Let $\Omega \subset \mathbb{R}^d$ be open and with $C^1$ boundary $\Gamma$. For any given point $x_0 \in \Gamma$ we know there's a neighborhood where $\Gamma$ is the graph of some $C^1$ function $\gamma : \mathbb{R}^{d - 1} \longrightarrow \mathbb{R}^d, x' \longmapsto \gamma ( x') = x_d$. We can use it to straighten the boundary with the local diffeomorphism
$ T ( x', x_d - \gamma ( x')) = ( x', x_d - \gamma ( x')), $
and its differential $D T$ has a nice $( d - 1) \times ( d - 1)$ identity matrix as first block and a bottom row $\nabla T_d = ( - \nabla \gamma, 1)$ which is proportional to the vector $\vec{n}$ normal to $\Gamma$ at each point, say $c ( x) \vec{n} ( x) = \nabla T_d ( x)$, where $c ( x) = - \| \nabla T_d ( x) \|$.
For my calculations in concrete examples with parametrized domains, etc., I want $\nabla T_d$ to actually be the outward pointing normal: I need this $c ( x)$ to be $- 1$. If I try to impose the condition after constructing $T$, then I have to integrate expressions which I'm just not capable of. I can try to throw it at some symbolic integration software, but there has to be some other way, right? In almost every book on PDEs it's stated that this $T$ may be normalized so as to have the property I mention. But how?