---
tag: 047NDTEPCV
kind: concept
name: Jacobian matrix
aliases: [jacobian, derivative-matrix]
msc: [26B10]
uses: [09EGC6F8PT, 0FNKFT8H6E]
sources:
  - url: https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
    locator: Jacobian matrix
same_as:
  wikidata: Q506041
  wikipedia: Jacobian matrix and determinant
---

## Summary

Let $U \subseteq \mathbb{R}^n$ be open. The Jacobian matrix of a map $f \colon U \to \mathbb{R}^m$ at a point $a$ is the $m \times n$ matrix of all first [[partial derivatives|09EGC6F8PT]] $\partial f_i / \partial x_j$ at $a$.
When $f$ is differentiable at $a$, it is the matrix of the [[total derivative|0FNKFT8H6E]] $Df(a)$ in the standard bases.