quality:correction, linguistic:paraphrasing, stylistic:style_transfer
Jacobian

The Jacobian matrix is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative.

Suppose f : R → R is a function such that each of its first-order partial derivatives exist on ℝ. Then the Jacobian matrix of f is defined to be an m × n matrix, denoted by J f (x) {\displaystyle \mathbf {J} _{\mathbb {f} }(\mathbb {x})} or simply J {\displaystyle \mathbf {J} }. The (i, j) th entry is J i j = ∂ f i / ∂ x j {\textstyle \mathbf {J} _{ij}={\partial f_{i}}/{\partial x_{j}}}. Explicitly J = = =. {\displaystyle \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}}.}