Gradient

Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, and let $a \in U$ be a point at which $f$ is differentiable. The gradient of $f$ at $a$ is the unique vector $\nabla f(a) \in \mathbb{R}^n$ such that $$Df(a)(v) = \nabla f(a) \cdot v \quad \text{for each } v \in \mathbb{R}^n,$$ where $\cdot$ is the dot product of Euclidean space. In coordinates, it is the vector of partial derivatives $$\nabla f(a) = \left(\frac{\partial f}{\partial x_1}(a), \dots, \frac{\partial f}{\partial x_n}(a)\right).$$ Other notations are $\operatorname{grad} f(a)$ and $\nabla f(a)$, where $\nabla$ is the del operator.