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 $(m\times n)$ matrix

of partial derivatives

where $x=(x^1,\dots ,x^n)$. 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 $f=(f^1(x),\dots ,f^m(x))$ 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 $(J_{p}f{)}_j^i=\frac{\partial f^i}{\partial x^j}{\mid }_x$.

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 $\det (J(f))$ 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 $(T_{p}g)(X_p)(f)=X_p(f\circ g)$, for $p\in M$, can in local coordinates be represented by a Jacobian matrix. Namely, if $(U,\varphi )\ni p$ and $(V,\psi )\ni g(p)$ are charts and $X_p=\sum X^i\frac{\partial }{\partial x^i}{\mid }_p$ (i.e. $X_p(f)={\sum }_i{X}_p^i\frac{\partial (f\circ {\varphi }^{-1})}{\partial x^i}{\mid }_{\varphi (p)}$ for all germs $f\in {ℱ}_p$), then

Related concepts

• differentiation, Hessian matrix, extremum