Gradient - Imported Knowlpedia Content

A gradient of a differentiable function $f:U\to \mathbb{R}$ (with $U\subseteq \mathbb{R}^n$ ) at a point $a\in U$ is the vector

\nabla f(a)=\left(\frac{\partial f}{\partial x_1}(a),\dots,\frac{\partial f}{\partial x_n}(a)\right),

formed from the partial derivatives of $f$ .

With respect to the standard inner product on $\mathbb{R}^n$ , the gradient encodes directional derivatives via $D_v f(a)=\nabla f(a)\cdot v$ whenever $f$ is differentiable at $a$ . Points where $\nabla f(a)=0$ are critical points.

Examples

If $f(x,y)=x^2+y^2$ , then $\nabla f(x,y)=(2x,2y)$ .
If $f(x,y,z)=x e^{yz}$ , then $\nabla f(x,y,z)=\big(e^{yz},\,xze^{yz},\,xye^{yz}\big)$ .