I'm looking for how to formally derive the following formula from the Wikipedia article regarding integration of an integrable function over a tubular neighborhood of a boundary of a set:
\begin{aligned} \int_{T(\partial\Omega,\mu)}g(x)dx=\int_{\partial\Omega}\int_{-\mu}^{\mu}g(u+\lambda N(u)){\rm det}(I-\lambda W_u)d\lambda dS_u. \end{aligned}
Here, $\Omega$ is a subset of $\mathbb{R}^n$, $\partial\Omega$ is the boundary of $\Omega$ and is $C^2$, $T(\partial\Omega,\mu)$ is the set of points within distance $\mu$ of $\partial\Omega$, $N(u)$ is the inward unit normal vector at $u\in\partial\Omega$, $W_u$ is the Hessian of the signed distance function of $\Omega$, and $g$ is an integrable function. $dS_u$ indicates that we are taking the surface integral.
The textbook cited by the article (Gilbarg and Trudinger (1983): Elliptic Partial Differential Equations of Second Order) states that, if $\Omega$ is bounded and $\partial\Omega$ is $C^2$, there exists $\mu>0$ such that the distance function $d$ of $\Omega$ is $C^2$ on $\{x\in\bar\Omega|d(x)<\mu\}$.
My questions are:
How do we show that there exists $\mu>0$ such that the signed distance function is $C^2$ on $T(\partial\Omega,\mu)$ from the above textbook's result? $T(\partial\Omega,\mu)$ contains regions both inside and outside of $\Omega$.
How do we derive the above formula regarding integration? I suppose we use change of variables $x=u-\lambda N(u)$ but do not know how to do it.