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:
$\quad \mathbf J_{\mathbf f} := \begin{pmatrix} \map {\dfrac {\partial f_1} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_1} {\partial x_n} } {\mathbf x} \\ \vdots & \ddots & \vdots \\ \map {\dfrac {\partial f_m} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_m} {\partial x_n} } {\mathbf x} \end{pmatrix}$
Jacobian Determinant
The Jacobian determinant of $\mathbf f$ at $\mathbf x$ is defined to be the determinant of the Jacobian matrix:
$\quad \map \det {\mathbf J_{\mathbf f} } := \begin {vmatrix} \map {\dfrac {\partial f_1} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_1} {\partial x_n} } {\mathbf x} \\ \vdots & \ddots & \vdots \\ \map {\dfrac {\partial f_m} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_m} {\partial x_n} } {\mathbf x} \end {vmatrix}$
Also known as
Note that both the Jacobian determinant and Jacobian matrix 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.
Also see
- Results about Jacobians can be found here.
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