MathLiveJacobian matrix
The Jacobian matrix collects all first-order partial derivatives of a vector-valued function into one m by n matrix. It is used to linearize a function near a point, to change variables in multiple integrals, and to study how small input changes affect outputs.
J_{\mathbf{f}}(\mathbf{x})=\begin{pmatrix} \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{pmatrix}What the symbols mean
| vector-valued function with m component functions | |
| input vector of n real variables | |
| first-order partial derivative of the i-th component with respect to the j-th variable | |
| Jacobian matrix of f evaluated at x |
About this formula
- FieldMathematics
- TopicMultivariable calculus
- LevelUniversity
- Also calledJacobi matrix, Jacobi system, Jacobian, Jacobian of a vector function, matrix of first partial derivatives, derivative matrix of a vector-valued function
- SourceWikidata Q506041
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