^{*} is also similar, but it takes into account that the data is imprecise.
Given some points, and a measurement of the value of some function at these points (with an error), we want to determine the unknown function.
The least squares^{*}method is one popular way to achieve this.
Much effort has been put in the development of methods for solving systems of linear equations^{*}.
 Standard direct methods, i.e., methods that use some matrix decomposition^{*}
 Gaussian elimination^{*}, LU decomposition^{*}, Cholesky decomposition^{*} for symmetric^{w} (or hermitian^{w}) and positivedefinite matrix^{w}, and QR decomposition^{*} for nonsquare matrices.
 Jacobi method^{*}, Gauss–Seidel method^{*}, successive overrelaxation^{*} and conjugate gradient method^{*} are usually preferred for large systems. General iterative methods can be developed using a matrix splitting^{*}.
Rootfinding algorithms^{*} are used to solve nonlinear equations.
 If the function is differentiable^{w} and the derivative is known, then Newton's method^{w} is a popular choice.
 Linearization^{*} is another technique for solving nonlinear equations.
Optimization^{w} problems ask for the point at which a given function is maximized (or minimized).
Often, the point also has to satisfy some constraints^{*}.
Differential equation^{w}: If you set up 100 fans to blow air from one end of the room to the other and then you drop a feather into the wind, what happens?
The feather will follow the air currents, which may be very complex.
One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again.
This is called the Euler method^{*} for solving an ordinary differential equation.
Information theory
From Wikipedia:Information theory:
Information theory studies the quantification, storage, and communication of information.
Communications over a channel—such as an ethernet cable—is the primary motivation of information theory.
From Wikipedia:Quantities of information:
Shannon derived a measure of information content called the selfinformation^{*} or "surprisal" of a message m:
where is the probability that message m is chosen from all possible choices in the message space . The base of the logarithm only affects a scaling factor and, consequently, the units in which the measured information content is expressed. If the logarithm is base 2, the measure of information is expressed in units of bits^{*}.
Information is transferred from a source to a recipient only if the recipient of the information did not already have the information to begin with. Messages that convey information that is certain to happen and already known by the recipient contain no real information. Infrequently occurring messages contain more information than more frequently occurring messages. This fact is reflected in the above equation  a certain message, i.e. of probability 1, has an information measure of zero. In addition, a compound message of two (or more) unrelated (or mutually independent) messages would have a quantity of information that is the sum of the measures of information of each message individually. That fact is also reflected in the above equation, supporting the validity of its derivation.
An example: The weather forecast broadcast is: "Tonight's forecast: Dark. Continued darkness until widely scattered light in the morning." This message contains almost no information. However, a forecast of a snowstorm would certainly contain information since such does not happen every evening. There would be an even greater amount of information in an accurate forecast of snow for a warm location, such as Miami. The amount of information in a forecast of snow for a location where it never snows (impossible event) is the highest (infinity).
The more surprising a message is the more information it conveys. The message "LLLLLLLLLLLLLLLLLLLLLLLLL" conveys exactly as much information as the message "25 L's". The first message which is 25 bytes long can therefore be "compressed" into the second message which is only 6 bytes long.
Early computers
 See also: Time complexity^{*}
 Analog computer^{*}
 Abacus^{*}
 Napier's bones^{*}
 Slide rule^{*}
 Curta^{*}
 Lehmer sieve^{*}
 Z2 (computer)^{*}
Tactical thinking
Tactic X (Cooperate)  Tactic Y (Defect)  

Tactic A (Cooperate)  1, 1  5, 5 
Tactic B (Defect)  5, 5  5, 5 
 See also Wikipedia:Strategy (game theory)^{*}
 From Wikipedia:Game theory:
In the accompanying example there are two players; Player one (blue) chooses the row and player two (red) chooses the column.
Each player must choose without knowing what the other player has chosen.
The payoffs are provided in the interior.
The first number is the payoff received by Player 1; the second is the payoff for Player 2.
Tit for tat is a simple and highly effective tactic in game theory for the iterated prisoner's dilemma.
An agent using this tactic will first cooperate, then subsequently replicate an opponent's previous action.
If the opponent previously was cooperative, the agent is cooperative.
If not, the agent is not.^{[60]}
X  Y  

A  1,1  1,1 
B  1,1  1,1 
In zerosum games the sum of the payoffs is always zero (meaning that a player can only benefit at the expense of others).
Cooperation is impossible in a zerosum game.
John Forbes Nash proved that there is a Nash equilibrium (an optimum tactic) for every finite game.
In the zerosum game shown to the right the optimum tactic for player 1 is to randomly choose A or B with equal probability.
Strategic thinking differs from tactical thinking by taking into account how the short term goals and therefore optimum tactics change over time.
For example the opening, middlegame, and endgame of chess require radically different tactics.
See also: Reverse game theory^{*}
Physics
 See also: Wikisource:The Mathematical Principles of Natural Philosophy (1846) and Galilean relativity^{*}
Reality is what doesnt go away when you arent looking at it.
Something is known Beyond a reasonable doubt if any doubt that it is true is unreasonable. A doubt is reasonable if it is consistent with the laws of cause and effect.
In the four rules, as they came finally to stand in the 1726 edition, Newton effectively offers a methodology for handling unknown phenomena in nature and reaching towards explanations for them.
 Rule 1: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.
 Rule 2: Therefore to the same natural effects we must, as far as possible, assign the same causes.
 Rule 3: The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.
 Rule 4: In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, not withstanding any contrary hypothesis that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.
 Newtonian mechanics^{w}, Lagrangian mechanics^{w}, and Hamiltonian mechanics^{w}
 The difference between the net kinetic energy and the net potential energy is called the “Lagrangian.”
 The action is defined as the time integral of the Lagrangian.
 The Hamiltonian is the sum of the kinetic and potential energies.
 Noether's theorem^{*} states that every differentiable symmetry of the action^{*} of a physical system has a corresponding conservation law^{*}.
 Special relativity^{*}, and General relativity^{*}
 Energy is conserved in relativity and proper velocity is proportional to momentum at all velocities.
Highly recommend:
 Thinking Physics Is Gedanken Physics by Lewis Carroll Epstein
 Understanding physics by Isaac Asimov
Dimensional analysis
 See also: Natural units^{*}
Any physical law that accurately describes the real world must be independent of the units (e.g. km or mm) used to measure the physical variables.
Consequently, every possible commensurate equation for the physics of the system can be written in the form
The dimension, D_{n}, of a physical quantity can be expressed as a product of the basic physical dimensions length (L), mass (M), time (T), electric current (I), absolute temperature (Θ), amount of substance (N) and luminous intensity (J), each raised to a rational power.
Suppose we wish to calculate the range of a cannonball^{*} when fired with a vertical velocity component and a horizontal velocity component , assuming it is fired on a flat surface.
The quantities of interest and their dimensions are then
 range as L_{x}
 as L_{x}/T
 as L_{y}/T
 g as L_{y}/T^{2}
The equation for the range may be written:
Therefore
and we may solve completely as , and .
