Jacobian matrix

A Jacobian matrix of a map $f=(f_1,\dots,f_m):U\to \mathbb{R}^m$ (with $U\subseteq \mathbb{R}^n$) at a point $a\in U$ is the $m\times n$ matrix

provided these partial derivatives exist.

When $f$ is differentiable at $a$, the Fréchet derivative $Df(a)$ is a linear map, and $Jf(a)$ is the matrix that represents $Df(a)$ in the standard bases of $\mathbb{R}^n$ and $\mathbb{R}^m$. For scalar-valued $f$, $Jf(a)$ is closely related (up to transpose conventions) to the gradient .

Examples:

• If $f(x,y)=(x^2y,\sin(x-y))$, then

  $$Jf(x,y)= \begin{pmatrix} 2xy & x^2\\ \cos(x-y) & -\cos(x-y) \end{pmatrix}.$$

• If $f(x,y,z)=x+y+z$, then $Jf(x,y,z)=\begin{pmatrix}1&1&1\end{pmatrix}$.