Definition. partial derivatives [br916]

Consider $\mathbf{f}: E \to \mathbb{R}^{m}$ where $E \subset \mathbb{R}^{n}$, Let $\{ \mathbf{e}_{1}, \dots, \mathbf{e}_{n} \}$ and $\{ \mathbf{u}_{1}, \dots, \mathbf{u}_{m} \}$ be the standard bases of $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$. The components of $\mathbf{f}$ are the real functions $f_{1}, \dots, f_{m}$ defined by $$ \mathbf{f}(\mathbf{x}) = \sum_{i=1}^{m} f_{i}(\mathbf{x}) \mathbf{u}_{i} \qquad (\mathbf{x} \in E), $$ or equivalently, by $f_{i}(\mathbf{x}) = \mathbf{f}(\mathbf{x}) \cdot \mathbf{u}_{i}, 1 \leq i \leq m$.

For, $\mathbf{x} \in E$, $1 \leq i \leq m, 1 \leq j \leq n$, we define $$ (D_{j}f_{i})(\mathbf{x}) = \lim_{ t \to 0 } \frac{f_{i}(\mathbf{x} + t \mathbf{e}_{j}) - f_{i}(\mathbf{x}) }{t}, $$ provided the limit exists. Writing $f_{i}(x_{1}, \dots, x_{n})$ in place of $f_{i}(\mathbf{x})$, we see that $D_{j}f_{i}$ is the derivative of $f_{i}$ with respect to $x_{j}$, keeping the other variables fixed. The notation $$ \frac{\partial f_{i}}{\partial x_{j}} $$ is therefore used in place of $D_{j}f_{i}$, and $D_{j}f_{i}$ is called a partial derivative.