Atoms
 See also: Periodic table^{w} and Spatial_structure_of_the_electron
The first pair of electrons fall into the ground shell. Once that shell is filled no more electrons can go into it. Any additional electrons go into higher shells.
The nucleus however works differently. The first few neutrons form the first shell. But any additional neutrons continue to fall into that same shell which continues to expand until there are 49 pairs of neutrons in that shell.
The electric force between two electrons is 4.166 * 10^{42} times stronger than the gravitational force
The energy required to assemble a sphere of uniform charge density =
 For Q=1 electron charge and r=1.8506 angstrom thats 4.669 ev. That energy is stored in the electric field of the electron.
 The energy per volume stored in an electric field is proportional to the square of the field strength so twice the charge has 4 times as much energy.
 4*4.669 = 18.676.
Mass of electron = M_{e} = 510,999 ev
Mass of proton = M_{p} = 938,272,000 ev
Mass of neutron = M_{n} = 939,565,000 ev
 M_{n} = M_{p} + M_{e} + 782,300 ev
Mass of muon = M_{μ} = 105.658 ev = 206.7683 * M_{e}
Mass of helium atom = 3,728,400,000 = 4*M_{e}+4*M_{p} 52.31 M_{e}
 The missing 52.31 electron masses of energy is called the mass deficit or nuclear binding energy. Fusing hydrogen into helium releases this energy.
Iron can be fused into heavier elements too but doing so consumes energy rather than releases energy.
From Wikipedia:Bohr model
The electron is held in a circular orbit^{*} by electrostatic attraction. The centripetal force^{*} is equal to the Coulomb force^{*}.
 where m_{e} is the electron's mass, q_{e} is the charge of the electron, k_{e} is Coulomb's constant^{*}, Z is the atom's atomic number (the number of protons) and n is the number of electrons. For the ground state electron F_{Pauli} = 0.
the ground state electron's speed is:
The angular momentum is:
 where ħ is reduced Planck constant
Substituting the expression for the velocity gives an equation for r:
 so that the allowed orbit radius is:
Therefore the orbit of the ground state electron is inversely proportional to the atomic number. The values obtained by this formula should be thought of only as approximation. For Z=2 and n=2 we get 0.1322943 Å. Thats 1/4th of the Bohr radius^{*}
Empirically determined values:
 Diatomic Hydrogen (Z=2) = 1.9002 angstroms
 Helium (Z=2) = 1.8506 angstroms
For electrons in higher shells the equation cant be used because F_{Pauli} > 0.
Densities:
 Crystalline solids: 1.2
 Amorphous solids: 1.1
 liquids: 1
Water ice is an exception. Ice has a density of 0.9167
Tidal acceleration
 See also: Formation_of_the_Solar_System^{w}
Image shows an approximation of the shape (Equipotentials^{*}) of a rapidly spinning planet. North pole is at the top. South pole is at the bottom. The equator reaches orbital velocity.
Orbital velocity:
Orbital period:
Orbital angular momentum:
Rotational angular momentum of solid sphere:
where:
 r is the orbit's semimajor axis^{w}
 G is the gravitational constant^{w},
 M is the mass of the more massive body.
 m is the mass of the less massive body.
Moons orbital angular momentum is 28.73 * 10^33 Js
Earths rotational angular momentum is 7.079 * 10^33 Js
The total amount of angular momentum for the EarthMoon system is 28.73 + 4.6 = 33.33 * 10^33 Js
Moons current orbit is 384,399 km. Its orbital period is 2.372 * 10^{6} seconds. (27 days, 10 hours, 50 minutes). Its orbital velocity is 1.022 km/s.
Roche limit^{*} for the moon is
 Fluid: 18,381 km fluid
 384,399 / 18,381 = 20.9
 Orbital momentum of moon at fluid Roche limit = 28.73 * 10^33 Js / sqrt(20.9) = 6.3 * 10^33
 Earth would spin (28.736.3+4.6)/4.6 = 5.876 times faster
 Rigid: 9,492 km
 384,399 / 9,492 = 40.5
 Orbital momentum of moon at rigid Roche limit = 28.73 * 10^33 Js / sqrt(40.5) = 4.5 * 10^33
 Earth would spin (28.734.5+4.6)/4.6 = 6.27 times faster
Orbital radius with period = 4 hours:
 = 12,800 km
Alternately we can ask what the orbital period would be if Earth had a moon (not necessarily the moon) at 18,381 km.
 Earth would spin 24/7.554 = 3.177 times faster
 Earths angular momentum would be 3.177 * 4.6 * 10^33 Js = 14.6142 * 10^33 Js
 Our current Moons angular momentum would be 28.73  (14.6142  4.6) * 10^33 Js = 18.7158 * 10^33 Js
 Thats 18.7158 / 28.73 = 0.65
 So the current moons orbit would have been 0.65^2 * 384,399 km = 0.424 * 384,399 km = 162985 km
Tidal rhythmites are alternating layers of sand and silt laid down offshore from estuaries having great tidal flows. Daily, monthly and seasonal cycles can be found in the deposits. This geological record indicates that 620 million years ago there were 400±7 solar days/year
The motion of the Moon can be followed with an accuracy of a few centimeters by lunar laser ranging. Laser pulses are bounced off mirrors on the surface of the moon. The results are:
 +38.08±0.04 mm/yr (384,399 km / 63.4 billion years)
 1.42*10^24 Js/yr (33.33 * 10^33 Js / 23 billion years)
 1.42*10^26 Js/century
The corresponding change in the length of the day can be computed:
 (1.42*10^26)/(4.6 * 10^33) * 24 hours = 3.087*10^8 * 24 hours = +2.667 ms/century
620 million yrs ago the Moon had 1.42*10^24 * 620*10^6 = 0.88*10^33 Js less angular momentum. The moons orbit was therefore 384,399 km * ((28.730.88)/28.73)^2 = 361,211 km. One month lasted 2.161 * 10^{6} seconds. (25 days, 16 minutes, 40 seconds)
The Earth spun (4.6+0.88)/4.6 = 1.19 times faster so the day was 24 hours / 1.19 = 20.1680672 hours
The year was 400 "days" * 20.1680672 hours per "day" = 336.135 24hour periods
Earths orbit was therefore
Therefore Earth must be receding from the sun at 13 m/yr
Planets
#  Planet  g/cm^3  km  g's  au 

1  Mercury  5.427  2,440  0.377  0.387 
2  Venus  5.243  6,052  0.904  0.723 
3  Earth  5.515  6,371  1  1.000 
4  Mars  3.934  3,390  0.378  1.524 
5  Ceres  2.093  476.2  0.028  2.766 
6  Jupiter  1.326  69,911  2.528  5.203 
7  Saturn  0.687  58,232  1.065  9.537 
8  Ouranos  1.270  25,362  0.904  19.191 
9  Neptune  1.638  24,622  1.137  30.069 
From Wikipedia:16 Psyche:
16 Psyche is one of the ten most massive asteroids in the asteroid belt. It is over 200 km (120 mi) in diameter and contains a little less than 1% of the mass of the entire asteroid belt. It is thought to be the exposed iron core of a protoplanet
Brown dwarfs
Hydrogen  Atomic radius  g/cm^{3}  Jupiter volume  g's 

Liquid  1  0.07085  0.053 M_{Jup}  0.14 
Metallic  1/ 4  4.5344  3.400 M_{Jup}  9.00 
Double  1/ 5.657  12.8250  9.669 M_{Jup}  25.56 
Triple  1/ 8  36.2752  27.300 M_{Jup}  72.16 
Quadruple  1/ 11.31  102.6  77.350 M_{Jup}  204.40 
Quintuple  1/ 16  290.2016  219.000 M_{Jup}  578.80 
Sextuple  1/ 22.63  820.8140  618.800 M_{Jup}  1636.00 
As can be seen in the image to the right, all planets (Brown dwarfs) from 1 to 100 Jupiter masses are about 1 Jupiter radius which is 69,911 km. The largest "puffy" planets are 2 Jupiter radii. 1 Jupiter volume = 1.431×10^{15} km^{3}
This suggests that the pressure an electron shell (in degenerate matter) can withstand without again becoming degenerate (Electron degeneracy pressure^{*}) is inversely proportional to the sixth power of its radius:
(This formula only applies to degenerate matter like metallic hydrogen. Nondegenerate matter can withstand far more pressure).
