---
tag: 0ZCSMYHTBP
kind: concept
name: Jacobian determinant
aliases: [jacobian-of-a-map]
msc: [26B10]
uses: [047NDTEPCV]
sources:
  - url: https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
    locator: Jacobian determinant
---

## Summary

The Jacobian determinant of a map from an open subset of $\mathbb{R}^n$ to $\mathbb{R}^n$ at a point is the determinant of its [[Jacobian matrix|047NDTEPCV]] there.
Its absolute value is the factor by which the map scales small volumes near the point. Its sign tells whether the map keeps or reverses orientation.