# Definition:Jacobian

## Definition

Let $U$ be an open subset of $\R^n$.

Let $\mathbf f = \left({f_1, f_2, \ldots, f_m}\right)^\intercal: U \to \R^m$ be a vector valued function, differentiable at $\mathbf x = \left({x_1, x_2, \ldots, x_n}\right)^\intercal \in U$.

### Jacobian Matrix

The Jacobian matrix of $\mathbf f$ at $\mathbf x$ is defined to be the matrix of partial derivatives:

$\displaystyle \mathbf J_{\mathbf f} := \begin{pmatrix} \dfrac {\partial f_1} {\partial x_1} \left({\mathbf x}\right) & \cdots & \dfrac {\partial f_1} {\partial x_n} \left({\mathbf x}\right) \\ \vdots & \ddots & \vdots \\ \dfrac {\partial f_m} {\partial x_1} \left({\mathbf x}\right) & \cdots & \dfrac {\partial f_m} {\partial x_n} \left({\mathbf x}\right) \end{pmatrix}$

### Jacobian Determinant

The Jacobian determinant of $\mathbf f$ at $\mathbf x$ is defined to be the determinant of the Jacobian matrix:

$\displaystyle \det \left({\mathbf J_{\mathbf f} }\right) := \begin{vmatrix} \dfrac {\partial f_1} {\partial x_1} \left({\mathbf x}\right) & \cdots & \dfrac {\partial f_1} {\partial x_n} \left({\mathbf x}\right) \\ \vdots & \ddots & \vdots \\ \dfrac {\partial f_m} {\partial x_1} \left({\mathbf x}\right) & \cdots & \dfrac {\partial f_m} {\partial x_n} \left({\mathbf x}\right) \end{vmatrix}$

## Also known as

Note that both concepts are often called just the Jacobian of $\mathbf f$ at $\mathbf x$.

It is advisable to use the full term for whichever is intended unless context makes it obvious which one is meant.

## Source of Name

This entry was named for Carl Gustav Jacob Jacobi.