Level Set

Definition (Level Set)

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be sufficiently differentiable. The Level Set at some constant $c\in \mathbb{R}$, is defined as

$$L_f(c)=\{x\in \text{dom}(f)|f(x)=c\}$$

where $\text{dom}(f)$ is the domain of $f$.

Theorem (Level Sets are Orthogonal with Gradients)

Level sets are always orthogonal to gradients. In particular, if $r(t)$ is a curve that lies entirely within the level set $L_f(c)$ for some $c\in \mathbb{R}$, then for any point $x=r(t)$ on that curve,

$$\nabla f(x)\cdot r^{\prime }(t)=\vec{0}$$

where $r^{\prime }(t)$ is the tangent direction along that level set. One important thing to note is that $\Delta f(x)\ne \vec{0}$.