Gradient as a vector field.

In vector calculus, gradient is the vector (or more specifically, the covector) made from the partial derivatives of a function with respect to each independent variable; as such, it is a special case of the Jacobian matrix. Intuitively, it can thought of as the direction of greatest slope of a graph. It can be calculated by taking the del operator of a scalar function. In three dimensions, it is equal to

\nabla f(x,y,z)=\frac{\part f}{\part x}\mathbf{\hat i}+\frac{\part f}{\part y}\mathbf{\hat j}+\frac{\part f}{\part z}\mathbf{\hat k}

In n dimensions, it is equal to

\nabla f(x_0,\ldots,x_n)=\frac{\part f}{\part x_0}e_0+\cdots+\frac{\part f}{\part x_n}e_n

with e being the unit vector.

See also