Jacobian matrix

The matrix of all first partial derivatives -- the total derivative written out in coordinates.

For $f:\mathbb{R}^n\to\mathbb{R}^m$ , the Jacobian matrix $Df$ is the $m\times n$ matrix whose $(i,j)$ entry is $\partial f_i/\partial x_j$ . It IS the total derivative, expressed concretely as an array of numbers once a coordinate basis is chosen.

Examples

For $f(u,v)=(u^2-v^2,\ 2uv)$ , $Df = \begin{pmatrix}2u & -2v\\ 2v & 2u\end{pmatrix}$ .