Jacobian matrix

Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f = (f_1, \dots, f_m) \colon U \to \mathbb{R}^m$, and let $a \in U$. Suppose that each partial derivative $\partial f_i / \partial x_j$ exists at $a$, for $1 \le i \le m$ and $1 \le j \le n$. The Jacobian matrix of $f$ at $a$ is the $m \times n$ matrix $$J_f(a) = \begin{pmatrix} \dfrac{\partial f_1}{\partial x_1}(a) & \cdots & \dfrac{\partial f_1}{\partial x_n}(a) \\ \vdots & \ddots & \vdots \\ \dfrac{\partial f_m}{\partial x_1}(a) & \cdots & \dfrac{\partial f_m}{\partial x_n}(a) \end{pmatrix},$$ whose entry in row $i$ and column $j$ is $\partial f_i / \partial x_j (a)$. Other notations are $Df(a)$, $f'(a)$ and $\dfrac{\partial (f_1, \dots, f_m)}{\partial (x_1, \dots, x_n)}$.

Remarks

If $f$ is differentiable at $a$, then $Df(a)(h) = J_f(a) \, h$ for each column vector $h \in \mathbb{R}^n$. Row $i$ of $J_f(a)$ is the transpose of the gradient of $f_i$ at $a$. For $m = 1$, the Jacobian matrix is the gradient written as a row. For the polar map $f(r, \theta) = (r \cos \theta, r \sin \theta)$, the Jacobian matrix is $\begin{pmatrix} \cos \theta & -r \sin \theta \\ \sin \theta & r \cos \theta \end{pmatrix}$. For $m = n$, its determinant is the Jacobian determinant.

Explanations

L1 human

The Jacobian matrix lists how each output changes with each input: one row for each output, one column for each input. It is the multivariable form of the derivative. It turns a small step $h$ in the input into the approximate step $J_f(a) h$ in the output.