# nLab Jacobian

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

For Jacobian in the sense of Jacobian variety (of an algebraic curve), see there (also more general intermediate Jacobians).

## Definition

If $f:{ℝ}^{n}\to {ℝ}^{m}$ is a ${C}^{1}$-differentiable map, between Cartesian spaces, its Jacobian matrix is the $\left(m×n\right)$ matrix

$J\left(f\right)\in {\mathrm{Mat}}_{m×n}\left({C}^{0}\left(ℝ,ℝ\right)\right)$J(f) \in Mat_{m \times n}(C^0(\mathbb{R}, \mathbb{R}))
$J\left(f{\right)}_{j}^{i}:=\frac{\partial {f}^{i}}{\partial {x}^{j}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,\dots ,m;j=1,\dots ,n,$J(f)^i_j := \frac{\partial f^i}{\partial x^j},\,\,\,\,\,\,\,i=1,\ldots,m; j = 1,\ldots,n,

where $x=\left({x}^{1},\dots ,{x}^{n}\right)$. Here the convention is that the upper index is a row index and the lower index is the column index; in particular ${R}^{n}$ is the space of real column vectors of length $n$.

In more general situation, if $f=\left({f}^{1}\left(x\right),\dots ,{f}^{m}\left(x\right)\right)$ is differentiable at a point $x$ (and possibly defined only in a neighborhood of $x$), we define the Jacobian ${J}_{p}f$ of map $f$ at point $x$ as a matrix with real values $\left({J}_{p}f{\right)}_{j}^{i}=\frac{\partial {f}^{i}}{\partial {x}^{j}}{\mid }_{x}$.

That is, the Jacobian is the matrix which describes the total derivative?.

If $n=m$ the Jacobian matrix is a square matrix, hence its determinant $\mathrm{det}\left(J\left(f\right)\right)$ is defined and called the Jacobian of $f$ (possibly only at a point). Sometimes one refers to Jacobian matrix rather ambigously by Jacobian as well.

## Properties

The chain rule may be phrased by saying that the Jacobian matrix of the composition ${R}^{n}\stackrel{f}{\to }{R}^{m}\stackrel{g}{\to }{R}^{r}$ is the matrix product of the Jacobian matrices of $g$ and of $f$ (at appropriate points).

If $g:M\to N$ is a ${C}^{1}$-map of ${C}^{1}$-manifolds, then the tangent map $Tg:TM\to TN$ defined point by point abstractly by $\left({T}_{p}g\right)\left({X}_{p}\right)\left(f\right)={X}_{p}\left(f\circ g\right)$, for $p\in M$, can in local coordinates be represented by a Jacobian matrix. Namely, if $\left(U,\varphi \right)\ni p$ and $\left(V,\psi \right)\ni g\left(p\right)$ are charts and ${X}_{p}=\sum {X}^{i}\frac{\partial }{\partial {x}^{i}}{\mid }_{p}$ (i.e. ${X}_{p}\left(f\right)={\sum }_{i}{X}_{p}^{i}\frac{\partial \left(f\circ {\varphi }^{-1}\right)}{\partial {x}^{i}}{\mid }_{\varphi \left(p\right)}$ for all germs $f\in {ℱ}_{p}$), then

$\left({T}_{p}g\right)\left({X}_{p}\right)=\sum _{i,j}{J}_{p}\left(\psi \circ g\circ {\varphi }^{-1}{\right)}_{i}^{j}{X}_{p}^{i}\frac{\partial }{\partial {y}^{j}}{\mid }_{g\left(p\right)}$(T_p g)(X_p) = \sum_{i,j} J_p(\psi \circ g\circ\phi^{-1})_i^j X^i_p \frac{\partial}{\partial y^j}|_{g(p)}

Revised on October 15, 2012 22:01:11 by Zoran Škoda (161.53.130.104)