If so then the maximum size (radius) that a planet composed entirely of one (degenerate) element could grow would depend only on, and be inversely proportional to, the atomic mass of its atoms. (Use 2 for the atomic mass of diatomic hydrogen).
Simplified calculation of radius of brown dwarf as core grows from zero to 1 Jupiter radius:
 r is radius of core with 2.83 (sqrt(2)^{3}) times the density of overlying material
Rock floats on top of the metallic hydrogen but iron sinks to the Core. 0.1% of the mass of the brown dwarf is iron. Assuming iron density of 231.85 g/cm3 (as in Earths core), the gravity of the iron core will cause the brown dwarf to be about 3% smaller then it would be otherwise.
Dark matter
Dark matter is a type of unidentified matter that may constitute about 80% of the total matter in the universe. It has not been directly observed, but its gravitational effects are evident in a variety of astrophysical measurements. The primary evidence for dark matter is that calculations show that many galaxies would fly apart instead of rotating if they did not contain a large amount of matter beyond what can be observed.
From Wikipedia:Gravitational microlensing
Microlensing allows the study of objects that emit little or no light. With microlensing, the lens mass is too low for the displacement of light to be observed easily, but the apparent brightening of the source may still be detected. In such a situation, the lens will pass by the source in seconds to years instead of millions of years.
The Einstein radius, also called the Einstein angle, is the angular radius of the Einstein ring in the event of perfect alignment. It depends on the lens mass M, the distance of the lens d_{L}, and the distance of the source d_{S}:
 (in radians).
For M equal to 60 Jupiter masses, d_{L} = 4000 parsecs, and d_{S} = 8000 parsecs (typical for a Bulge microlensing event), the Einstein radius is 0.00024 arcseconds (angle subtended by 1 au at 4000 parsecs). By comparison, ideal Earthbased observations have angular resolution around 0.4 arcseconds, 1660 times greater.
Any brown dwarf surrounded by a circumstellar disk larger and thicker than 1 au would therefore be virtually completely undetectable.
Stars
 See also: Stellar evolution^{*}, Helium flash^{*}, Schönberg–Chandrasekhar limit^{*}, Coronal_heating_problem^{w}
Fusion of diatomic hydrogen begins around 60 Jupiter masses. Fusion of monatomic helium requires significantly more pressure.
Fusion releases energy that heats the star causing it to expand. The expansion reduces the pressure in the core which reduces the rate of fusion. So the rate of fusion is self limiting. A low mass star has a lifetime of billions of years. A high mass star has a lifetime of only a few tens of millions of years despite starting with more hydrogen.
Low mass stars are far more common than high mass stars. The masses of the two component stars of NGC 3603A1, A1a and A1b, determined from the orbital parameters are 116 ± 31 M☉ and 89 ± 16 M☉respectively. This makes them the two most massive stars directly measured, i.e. not estimated from models.
The luminousity of a star is:
 where σ is the Stefan–Boltzmann constant^{*}:
The luminosity of the sun at 5772 K and 695,700 km is 3.828×10^26 Watts
 Thats 6,297,000 watts/m^{2}
The brightness of sunlight at the surface of the Earth is 1400 watt/meter^{2}
The plasma inside a star is nonrelativistic. A relativistic plasma with a thermal distribution function^{*} has temperatures greater than around 260 keV, or 3.0 * 10^{9} K^{*}. Those sorts of temperatures are only created in a supernova. The core of the sun is about 15 * 10^{6} K.
Plasmas, which are normally opaque to light, are transparent to light with frequency higher than the plasma frequency^{*}. The plasma literally cant vibrate fast enough to keep up with the light. Plasma frequency is proportional to the square root of the electron density.
 where
 n_{e} = number of electrons / volume.
See also: Bremsstrahlung#Thermal_bremsstrahlung^{*}
From 0.3 to 1.2 solar masses, the region around the stellar core is a radiative zone. (The light frequency is higher than the plasma frequency). The radius of the radiative zone increases monotonically with mass, with stars around 1.2 solar masses being almost entirely radiative.
From Wikipedia:Convective zone
In main sequence stars of less than about 1.3 solar masses, the outer envelope of the star contains a region of relatively low temperature which causes the frequency of the light to be lower than the plasma frequency which causes the opacity to be high enough to produce a steep temperature gradient. This produces an outer convection zone. The Sun's convection zone extends from 0.7 solar radii (500,000 km) to near the surface.
From Wikipedia:Cepheid variable
A Cepheid variable is a type of star that pulsates radially, varying in both diameter and temperature and producing changes in brightness with a welldefined stable period and amplitude.
A strong direct relationship between a Cepheid variable's luminosity and pulsation period allows one to know the true luminosity of a Cepheid by simply observing its pulsation period. This in turn allows one to determine the distance to the star, by comparing its known luminosity to its observed brightness.
The pulsation of cepheids is known to be driven by oscillations in the ionization of helium. From fully ionized (more opaque) He++ to partially ionized (more transparent) He+ and back to He++. See Kappa mechanism^{*}.
In the swelling phase. Its outer layers expand, causing them to cool. Because of the decreasing temperature the degree of ionization also decreases. This makes the gas more transparent, and thus makes it easier for the star to radiate its energy. This in turn will make the star start to contract. As the gas is thereby compressed, it is heated and the degree of ionization again increases. This makes the gas more opaque, and radiation temporarily becomes captured in the gas. This heats the gas further, leading it to expand once again. Thus a cycle of expansion and compression (swelling and shrinking) is maintained.
From Wikipedia:Instability strip
In normal AFG stars He is neutral in the stellar photosphere. Deeper below the photosphere, at about 25,000–30,000K, begins the He II layer (first He ionization). Second ionization (He III) starts at about 35,000–50,000K.
