Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Gradient
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
=== Jacobian === {{Main|Jacobian matrix and determinant}} The [[Jacobian matrix]] is the generalization of the gradient for vector-valued functions of several variables and [[differentiable map]]s between [[Euclidean space]]s or, more generally, [[manifold]]s.<ref>{{harvtxt|Beauregard|Fraleigh|1973|pp=87,248}}</ref><ref>{{harvtxt|Kreyszig|1972|pp=333,353,496}}</ref> A further generalization for a function between [[Banach space]]s is the [[Fréchet derivative]]. Suppose {{math|'''f''' : '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup>}} is a function such that each of its first-order partial derivatives exist on {{math|ℝ<sup>''n''</sup>}}. Then the Jacobian matrix of {{math|'''f'''}} is defined to be an {{math|''m''×''n''}} matrix, denoted by <math>\mathbf{J}_\mathbb{f}(\mathbb{x})</math> or simply <math>\mathbf{J}</math>. The {{math|(''i'',''j'')}}th entry is <math display="inline">\mathbf J_{ij} = {\partial f_i} / {\partial x_j}</math>. Explicitly <math display="block">\mathbf J = \begin{bmatrix} \dfrac{\partial \mathbf{f}}{\partial x_1} & \cdots & \dfrac{\partial \mathbf{f}}{\partial x_n} \end{bmatrix} = \begin{bmatrix} \nabla^\mathsf{T} f_1 \\ \vdots \\ \nabla^\mathsf{T} f_m \end{bmatrix} = \begin{bmatrix} \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n}\\ \vdots & \ddots & \vdots\\ \dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n} \end{bmatrix}.</math>
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)