Definition:Jacobian/Determinant

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

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

The Jacobian determinant of $f$ at $x$ is defined to be the determinant of the Jacobian matrix:
 * $\displaystyle J_f := \begin{vmatrix}

\frac{\partial f_1}{\partial x_1} \left({x}\right) & \cdots & \frac{\partial f_1}{\partial x_n} \left({x}\right) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} \left({x}\right) & \cdots & \frac{\partial f_m}{\partial x_n} \left({x}\right) \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 $f$ at $x$.

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.

Also see

 * Jacobian Matrix