I am getting a bit confused. In the definition of $C^1$ manifold in Renardy and Rogers, they say that $\partial\Omega$ is of class $C^1$ if every point on $\partial\Omega$ has a neighbourhood within which $\partial\Omega$ can be represented as a graph of a $C^1$ function.
There is no transformation of the coordinate system in this definiton. BUT when they define Lipschitz domain, there is an affine transformation involved of the form $\tilde X = AX + C$ where $A$ is a matrix and $C$ is a vector.
The transformation messes things up for me (because it is present in the surface integral of functions on $\partial\Omega$) and I'd prefer not to use it.
Am I right that the transformation is not needed when we have $C^1$ domain, and is only needed when all we know of the domain is that it is Lipschitz?