The Jacobi Matrix: Using \(\mathrm{d}f(x)h = \frac{\partial f}{\partial x^1}(x) h^1 + \cdots + \frac{\partial f}{\partial x^m}(x) h^m\), we derive the coordinate representation of the differential for a vector-valued function \( f : E \to \mathbb{R}^n \), with \( E \subset \mathbb{R}^m \), as \[ \mathrm{d}f(x) h = \begin{pmatrix} \mathrm{d}f^1(x) h \\ \vdots \\ \mathrm{d}f^n(x) h \end{pmatrix} = \begin{pmatrix} \partial_i f^1(x) h^i \\ \vdots \\ \partial_i f^n(x) h^i \end{pmatrix} = \begin{pmatrix} \frac{\partial f^1}{\partial x^1}(x) & \cdots & \frac{\partial f^1}{\partial x^m}(x) \\ \vdots & \ddots & \vdots \\ \frac{\partial f^n}{\partial x^1}(x) & \cdots & \frac{\partial f^n}{\partial x^m}(x) \end{pmatrix} \begin{pmatrix} h^1 \\ \vdots \\ h^m \end{pmatrix} \]

The matrix \(\left( \partial_i f^j(x) \right), (i = 1, \dots, m,\ j = 1, \dots, n)\) of partial derivatives of the coordinate functions of a given mapping at the point \( x \in E \), is called the Jacobi matrix or the Jacobian of the mapping at that point