Category:Definitions/Jacobian Determinants

This category contains definitions related to Jacobian Determinants.
Related results can be found in Category:Jacobian Determinants.

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

$\quad \map \det {\mathbf J_{\mathbf f} } := \begin {vmatrix} \map {\dfrac {\partial f_1} {\partial x_1} } {\mathbf x} & \map {\dfrac {\partial f_1} {\partial x_2} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_1} {\partial x_n} } {\mathbf x} \\ \map {\dfrac {\partial f_2} {\partial x_1} } {\mathbf x} & \map {\dfrac {\partial f_2} {\partial x_2} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_2} {\partial x_n} } {\mathbf x} \\ \vdots & \vdots & \ddots & \vdots \\ \map {\dfrac {\partial f_n} {\partial x_1} } {\mathbf x} & \map {\dfrac {\partial f_n} {\partial x_2} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_n} {\partial x_n} } {\mathbf x} \end {vmatrix}$