Suppose $f\in C^3(\partial U)$ where $U \subset \mathbb R^3$ is an open, bounded and connected set with regular boundary. For some $s>0$ and $\epsilon>0$ sufficiently small, we define
$A_{\pm}:=\bigg \{x_0\pm t\frac{\nabla f(x_0)}{|\nabla f(x_0)|}\;,\;0\leq t \leq 2\epsilon\;,\;x_0\in\{f=s\} \bigg \}$
We're interested in calculating $|A_{+} \cup A_{-}|$. The professor mentioned really quickly in class that
$|A_{+} \cup A_{-}|=\int \limits_{\partial U} \int \limits_{-2\epsilon}^{2\epsilon} J(x_0,t)\;dt\;dx_0\;\quad(*)\quad $ where $J$ denotes the Jacobian transformation.
QUESTIONS:
- I can not understand the formula in $(*)$. Shouldn't it be instead $|A_{+} \cup A_{-}|=\int \limits_{\{f=s\}} \int \limits_{-2\epsilon}^{2\epsilon} J(x_0,t)\;dt\;dx_0\;$? But still, how do we compute such an integral?
- I know that when we compute an integral with parametrization, then a Jacobian will appear. However I don't see here which were the initial parameters in order to have a picture of what this Jacobian looks like. For instance, is it bounded? If so, does the regularity of $f$ play a role?
I am getting really puzzled when it comes to surface integrals. Maybe this is a silly question to make, but I am completely stuck. Any help is much appreciated.
Many thanks in advance!