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Definition. Jacobian matrix [math-0014]

Let f:U⊂Rn→Rmf : U \subset \R^n \to \R^m be a map, hence we can view each mm component is a function Rn→R\R^n \to \R :

f(x1,x2,…,xn)=(f1(x1,…,xn)f2(x1,…,xn)⋮fm(x1,…,xn))f(x^1, x^2, \dots, x^n) = \begin{pmatrix}f_1\left(x^1,\ldots,x^{n}\right)\\ f_2\left(x^1,\ldots,x^{n}\right)\\ \vdots\\ f_{m}\left(x^1,\ldots,x^{n}\right) \end{pmatrix}

with respect to standard bases, and a∈Rna \in \R^n , Df∣aDf\left|_{a}\right. is given by the m×nm \times n matrix of partial derivatives (the Jacobian matrix) in the following sense

Df∣av=(∂f1∂x1(a)∂f1∂x2(a)⋯∂f1∂xn(a)∂f2∂x1(a)∂f2∂x2(a)⋯∂f2∂xn(a)⋮⋮⋱⋮∂fm∂x1(a)∂fm∂x2(a)⋯∂fm∂xn(a))⏞n()}m(v1v2⋮vn)Df\left|_{a}\right.v=\overbrace{\begin{pmatrix}\frac{\partial f_1}{\partial x^1}\left(a\right) & \frac{\partial f_1}{\partial x^2}\left(a\right) & \cdots & \frac{\partial f_1}{\partial x^{n}}\left(a\right)\\ \frac{\partial f_2}{\partial x^1}\left(a\right) & \frac{\partial f_2}{\partial x^2}\left(a\right) & \cdots & \frac{\partial f_2}{\partial x^{n}}\left(a\right)\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial f_{m}}{\partial x^1}\left(a\right) & \frac{\partial f_m}{\partial x^2}\left(a\right) & \cdots & \frac{\partial f_m}{\partial x^{n}}\left(a\right)\end{pmatrix}}^{n}\left.\vphantom{ \begin{pmatrix} \\ \\ \\ \\ \end{pmatrix} }\right\rbrace m\begin{pmatrix}v^1\\ v^2\\ \vdots\\ v^{n}\end{pmatrix}

Or using the index notation, so ω=Df(a)v\omega=Df\left(a\right)v can be expressed as:

ωi=∑j∂fi∂xj(a)vj\omega^{i}=\sum_{j}\frac{\partial f_{i}}{\partial x^{j}}\left(a\right)v_{}^{j}