Now consider a function
$f:\mathbb{R}^m\to\mathbb{R}^n$.
Write $f_i$ for the $i^{th}$ coordinate of $f$, so we can write $f(x)=(f_1(x),f_2(x),\ldots,f_n(x))$,
where $x\in\mathbb{R}^m$.
If each coordinate function $f_i$ is differentiable in each variable we can form the {\em Jacobian matrix} of $f$:
$$\begin{pmatrix}\nabla f_1\\ \vdots \\ \nabla f_n\end{pmatrix}.$$


In this matrix, the element in the $i^th$ row and $j^th$ column is $\frac{\partial f_i}{\partial x_j}$.

\begin{xmpl}

For the function $$f(x,y)=\begin{pmatrix}
  x^2 +y \\
  x y \\
  x
\end{pmatrix}
=
\begin{pmatrix}
  f_1(x,y) \\
  f_2(x,y) \\
  f_3(x,y)
\end{pmatrix},$$

the Jacobian matrix of $f$ is the matrix

$$J= \begin{bmatrix}
      \nabla f_1 \\
      \nabla f_2 \\
      \nabla f_3
      \end{bmatrix}
=
\begin{bmatrix}
      2x & 2y \\
      y & x \\
      1 & 0
      \end{bmatrix}.
$$
\end{xmpl}