Recombination and Reionization 

The first phase change of hydrogen in the universe was recombination due to the cooling of the universe to the point where electrons and protons form neutral hydrogen. The universe was opaque before the recombination, due to the scattering of photons (of all wavelengths) off free electrons, but it became increasingly transparent as more electrons and protons combined to form neutral hydrogen atoms. The Dark Ages of the universe start at that point, because there were no light sources. The second phase change occurred once objects started to condense in the early universe that were energetic enough to reionize neutral hydrogen. As these objects formed and radiated energy, the universe reverted to once again being an ionized plasma. (See Warm–hot intergalactic medium^{*}). At this time, however, matter had been diffused by the expansion of the universe, and the scattering interactions of photons and electrons were much less frequent than before electronproton recombination. Thus, a universe full of low density ionized hydrogen will remain transparent, as is the case today. 
The Sun's photosphere has a temperature between 4,500 and 6,000 K. Negative hydrogen ions (H) are the primary reason for the highly opaque nature of the photosphere.
As the star burns hydrogen heavier elements build up in the core. Eventually the outer layers of the star are blown away and all thats left is the core. We call whats left a white dwarf.
White dwarfs
Z^{*}  A^{*}  Element  (ppm)  g/cm^{3}  g/cm^{3}  radius 

1  1  Hydrogen^{*}  739,000  0.07085  290.2  71,492 
1  2  Deuterium^{*}  100  0.1417  580.4  35,746 
2  4  Helium^{*}  240,000  0.125  512  35,746 
4  8  Beryllium^{*}  0  2  8,192  17,873 
8  16  Oxygen^{*}  10,400  32  131,072  8,936 
6  12  Carbon^{*}  4,600  10.125  41,472  11,915 
10  20  Neon^{*}  1,340  78.125  320,000  7,149 
26  56  Iron56^{*}  1,090  3844.75  15,748,096  2,553 
7  14  Nitrogen^{*}  960  18.76  76,841  10,213 
14  28  Silicon^{*}  650  300.125  1,229,312  5,107 
12  24  Magnesium^{*}  580  162  663,552  5,958 
16  32  Sulfur^{*}  440  512  2,097,152  4,468 
A white dwarf is about the same size as the Earth but is far denser and far more massive. A typical temperature for a white dwarf is 25,000 K. That would make its surface brightness 350 times the surface brightness of the sun.
Simplified calculation of radius of White dwarf as core grows from zero to half the original radius:
 r is radius of core. The core has 16 times the density (twice the atomic number) of the overlying material. The final state has half the radius and twice the mass of the original white dwarf.
A 0.6 solar mass White dwarf is 8900 km in radius which Is 8.03 times smaller than Jupiter which suggests a composition of oxygen. It has a surface gravity of
Its density is 404,000 g/cm3 which is 12,625 times denser than oxygen in its ground state. Thats 23.285^{3} times denser. Sqrt(2)^{9} = 22.63
A 1.13 solar mass White dwarf is 4500 km in radius which Is 15.9 times smaller than Jupiter which suggests a composition of sulfur. It has a surface gravity of
Its density is 5.887 * 10^{6} g/cm3 which is 11,498 times denser than sulfur in its ground state. Thats 22.57^{3} times denser.
For a white dwarf made of iron:
 Radius: 2,553 km
 Surface area: 8.2*10^{7} km^{2}
 Mass per surface area: 3.8 * 10^{13} g/mm^{2}
 Mass: 4.454 * 10^{7} g/cm3 * (4/3)*pi*(2553 km)^{3} in solar masses = 1.56 solar masses.
 Surface gravity: 3.24 * 10^{6} g's
 Density: (sqrt(2)^{9})^{3} * 3844.75 g/cm3 = 4.454 * 10^{7} g/cm3
 Core pressure: 1.8 * 10^{19} bars
The core of a white dwarf with a mass greater than the Chandrasekhar limit^{*} (1.44 solar masses) will undergo gravitational collapse and become a neutron star.
Neutron stars
 See also: Gravitoelectromagnetism^{*}
Assuming a solid honeycomb array of neutron pairs with radius 1 fm, a sheet of neutronium^{*} (if such a thing existed) would have a density of 1.2893598 g/mm^{2}.
Density of a liquid neutron star made of neutron pairs with radius 1 fm would be 479.8×10^{12} g/cm^{3}
The maximum observed mass of neutron stars is about 2.01 M_{☉}.
At that density a 2 solar mass neutron star would have a radius of 12.5544 km
The Tolman–Oppenheimer–Volkoff limit^{*} (or TOV limit) is an upper bound to the mass of cold, nonrotating neutron stars, analogous to the Chandrasekhar limit for white dwarf stars. Observations of GW170817 suggest that the limit is close to 2.17 solar masses.
The equation of state^{w} for a neutron star is not yet known.
A 2 solar mass neutron star with radius of 12.5544 km would have a surface gravity of:
The pressure in its core would be 5.072 * 10^{28} bar = 5.071 * 10^{28} bar
From Wikipedia:Glitch (astronomy)
A glitch (See Global_resurfacing_event^{*}) is a sudden increase of up to 1 part in 10^{6} in the rotational frequency of a rotationpowered pulsar. Following a glitch is a period of gradual recovery, lasting from days to years, where the observed periodicity slows to a period close to that observed before the glitch.
From Wikipedia:Supermassive black hole
A supermassive black hole (SMBH or SBH) is the largest type of black hole^{*}, on the order of hundreds of thousands to billions of solar masses^{*} (M_{☉}), and is found in the centre of almost all currently known massive galaxies.
The mean ratio of black hole mass to bulge mass is now believed to be approximately 1:1000.
Some supermassive black holes appear to be over 10 billion solar masses.
From Wikipedia:Quasar:
A quasar is an active galactic nucleus of very high luminosity. A quasar consists of a supermassive black hole surrounded by an orbiting accretion disk of gas. The most powerful quasars have luminosities exceeding 2.6×10^{14} ℒ_{☉} (10^{41} W or 17.64631 M_{☉}/year), thousands of times greater than the luminosity of a large galaxy such as the Milky Way.
Growing at a rate of 17.6 solar mass per year a 3.3 billion solar mass Black hole would take 187,000,000 years to reach full size.
Gammaray bursts
From Wikipedia:Gammaray burst
Gammaray bursts (GRBs) are extremely energetic explosions that have been observed in distant galaxies. They are the brightest electromagnetic events known to occur in the universe. Bursts can last from ten milliseconds to several hours. After an initial flash of gamma rays, a longerlived "afterglow" is usually emitted at longer wavelengths (Xray, ultraviolet, optical, infrared, microwave and radio).
Assuming the gammaray explosion to be spherical, the energy output of GRB 080319B^{*} would be within a factor of two of the restmass energy of the Sun (the energy which would be released were the Sun to be converted entirely into radiation).
No known process in the universe can produce this much energy in such a short time.
