Gradient

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

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