Jacobian

This entry is about the concept in differential geometry. For the concept of Jacobian variety see there.

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

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

$J(f) \in Mat_{m \times n}(C^0(\mathbb{R}, \mathbb{R}))$

$J(f)^i_j := \frac{\partial f^i}{\partial x^j},\,\,\,\,\,\,\,i=1,\ldots,m; j = 1,\ldots,n,$

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

In more general situation, if $f = (f^1(x),\ldots,f^m(x))$ 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 $(J_p f)^i_j = \frac{\partial f^i}{\partial x^j}|_x$.

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.

The chain rule may be phrased by saying that the Jacobian matrix of the composition $\mathbf{R}^n\stackrel{f}\to\mathbf{R}^m\stackrel{g}\to\mathbf{R}^r$ 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 $T g: T M\to T N$ 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 $(U,\phi)\ni p$ and $(V,\psi)\ni g(p)$ are charts and $X_p = \sum X^i\frac{\partial}{\partial x^i}|_p$ (i.e. $X_p(f) = \sum_i X^i_p \frac{\partial (f\circ \phi^{-1})}{\partial x^i}|_{\phi(p)}$ for all germs $f\in \mathcal{F}_p$), then

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

Revised on July 18, 2015 04:52:59
by Urs Schreiber
(94.118.161.144)