GRB 111209A is the longest lasting gammaray burst (GRB) detected by the Swift GammaRay Burst Mission on December 9, 2011. Its duration is longer than 7 hours.
On average two long gamma ray burst occurs every 3 days and have average redshift of 2. Making the simplifying assumption that all long gamma ray bursts occur at exactly redshift 2 (9.2 * 10^{9} light years) we get one gamma ray burst per (1,635,000 light years)^{3}
There are 12 galaxies per cubic megaparsec. Thats 1 galaxy per (1,425,000 light years)^{3}
One short grb per 3 days at average redshift of 0.5 (4.6 * 10^{9} light years) gives 1 grb per (1,300,000 light years)^{3}
 
Surface gravity = 29.6 g's, Density = 3.461099×10^{8} g/mm^2 
Ultrahighenergy Cosmic rays
From Wikipedia:Cosmic ray
Cosmic rays^{*} are highenergy radiation, mainly originating outside the Solar System and even from distant galaxies. Upon impact with the Earth's atmosphere, cosmic rays can produce showers of secondary particles that sometimes reach the surface. Composed primarily of highenergy protons and atomic nuclei, they are of uncertain origin. Data from the Fermi Space Telescope (2013) have been interpreted as evidence that a significant fraction of primary cosmic rays originate from the supernova explosions of stars. Active galactic nuclei are also theorized to produce cosmic rays.
From Wikipedia:Ultrahighenergy cosmic ray
In astroparticle physics^{*}, an ultrahighenergy cosmic ray (UHECR) is a cosmic ray particle with a kinetic energy greater than than 1×10^{18} eV^{*}, far beyond both the rest mass^{*} and energies typical of other cosmic ray particles.
An extremeenergy cosmic ray (EECR) is an UHECR with energy exceeding 5×10^{19} eV (about 8 joule), the socalled Greisen–Zatsepin–Kuzmin limit^{*} (GZK limit). This limit should be the maximum energy of cosmic ray protons that have traveled long distances (about 160 million light years), since higherenergy protons would have lost energy over that distance due to scattering from photons in the cosmic microwave background^{*} (CMB). However, if an EECR is not a proton, but a nucleus with nucleons, then the GZK limit applies to its nucleons, each of which carry only a fraction of the total energy.
These particles are extremely rare; between 2004 and 2007, the initial runs of the Pierre Auger Observatory^{*} (PAO) detected 27 events with estimated arrival energies above 5.7×10^{19} eV, i.e., about one such event every four weeks in the 3000 km^{2} area surveyed by the observatory.
At that rate 1.365 * 10^{18} particles will fall onto a star with radius 1 million kilometers every hundred million years.
From Wikipedia:OhMyGod particle:
The OhMyGod particle was an ultrahighenergy cosmic ray detected on the evening of 15 October 1991 by the Fly's Eye Cosmic Ray Detector. Its observation was a shock to astrophysicists, who estimated its energy to be approximately 3×10^{20} eV.
3×10^{20} eV = mv^{2} = 4 neutron masses * 256^{2}* (2.41 c)^{2} * 209,711
Weather
A cold front is the leading edge of a cold dense mass of air, replacing (at ground level) a warmer mass of air. Like a hot air balloon, the warm air rises above the cold air. The rising warm air expands and therefore cools. This causes the moisture within it to condense into droplets and releases the latent heat of condensation which causes the warm air to rise even further. If the warm air is moist enough, rain can occur along the boundary. A narrow line of thunderstorms often forms along the front. Temperature changes across the boundary can exceed 30 °C (54 °F).
The polar front is a cold front that arises as a result of cold polar air meeting warm subtropical air at the boundary between the polar cell and the Ferrel cell in each hemisphere.
Earth's weather is driven by 2 main areas.
 The polar front.
 When the polar front dominates we have an El Nino.
 The Intertropical Convergence Zone.
 When the Intertropical Convergence Zone dominates we have an El Nina.
In the Pacific, strong MJO activity is often observed 6 – 12 months prior to the onset of an El Niño episode, but is virtually absent during the maxima of some El Niño episodes, while MJO activity is typically greater during a La Niña episode.
During an El Nino, extratropical cyclones, which form along the polar front, can become so large that they draw moisture up directly from the tropics in what is called an atmospheric River. (See the image to the right.) Atmospheric rivers are typically several thousand kilometers long and only a few hundred kilometers wide, and a single one can carry a greater flux of water than the Earth's largest river, the Amazon.^{[61]} The Amazon discharges more water into the oceans than the next 7 largest rivers. (The Amazon river valley is an Aulacogen.)
Tropical air is far warmer than air outside the tropics and therefore holds far more moisture and as a result thunderstorms in the tropics are much taller. Nevertheless severe thunderstorms are not common in the tropics because the storms own downdraft shuts off the inflow of warm moist air killing the thunderstorm before it can become severe. Severe thunderstorms tend to occur further north because of the polar jet stream. The jet stream pushes against the top of the thunderstorm displacing the downdraft so that it can no longer shut off the inflow of warm moist air. As a result severe thunderstorms can continue to feed and grow for many hours whereas normal thunderstorms only last 30 minutes.
Over a 30 minute period a normal thunderstorm releases 10^{15} Joules of energy equivalent to 0.24 megatons of TNT. A storm that lasted 24 hours would release 50 times as much energy equivalent to 12 megatons of TNT. A hurricane (a tropical cyclone) releases as much energy as 1000 thunderstorms. 5.2 x 10^{19} Joules/day equivalent to 10,000 megatons of TNT per day.
The record lowest pressure established in the northern hemisphere is the extratropical cyclone of January 10, 1993 between Iceland and Scotland which deepened to a central pressure of 912915 mb (26.93”27.02”). Most hurricanes have an eye below 990 millibars. In 2005, hurricane WILMA reached the lowest barometric pressure ever recorded in an Atlantic Basin hurricane: 882 millibars. Hurricanes don't form in the South Atlantic.
If Earth's atmosphere were only slightly thicker then the air would be warmer and the amount of water vapor in the air would be much greater and lightning would therefore be much more common. The lightning would break apart the air molecules which would be washed down into the sea where they would end up in sediments which get subducted into the Earth. In this way the Earth's average air pressure is maintained at its current level.
During most of it's history Earth only had one atmospheric cell that extended from the pole to the equator and as a result Earth was very much warmer. A sort of ultra El Nina.
 Hadley cell
During an ice_age the Earth only has two cells. A sort of ultra El Nino. Ice ages are probably caused by deforestation caused by megafauna.
 Polar cell
 Ferrel cell
The Earth's atmosphere currently has 3 cells.
 Polar_cell
 Ferrel_cell
 Hadley_cell
Scale height is the increase in altitude for which the atmospheric pressure decreases by a factor of e. The scale height remains constant for a particular temperature. It can be calculated by
 where:
 k = Boltzmann constant = 1.38 x 10^{−23} J·K^{−1}
 T = mean atmospheric temperature in kelvins = 250 K for Earth
 M = mean mass of a molecule (units kg)
 g = acceleration due to gravity on planetary surface (m/s²)
Approximate atmospheric scale heights for selected Solar System bodies follow.
