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A Jacobian matrix, sometimes simply called a Jacobian, is a matrix of first order partial derivatives (in some cases, the term "Jacobian" also refers to the determinant of the Jacobian matrix).

For a function \mathbf f:\R^n\to\R^m , the Jacobian is the following m\times n matrix:

\mathbf J=\frac{\part(u_1,\ldots,u_m)}{\part(x_1,\ldots,x_n)}=
\begin{bmatrix}
\dfrac{\part u_1}{\part x_1}&\cdots&\dfrac{\part u_1}{\part x_n}\\
\vdots&\ddots &\vdots\\
\dfrac{\part u_m}{\part x_1}&\cdots&\dfrac{\part u_m}{\part x_n}
\end{bmatrix}

or, in Einstein notation,

\mathbf J_i^j=\part^j f_i

Note that in some conventions, the Jacobian is the transpose of the above matrix.

Jacobians where m=n are square matrices, and are commonly used when changing coordinates, especially when taking multiple integrals and determining whether complex functions are holomorphic. For example, a Jacobian representing a change in variables from x to x(u,v) and y to y(u,v) in two dimensions is represented as

\mathbf J=\frac{\part(u,v)}{\part(x,y)}=
\begin{bmatrix}
\dfrac{\part u}{\part x}&\dfrac{\part u}{\part y}\\\\
\dfrac{\part v}{ \part x}&\dfrac{\part v}{\part y} 
\end{bmatrix}

A Jacobian matrix is what is usually meant by the derivative of higher-dimensional functions; indeed, differentiability in the components of a Jacobian guarantees differentiability in the function itself. In the case of a multivariable function f:\R^N\to\R , the Jacobian matrix with respect to the input variables is simply the gradient of the function. The Jacobian is also related to the Hessian matrix by

\mathbf{H}(f)=\mathbf{J}(\nabla f)

Applications

Jacobian matrices are useful in integration when changing coordinate systems. For example, given a two dimensional coordinate transformation, the double integral of f(x,y) becomes

\begin{align}\iint_D f(x,y)\,dx\,dy&=\iint_{D'}f\big(x(u,v),y (u,v)\big)\left|\dfrac{\part(x,y)}{\part(u,v)}\right|\,du\,dv\\
&=\iint_{D'}f\big(x(u,v),y(u,v)\big)\left|\dfrac{\part(u,v)}{\part(x,y)}\right|^{-1}\,du\,dv\end{align}

When working with one independent variable, this becomes

\left|\dfrac{\part u}{\part x}\right|=\left|\dfrac{dx}{du}\right|^{-1}=\dfrac{du}{dx}

which, when used to compute an integral, yields the formula known as integration by substitution.

Examples

Jacobian matrices are useful in integration when changing coordinate systems. For example, given a two dimensional coordinate transformation, the double integral of f(x,y) becomes

\begin{align}\iint_D f(x,y)\,dx\,dy&=\iint_{D'}f\big(x(u,v),y (u,v)\big)\left|\dfrac{\part(x,y)}{\part(u,v)}\right|\,du\,dv\\
&=\iint_{D'}f\big(x(u,v),y(u,v)\big)\left|\dfrac{\part(u,v)}{\part(x,y)}\right|^{-1}\,du\,dv\end{align}

When working with one independent variable, this becomes

\left|\dfrac{\part u}{\part x}\right|=\left|\dfrac{dx}{du}\right|^{-1}=\dfrac{du}{dx}

which, when used to compute an integral, yields the formula known as integration by substitution. Find the area of a circle of radius a by transforming from Cartesian coordinates to Polar coordinates.

A=\iint_D dA=\iint_D dx\,dy=\iint_{D'}\left|\dfrac{\part(x,y)}{\part(r,\theta)}\right|\,dx\,dy

Since x=r\cos(\theta) and y=r\sin(\theta) , the Jacobian determinant becomes

\begin{align}
\left|\dfrac{\part(x,y)}{\part(r,\theta)}\right|=
\begin{vmatrix}\cos(\theta)&-r\sin(\theta)\\\sin(\theta)&r\cos(\theta)\end{vmatrix}
&=\big(\cos(\theta)\big)\big(r\cos(\theta)\big)-\big(\sin(\theta)\big)\big(-r\sin(\theta)\big)\\
&=r\big(\cos^2(\theta)+\sin^2(\theta)\big)=r
\end{align}

The integral now becomes

\iint_{D'}r\,dr\,d\theta

Since it is now in polar coordinates, we can add the bounds as

\int\limits_0^{2\pi}\int\limits_0^a r\,dr\,d\theta

Now we can integrate it.

\int\limits_0^{2\pi}\int\limits_0^a r\,dr\,d\theta=\int\limits_0^{2\pi}\frac{a^2}{2}d\theta=\pi a^2

Which is to be expected. We have also found that the differential of dA is r\,dr\,d\theta .

This same method can be used to find the volume of a sphere in spherical coordinates. Since

\begin{align}
x&=r\sin(\phi)\cos(\theta)\\y&=r\sin(\phi)\sin(\theta)\\z&=r\cos(\phi)
\end{align}

the Jacobian determinant evaluates to:

\left|\dfrac{\part(x,y,z)}{\part(r,\phi,\theta)}\right|=
\begin{vmatrix}
\sin(\phi)\cos(\theta)&r\cos(\phi)\cos(\theta)&-r\sin(\phi)\sin(\theta)\\
\sin(\phi)\sin(\theta)&r\cos(\phi)\sin(\theta)&r\sin(\phi)\cos(\theta)\\
\cos(\phi)&-r\sin(\phi)&0
\end{vmatrix}=
r^2\sin(\phi)

This means that for any function in spherical coordinates, the volume element is

dV=r^2\sin(\phi)\,dr\,d\theta\,d\phi

The integral is therefore

\begin{align}\iiint_V dV&=\int\limits_0^\pi\int\limits_0^{2\pi}\int\limits_0^a r^2\sin(\phi)\,dr\,d\theta\,d\phi=\int\limits_0^\pi\sin(\phi)\,d\phi\int\limits_0^{2\pi}d\theta\int\limits_0^a r^2dr\\
&=(2)(2\pi)\frac{a^3}{3}=\frac43\pi a^3\end{align}

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