# Definition:Jacobian/Matrix

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## Definition

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

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

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

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

## Also known as

This concept is often called just the Jacobian of $\mathbf f$ at $\mathbf x$.

However, this can allow it to be confused with the Jacobian determinant, so it is advised to use the full name unless context establishes which is meant.

Some sources present this as $J_f$, but boldface for matrices is usual, and standard on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Source of Name

This entry was named for Carl Gustav Jacob Jacobi.