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.



Defining surface integral on boundary of $C^1$-domain
Take any piecewise continuous function $f$ defined on the boundary $\partial \Omega$, and extend $f$ to be constant along all lines perpendicular to the boundary, at least until those lines collide with one another. If $\partial \Omega$ is compact, this extends $f$ to some $\varepsilon$-neighborhood, as long as $\partial \Omega$ is $C^1$. Then use usual Lebesgue measure on that neighborhood to integrate the extended $f$. Then divide by $2 \varepsilon$ and take the limit as $\varepsilon \to 0$. It is easy to check that this recovers the usual Lebesgue integral on $\partial \Omega$, by taking charts. In particular, this definition extends uniquely to integrable functions $f$, essentially by Fubini's theorem. However, proving that there is a limit as $\varepsilon \to 0$ probably requires charts.