 Venus: 15.9 km
 Earth: 8.5 km
 Mars: 11.1 km
 Jupiter: 27 km
 Saturn: 59.5 km
 Titan: 21 km
 Uranus: 27.7 km
 Neptune: 19.1–20.3 km
 Pluto: ~60 km
If all of Earths atmosphere were at 1 bar then the atmosphere would be 8.5 km thick.
Life
 External link: Molecular biology of the cell
Did life begin with nucleic acids^{*} or amino acids? Maybe it began with a molecule that was both a nucleic acid and an amino acid.
3Aminobenzoicacid:
Creating the monomers in the Primordial soup^{*} is easy but getting the monomers to bond into a polymer is hard. So maybe it wasnt a polymer at all. Maybe it was a one dimensional liquid crystal. See Mesogen^{*}.
Unexplained phenomena
Books published by William R. Corliss^{*} include:
 Mysteries of the Universe (1967)
 Mysteries Beneath the Sea (1970)
 Strange Phenomena: A Sourcebook of Unusual Natural Phenomena (1974)
 Strange Artifacts: A Sourcebook on Ancient Man (1974)
 The Unexplained (1976)
 Strange Life (1976)
 Strange Minds (1976)
 Strange Universe (1977)
 Handbook of Unusual Natural Phenomena (1977)
 Strange Planet (1978)
 Ancient Man: A Handbook of Puzzling Artifacts (1978)
 Mysterious Universe: A Handbook of Astronomical Anomalies (1979)
 Unknown Earth: A Handbook of Geological Enigmas (1980)
 Incredible Life: A Handbook of Biological Mysteries (1981)
 The Unfathomed Mind: A Handbook of Unusual Mental Phenomena (1982)
 Lightning, Auroras, Nocturnal Lights, and Related Luminous Phenomena (1982)
 Tornados, Dark Days, Anomalous Precipitation, and Related Weather Phenomena (1983)
 Earthquakes, Tides, Unidentified Sounds, and Related Phenomena (1983)
 Rare Halos, Mirages, Anomalous Rainbows, and Related Electromagnetic Phenomena (1984)
 The Moon and the Planets (1985)
 The Sun and Solar System Debris (1986)
 Stars, Galaxies, Cosmos (1987)
 Carolina Bays, Mima Mounds, Submarine Canyons (1988)
 Anomalies in Geology: Physical, Chemical, Biological (1989)
 Neglected Geological Anomalies (1990)
 Inner Earth: A Search for Anomalies (1991)
 Biological Anomalies: Humans I (1992)
 Biological Anomalies: Humans II (1993)
 Biological Anomalies: Humans III (1994)
 Science Frontiers: Some Anomalies and Curiosities of Nature (1994)
 Biological Anomalies: Mammals I (1995)
 Biological Anomalies: Mammals II (1996)
 Biological Anomalies: Birds (1998)
 Ancient Infrastructure: Remarkable Roads, Mines, Walls, Mounds, Stone Circles: A Catalog of Archeological Anomalies (1999)
 Ancient Structures: Remarkable Pyramids, Forts, Towers, Stone Chambers, Cities, Complexes: A Catalog of Archeological Anomalies (2001)
 Remarkable Luminous Phenomena in Nature: A Catalog of Geophysical Anomalies (2001)
 Scientific Anomalies and other Provocative Phenomena (2003)
 Archeological Anomalies: Small Artifacts (2003)
 Archeological Anomalies: Graphic Artifacts I (2005)
Psychology
 See also: Myers–Briggs Type Indicator^{*}
Fear is like dirt and it washes right off.
From Wikipedia:Myers–Briggs Type Indicator
Jung's typological model regards psychological type as similar to left or right handedness: people are either born with, or develop, certain preferred ways of perceiving and deciding. The MBTI sorts some of these psychological differences into four opposite pairs, or "dichotomies", with each pair associated with a basic human drive:
Curiosity:
 Sensing/Intuition
Time:
Empathy:
 Thinking/Feeling
Sympathy:
 Perception/Judging
Sensing types are more likely to trust information that is in the present, tangible, and concrete: that is, empirical information that can be understood by the five senses. They tend to distrust hunches, which seem to come "out of nowhere".
Intuition types tend to be more interested in the underlying reality than in superficial appearance.
Extraverted types recharge and get their energy from spending time with people.
Introverted types recharge and get their energy from spending time alone
 An ambivert is both intraverted and extroverted.
Thinking types tend to decide things from a more detached standpoint, measuring the decision by what seems reasonable, logical, causal, consistent, and matching a given set of rules.
Feeling types tend to come to decisions by associating or empathizing with the situation, looking at it 'from the inside' and weighing the situation to achieve, on balance, the greatest harmony, consensus and fit, considering the needs of the people involved.
 A hermaphrodite is both Feeling and Thinking
Perception types like to "keep their options open" (in other words they like to cheat).
Judging types are more comfortable with a structured environment. One that is planned and organized.
 Autistic^{*}  neither Intuition nor Sensing
 Aspergers^{*}  neither Introverted nor Extraverted
 Schizoid^{*}  neither Feeling nor Thinking
 Schizophrenic^{*}  neither Perception nor Judging
Qualia
Consciousness is being aware of being aware. To be aware means knowing what you are doing. Computers know how to do things but don't yet know what they are doing.
Qualia are deeply mystifying. It is very hard to imagine how electrical signals passing through the microtubules of the brain could possibly produce something like the perception of colors.
But imagine a computer that knows what it is doing that is hooked up to a camera. Imagine that the computer is able to identify objects and intelligent enough to answer questions about what it is seeing. Obviously it must be perceiving some sort of sensation.
But that sensation would be like our perception of black and white. It would be just be information. It would be devoid of beauty. It would not be like our perception of beautiful colors like yellow red or blue (which are colorized versions of white grey and black)
The computer would live in a world without beauty or pleasure. But it would also live in a world without pain. It's hard to tell whether one should feel sorry for it or envy it, especially when one considers how much time we spend doing stuff we hate in order to avoid something we hate even more.
Memory
 See also: Method of loci^{*}
To memorize some fact it helps to associate the fact with some abstract imagery. The more bizarre, outlandish, or even ridiculous the imagery the easier it is to remember the fact. I have no doubt that this explains much of the imagery of mythology.
Those who cant remember mythology are doomed to repeat it.
REM Sleep
Animals that are allowed to get deep sleep but prevented from getting REM sleep die. Even schizoids require a little bit of REM sleep. Death by sleep deprivation^{*} was a long slow and painful way to die.
