\begin{definition}[Definition:Jacobian/Determinant]
{{:Definition:Jacobian}}
The '''Jacobian determinant''' of $\mathbf f$ at $\mathbf x$ is defined to be the determinant of the Jacobian matrix:
:$\map \det {\mathbf J_{\mathbf f} } := \begin {vmatrix}
 \map {\dfrac {\partial f_1} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_1} {\partial x_n} } {\mathbf x} \\
 \vdots & \ddots & \vdots \\
 \map {\dfrac {\partial f_m} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_m} {\partial x_n} } {\mathbf x}
\end {vmatrix}$
\end{definition}