Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to \mathbb{R}$ is defined: $$\int_{\partial\Omega} fdS = ?$$ without the use of a transformation of coordinates in the sense that for Lipschitz domains $\Omega$, the surface integral involves a transformation of coordinates via a rotation and translation, which I believe is unnecessary for a $C^1$ domain. I want to avoid this transformation which causes problems for something I am doing.

I have not seen such a definition yet which does not involve a transformation.