Orthogonal Gradient Theorem

Definition

For a function $f:{\mathbb{R}}^n\to \mathbb{R}$ differentiable at the point $x_0$ satisfying $\nabla f(x_0)\ne 0$. $L_c(f)\perp \nabla f(x_0)\iff L_c(f)\cdot \nabla f(x_0)=0,\forall x_0\in L_c(f)$

The Gradient vector a function is perpendicular to its Level Set.

Proof

Suppose a function $f:{\mathbb{R}}^n\to \mathbb{R}$ and a point $x_0\in L_c(f)$

By the definition of Level Set, $\forall t,f(x_0(t))=c$ where $x_0(t)$ is a parametrization of a curve in the level set passing through $x_0$.

By differentiating both sides with respect to $t$ $\frac{df}{dt}=(\frac{df}{d{x}_1},\frac{df}{d{x}_1},\dots ,\frac{df}{d{x}_n})\cdot (\frac{d{x}_1}{dt},\frac{d{x}_2}{dt},\dots ,\frac{d{x}_n}{dt})=\nabla f\cdot x_0^{\prime }(t)=0$