Suppose $f\in C^3(M)$ where $M \subset \mathbb R^3$ is a $2$-dimensional manifold. For some $s>0$ and $\epsilon>0$ sufficiently small, we define
$A:=\bigg \{x_0+t\frac{\nabla f(x_0)}{|\nabla f(x_0)|}\;,\;0\leq t \leq \frac{\epsilon}{2}\;,\;x_0\in\{f=s\} \bigg \}$
I want to understand what exactly $A$ contains. Intuitively, I realize $A$ as follows:
- $+\frac{\nabla f(x_0)}{|\nabla f(x_0)|}$ is the unit normal vector in $x_0 \in \{f=s\}$ pointing to the set $\{f>s \}$
- $x_0$ is some point of the level set $\{f=s \}$
Therefore, I imagine that as we "walk" through $t\frac{\nabla f(x_0)}{|\nabla f(x_0)|}$, the set $A$ includes all level sets starting from $\{f=s\}$ up to distance $\frac{\epsilon}{2}$ from it. Is this correct or am I missing something here?
I would appreciate if somebody could enlighten me here! Many thanks in advance.