Definition:Jacobian/Matrix
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 $\mathbf x = \paren {x_1, x_2, \ldots, x_n}^\intercal \in U$.
The Jacobian matrix of $\mathbf f$ at $\mathbf x$ is defined to be the matrix of partial derivatives:
$\quad \mathbf J_{\mathbf f} := \begin{pmatrix} \map {\dfrac {\partial f_1} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_1} {\partial x_n} } {\mathbf x} \\ \vdots & \ddots & \vdots \\ \map {\dfrac {\partial f_m} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_m} {\partial x_n} } {\mathbf x} \end{pmatrix}$
Also known as
The Jacobian matrix is often called just the Jacobian of $\mathbf f$ at $\mathbf x$.
However, this can allow it to be confused with the Jacobian determinant, so it is advised to use the full name unless context establishes which is meant.
Some sources present this as $J_f$, but boldface for matrices is usual, and standard on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Also see
- Results about Jacobian matrices can be found here.
Source of Name
This entry was named for Carl Gustav Jacob Jacobi.
Sources
- 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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Jacobian matrix
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Jacobian matrix