# Definition:Jacobian/Determinant

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

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

## Also defined as

Sometimes the absolute value of the determinant of the Jacobian matrix is the intended meaning.

## 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 matrix, so it is advised to use the full name unless context establishes which is meant.

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

Other sources present it as either $\mathbf J_{\mathbf f}$ or $J_f$, allowing a further source of confusion between this and the Jacobian matrix.

## Source of Name

This entry was named for Carl Gustav Jacob Jacobi.