Definition:Jacobian

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 $\mathbf x = \paren {x_1, x_2, \ldots, x_n}^\intercal \in U$.

Jacobian Matrix

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

$\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}$

Jacobian Determinant

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}$

Also known as

Note that both concepts are often called just the Jacobian of $\mathbf f$ at $\mathbf x$.

It is advisable to use the full term for whichever is intended unless context makes it obvious which one is meant.

Source of Name

This entry was named for Carl Gustav Jacob Jacobi.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Jacobian or Jacobian determinant
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Jacobian
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Jacobian