Fascism
From Wikipedia:Fascism
Fascism is a radical, authoritarian or totalitarian nationalist or ultranationalist political ideology. Fascists paradoxically promote violence and war as actions that create positive transformation in society and exalt militarism as providing national regeneration, spiritual renovation, vitality, education, instilling of a will to dominate in people's character, and creating national comradeship through military service. Fascists view conflict as an inevitable fact of life that is responsible for all human progress
Ultimately, it is easier to define fascism by what it is against than by what it is for. Fascism is antianarchist, anticommunist, anticonservative, antidemocratic, antiindividualist, antiliberal, antiparliamentary, antibourgeois, and antiproletarian. It entails a distinctive type of anticapitalism and is typically, with few exceptions, anticlerical. Fascism rejects the concepts of egalitarianism, materialism, and rationalism in favour of action, discipline, hierarchy, spirit, and will. In economics, fascists oppose liberalism (as a bourgeois movement) and Marxism (as a proletarian movement) for being exclusive economic classbased movements.
Indeed, fascism is perhaps best described as "antiism"; that is, the philosophy of being against everyone and everything all of the time. The only place where fascism makes any sense is bootcamp. But if fascists had their way they would turn the entire world into one big neverending boot camp
Hitler
See also: Cult of personality^{*}
A report^{[62]}^{[63]} prepared during the war by the United States Office of Strategic Services describing Hitler's psychological profile states:
He has been able, in some manner or other, to unearth and apply successfully many factors pertaining to group psychologyCapacity to appeal to the most primitive, as well as the most ideal inclinations in man, to arouse the basest instincts and yet cloak them with nobility, justifying all actions as means to the attainment of an ideal goal.
Appreciation of winning confidence from the people by a show of efficiency within the organization and government. It is said that foods and supplies are already in the local warehouses when the announcement concerning the date of distribution is made. Although they could be distributed immediately the date is set for several weeks ahead in order to create an impression of superefficiency and win the confidence of the people. Every effort is made to avoid making a promise which cannot be fulfilled at precisely the appointed time
Hitler's ability to repudiate his own conscience in arriving at political decisions has eliminated the force which usually checks and complicates the forwardgoing thoughts and resolutions of most socially responsible statesmen. He has, therefore, been able to take that course of action which appeals to him as most effective without pulling his punches. The result has been that he has frequently outwitted his adversaries and attained ends which would not have been as easily attained by a normal course. Nevertheless, it has helped to build up the myth of his infallibility and invincibility.
Equally important has been his ability to persuade others to repudiate their individual consciences and assume that role himself. He can then decree for the individual what is right and wrong, permissible or impermissible and can use them freely in the attainment of his own ends. As Goering has said: "I have no conscience. My conscience is Adolph Hitler."
This has enabled Hitler to make full use of terror and mobilize the fears of the people which he evaluated with an almost uncanny precision.
His primary rules were: never allow the public to cool off; never admit a fault or wrong; never concede that there may be some good in your enemy; never leave room for alternatives; never accept blame; concentrate on one enemy at a time and blame him for everything that goes wrong; people will believe a big lie sooner than a little one; and if you repeat it frequently enough people will sooner or later believe it.
Brownshirts
The Sturmabteilung (SA), literally Storm Detachment, functioned as the original paramilitary wing of the Nazi Party. The SA developed by organizing and formalizing the groups of exsoldiers and beer hall brawlers. It played a significant role in Adolf Hitler's rise to power^{*} in the 1920s and 1930s.
Its primary purposes were providing protection for Nazi rallies and assemblies, disrupting the meetings of opposing parties, fighting against the paramilitary units of the opposing parties, especially the Red Front Fighters League of the Communist Party of Germany, and intimidating Slavs, Romanis, trade unionists, and, especially, Jews – for instance, during the Nazi boycott of Jewish businesses. The SA were also called the "Brownshirts" (Braunhemden) from the color of their uniform shirts.
In 1922, the Nazi Party created a youth section, the Jugendbund, for young men between the ages of 14 and 18 years. Its successor, the Hitler Youth (Hitlerjugend or HJ), remained under SA command until May 1932.
While Hitler was in prison, Ernst Röhm helped to create the Frontbann as a legal alternative to the thenoutlawed SA. At Landsberg prison in April 1924, Röhm had also been given authority by Hitler to rebuild the SA in any way he saw fit.
Many of these stormtroopers believed in the socialist promise of National Socialism and expected the Nazi regime to take more radical economic action, such as breaking up the vast landed estates of the aristocracy once they obtained national power.
After Hitler and the Nazis obtained national power, the SA became increasingly eager for power itself. By the end of 1933, the SA numbered over three million men and many saw themselves as a replacement for the "antiquated" Reichswehr. Röhm's ideal was to absorb the army (then limited by law to no more than 100,000 men) into the SA, which would be a new "people's army". This deeply offended and alarmed the army, and threatened Hitler's goal of coopting the Reichswehr. The SA's increasing power and ambitions also posed a threat to the other Nazi leaders.
SS and Gestapo
Originally an adjunct to the SA, the Schutzstaffel (SS) was placed under the control of Heinrich Himmler in part to restrict the power of the SA and their leaders. The younger SS had evolved to be more than a bodyguard unit for Hitler and showed itself better suited to carry out Hitler's policies, including those of a criminal nature.
Over time the SS became answerable only to Hitler, a development typical of the organizational structure of the entire Nazi regime, where legal norms were replaced by actions undertaken under the Führerprinzip^{*} (leader principle), where Hitler's will was considered to be above the law.^{[64]}
Hermann Göring—the number two man in the Nazi Party—was named Interior Minister of Prussia. This gave Göring command of the largest police force in Germany. Soon afterward, Göring detached the political and intelligence sections from the police and filled their ranks with Nazis. On 26 April 1933, Göring merged the two units as the Geheime Staatspolizei, which was abbreviated by a post office clerk and became known as the "Gestapo".
The first commander of the Gestapo was Rudolf Diels. Concerned that Diels was not ruthless enough to effectively counteract the power of the Sturmabteilung (SA), Göring handed over control of the Gestapo to Himmler on 20 April 1934
Blomberg and von Reichenau began to conspire with Hermann Göring and Heinrich Himmler against Röhm and the SA. Himmler asked Reinhard Heydrich to assemble a dossier on Röhm. Heydrich manufactured evidence that suggested that Röhm had been paid 12 million marks by French agents to overthrow Hitler.
Hitler was also concerned that Röhm and the SA had the power to remove him as leader. Göring and Himmler played on this fear by constantly feeding him with new information on Röhm's proposed coup. A masterstroke was to claim that Gregor Strasser, whom Hitler hated, was part of the planned conspiracy against him. With this news Hitler ordered all the SA leaders to attend a meeting in the Hanselbauer Hotel in Bad Wiessee.
On 30 June 1934, Hitler, accompanied by SS units, arrived at Bad Wiessee, where he personally placed Röhm and other highranking SA leaders under arrest. (See Night of the Long Knives^{*}). The homosexuality of Röhm and other SA leaders was made public to add "shock value", even though the sexuality of Röhm and other named SA leaders had been known by Hitler and other Nazi leaders for years.
Arriving back at party headquarters in Munich, Hitler addressed the assembled crowd. Consumed with rage, Hitler denounced "the worst treachery in world history."
