Theorem. existence of partial derivatives [br917]
Theorem. existence of partial derivatives [br917]
Suppose $\mathbf{f}$ maps an open set $E \subset \mathbb{R}^{n}$ into $\mathbb{R}^{m}$, and $\mathbf{f}$ is differentiable at a point $\mathbf{x} \in E$. Then the partial derivative $(D_{j}f_{i})(\mathbf{x})$ exist, and $$ \mathbf{f}'(\mathbf{x})\mathbf{e}_{j} = \sum_{i=1}^{m}(D_{j}f_{i})(\mathbf{x})\mathbf{u}_{i} \qquad (1 \leq j \leq n). $$
If $\mathbf{F}(\mathbf{x})$ is a scalar-valued function, then $$D \mathbf{F}(\mathbf{x}) = \begin{bmatrix}D_{1}\mathbf{F} & D_{2}\mathbf{F} & \cdots & D_{n}\mathbf{F}\end{bmatrix}$$ It’s also called the gradient of $\mathbf{F}$.