Gradient

The vector of first partial derivatives ∇f of a scalar-valued function.

Gradient

Let $U\subseteq\mathbb{R}^k$ be open and let $f:U\to\mathbb{R}$ be differentiable (at least with existing partial derivatives ) at $a\in U$ . The gradient of $f$ at $a$ is the vector

\nabla f(a) := \left(\frac{\partial f}{\partial x_1}(a),\dots,\frac{\partial f}{\partial x_k}(a)\right)\in\mathbb{R}^k.

When $f$ is differentiable at $a$ , $\nabla f(a)$ represents the linear functional “best approximating” changes in $f$ near $a$ , via

f(a+h)\approx f(a)+\langle \nabla f(a),h\rangle.

The gradient points in the direction of steepest increase (see directional derivative ).

Examples:

If $f(x,y)=x^2+y^2$ , then $\nabla f(x,y)=(2x,2y)$ .
If $f(x)=\langle c,x\rangle$ , then $\nabla f(x)=c$ (constant).
If $f$ has a local extremum at an interior point and is differentiable there, then $\nabla f(a)=0$ (a necessary condition).