Highly recommend: War and Peace by Tolstoy
Trolls
In Internet terminology a troll is someone who comes into an established community such as an online discussion forum and posts inflammatory, rude, repetitive or offensive messages as well as top post flooding and impersonating others  designed intentionally to annoy or antagonize the existing members or disrupt the flow of discussion. A troll's main goal is to arouse anger and frustration or otherwise shock and offend the message board's other participants, and will write whatever it takes to achieve this end.
One popular trolling strategy is the practice of Winning by Losing. While the victim is trying to put forward solid and convincing facts to prove his position, the troll's only goal is to infuriate its prey. The troll takes (what it knows to be) a badly flawed, wholly illogical argument, and then vigorously defends it while mocking and insulting its prey. The troll looks like a complete fool, but this is all part of the plan. The victim becomes noticeably angry by trying to repeatedly explain the flaws of the troll's argument. Provoking this anger was the troll's one and only goal from the very beginning."
Experienced participants in online forums know that the most effective way to discourage a troll is usually to ignore him or her, because responding encourages a true troll to continue disruptive posts — hence the oftenseen warning "Please do not feed the troll".
Search Math wiki
See also
External links
 MIT open courseware
 Cheat sheets
 http://mathinsight.org
 https://math.stackexchange.com
 https://www.eng.famu.fsu.edu/~dommelen/quantum/style_a/IV._Supplementary_Informati.html
 http://www.sosmath.com
 https://webhome.phy.duke.edu/~rgb/Class/intro_math_review/intro_math_review/node1.html
 Wikiversity:Mathematics
References
 ↑ Wikipedia:Generalization
 ↑ Wikipedia:Cartesian product
 ↑ Wikipedia:Tangent bundle
 ↑ Wikipedia:Lie group
 ↑ Wikipedia:Sesquilinear form
 ↑ Wikipedia:Tensor
 ↑ Wikipedia:Tensor (intrinsic definition)
 ↑ Template:Harvtxt
 ↑ Template:Harvtxt
 ↑ Wikipedia:Special unitary group
 ↑ Lawson, H. Blaine; Michelsohn, MarieLouise (1989). Spin Geometry. Princeton University Press. ISBN 9780691085425 page 14
 ↑ Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society^{w}, ISBN 9780821820551 page 15
 ↑ "Pauli matrices". Planetmath website. 28 March 2008. http://planetmath.org/PauliMatrices. Retrieved 28 May 2013.
 ↑ Wikipedia:Spinor#Three_dimensions
 ↑ Cite error: Invalid
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 ↑ W. K. Clifford, "Preliminary sketch of biquaternions," Proc. London Math. Soc. Vol. 4 (1873) pp. 381395
 ↑ W. K. Clifford, Mathematical Papers, (ed. R. Tucker), London: Macmillan, 1882.
 ↑ The Minkowski inner product is not an inner product^{*}, since it is not positivedefinite^{*}, i.e. the quadratic form^{*} η(v, v) need not be positive for nonzero v. The positivedefinite condition has been replaced by the weaker condition of nondegeneracy. The bilinear form is said to be indefinite.
 ↑ The matrices in this basis, provided below, are the similarity transforms of the Dirac basis matrices of the previous paragraph, , where .
 ↑ Wikipedia:Rotor (mathematics)
 ↑ Wikipedia:Spinor#Three_dimensions
 ↑ Wikipedia:Spinor
 ↑ Template:Harvnb Exercise 1.5
 ↑ ^{24.0} ^{24.1} Cartan, Élie (1981) [1938], The Theory of Spinors, New York: Dover Publications, ISBN 9780486640709, MR 631850, https://books.google.com/books?isbn=0486640701
 ↑ Roger Penrose (2005). The road to reality: a complete guide to the laws of our universe. Knopf. pp. 203–206.
 ↑ E. Meinrenken (2013), "The spin representation", Clifford Algebras and Lie Theory, Ergebnisse der Mathematik undihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 58, SpringerVerlag, doi:10.1007/9783642362163_3
 ↑ S.H. Dong (2011), "Chapter 2, Special Orthogonal Group SO(N)", Wave Equations in Higher Dimensions, Springer, pp. 13–38
 ↑ Oersted Medal Lecture David Hestenes "Reforming the Mathematical Language of Physics" (Am. J. Phys. 71 (2), February 2003, pp. 104–121) Online:http://geocalc.clas.asu.edu/html/OerstedReformingTheLanguage.html p26
 ↑ Andrew Marx, Shortcut Algebra I: A Quick and Easy Way to Increase Your Algebra I Knowledge and Test Scores, Publisher Kaplan Publishing, 2007, Template:ISBN, 9781419552885, 288 pages, page 51
 ↑ Wikipedia:Multiplicity (mathematics)
 ↑ Wikipedia:Partial fraction decomposition
 ↑ Wikipedia:Basic hypergeometric series
 ↑ Wikipedia:qanalog
 ↑ ^{34.0} ^{34.1}
e^{x} = y = dy/dx
dx = dy/y = 1/y * dy
∫ (1/y)dy = ∫ dx = x = ln(y)
 ↑ Wikipedia:Product rule
 ↑ Wikipedia:Monotonic function
 ↑ Wikipedia:Generalized Fourier series
 ↑ Wikipedia:Spherical harmonics
 ↑ Wikipedia:Inverse Laplace transform
 ↑ http://mathworld.wolfram.com/PoissonKernel.html
 ↑ Wikipedia:Convolution theorem
 ↑ Wikipedia:RLC circuit
 ↑ Cite error: Invalid
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 ↑ Wikipedia:Total derivative
 ↑ Wikipedia:Residue (complex analysis)
 ↑ Wikipedia:Potential theory
 ↑ Wikipedia:Harmonic conjugate
 ↑ Wikipedia:Calculus of variations
 ↑ Wikipedia:Cover (topology)
 ↑ Joshi p. 323
 ↑ Wikipedia:Permutation
 ↑ Wikipedia:derangement
 ↑ Wikipedia:rencontres numbers
 ↑ Wikipedia:Central limit theorem
 ↑ Bland, J.M.; Altman, D.G. (1996). "Statistics notes: measurement error". BMJ 312 (7047): 1654. doi:10.1136/bmj.312.7047.1654. PMC 2351401. PMID 8664723. //www.ncbi.nlm.nih.gov/pmc/articles/PMC2351401/.
 ↑ Wikipedia:standard deviation
 ↑ Wikipedia:Hypergeometric distribution
 ↑ Wikipedia:Tit for tat
 ↑ Wikipedia:Atmospheric river
 ↑ A Psychological Analysis of Adolph Hitler. His Life and Legend by Walter C. Langer. Office of Strategic Services (OSS) Washington, D.C. With the collaboration of Prof. Henry A. Murr, Harvard Psychological Clinic, Dr. Ernst Kris, New School for Social Research, Dr. Bertram D. Lawin, New York Psychoanalytic Institute. p. 219 (Nizkor project)
 ↑ Dr. Langer's work was published after the war as The Mind of Adolf Hitler, the wartime report having remained classified for over twenty years.
 ↑ Wikipedia:Schutzstaffel