# Jacobian matrix

The Jacobian matrix $[\mathbf{J}f(\mathbf{a})]$ of a function $f\colon\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ at the point $\mathbf{a}$ with respect to some choice of bases for $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$ is the matrix of the linear map from $\mathbb{R}^{n}$ into $\mathbb{R}^{m}$ that generalizes the definition of the derivative of a function on $\mathbb{R}$. It can be defined as the matrix of the linear map $D$, such that

 $\lim_{\mathbf{h}\rightarrow\mathbf{0}}\frac{\|f(\mathbf{a+h})-f(\mathbf{a})-D(% \mathbf{h})\|}{\|\mathbf{h}\|}=0$

The linear map that satisfies the above limit is called the derivative of $f$ at $\mathbf{a}$. It is easy to show that the Jacobian matrix of a given differentiable function at $\mathbf{a}$ with respect to chosen bases is just the matrix of http://planetmath.org/node/841partial derivatives  of the component  functions of $f$ at $\mathbf{a}$:

 $[\mathbf{J}f(\mathbf{x})]=\left[\begin{array}[]{ccc}D_{1}f_{1}(\mathbf{x})&% \dots&D_{n}f_{1}(\mathbf{x})\\ \vdots&\ddots&\vdots\\ D_{1}f_{m}(\mathbf{x})&\dots&D_{n}f_{m}(\mathbf{x})\\ \end{array}\right]$

A more concise way of writing it is

 $[\mathbf{J}f(\mathbf{x})]=[\overrightarrow{D_{1}f}\;,\cdots\;,\overrightarrow{% D_{n}f}]=\left[\begin{array}[]{c}\nabla f_{1}\\ \vdots\\ \nabla f_{m}\end{array}\right]$

where $\overrightarrow{D_{n}\mathbf{f}}$ is the partial derivative with respect to the $n$’th variable and $\nabla f_{m}$ is the gradient  of the $m$’th component of $\mathbf{f}$.

Note that the Jacobian matrix represents the matrix of the derivative $D$ of $f$ at $\mathbf{x}$ iff $f$ is differentiable  at $\mathbf{x}$. Also, if $f$ is differentiable at $\mathbf{x}$, then the directional derivative  in the direction $\vec{v}$ is $D(\vec{v})=[\mathbf{J}f(\mathbf{x})]\vec{v}$.

Given local coordinates for some real submanifold $M$ of $\mathbb{R}^{n}$, it is easy to show that the effect of a change of coordinates on volume forms  is a local scaling of the volume form by the determinant  of the Jacobian matrix of the derivative of the backwards change of coordinates, which is called the inverse Jacobian  . The determinant of the inverse Jacobian is thus commonly seen in integration over a change of coordinates.

 $\int_{\Omega_{x}}f(x)d\Omega_{x}=\int_{\Omega_{\xi}}f(\xi)|J^{-1}|d\Omega_{\xi}$

where $x\in\Omega_{x}\subset M$, $\xi\in\Omega_{\xi}\subset M$, and $\mathbf{J}=\nabla\xi(x)$ is the derivative of the function $\xi(x)$ which maps points in $\Omega_{x}$ to points in $\Omega_{\xi}$.

 Title Jacobian matrix Canonical name JacobianMatrix Date of creation 2013-03-22 11:58:33 Last modified on 2013-03-22 11:58:33 Owner PhysBrain (974) Last modified by PhysBrain (974) Numerical id 18 Author PhysBrain (974) Entry type Definition Classification msc 26B10 Synonym derivative of a multivariable function Synonym derivative of a vector-valued function Synonym Jacobi matrix Related topic PartialDerivative Related topic Derivative2 Related topic DerivativeNotation Related topic Gradient Related topic ChainRuleSeveralVariables Related topic DirectionalDerivative Related topic JordanBanachAndJordanLieAlgebras Defines Jacobian