PlanetMath: Jacobian matrix

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Jacobian matrix

(Definition)

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 , such that

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

The linear map that satisfies the above limit is called the derivative of

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 partial derivatives of the component functions of

at $\mathbf {a}$ :

$\displaystyle [\mathbf{J}f(\mathbf {x})]=\left[\begin{array}{ccc} D_1f_1(\mathb... ...dots\ D_1f_m(\mathbf {x}) & \dots & D_nf_m(\mathbf {x})\ \end{array}\right]$

A more concise way of writing it is

$\displaystyle [\mathbf{J}f(\mathbf {x})]=[\overrightarrow{D_1 f}\;,\cdots\;, \o... ..._n f}]=\left[\begin{array}{c}\nabla f_1\\ \vdots\\ \nabla f_m\end{array}\right]$

where $\overrightarrow{D_n\mathbf {f}}$ is the partial derivative with respect to the

'th variable and $\nabla f_m$ is the gradient of the

'th component of $\mathbf{f}$ .

Note that the Jacobian matrix represents the matrix of the derivative of at $\mathbf {x}$ iff is differentiable at $\mathbf {x}$ . Also, if 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 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.

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

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 $\Omega_\xi$ .

"Jacobian matrix" is owned by PhysBrain. [ full author list (3) | owner history (2) ]

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Other names:

derivative of a multivariable function, derivative of a vector-valued function, Jacobi matrix

Also defines:

Jacobian

Keywords:

Jacobian, multivariable derivative, directional derivative, derivative matrix

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Cross-references: maps, inverse, determinant, scaling, volume forms, change of coordinates, real submanifold, local coordinates, directional derivative, differentiable, iff, represents, gradient, variable, partial derivative, component, differentiable function, limit, derivative, linear map, matrix, bases, point, function
There are 39 references to this entry.

This is version 14 of Jacobian matrix, born on 2001-11-14, modified 2007-01-14.
Object id is 842, canonical name is JacobianMatrix.
Accessed 59432 times total.

Classification:

AMS MSC:

26B10 (Real functions :: Functions of several variables :: Implicit function theorems, Jacobians, transformations with several variables)

Pending Errata and Addenda

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