I'm aware of Gauss's theorem (aka the divergence theorem) for compact subsets $K$ of $\mathbb R^d$ with "$C^1$-boundary"$^1$.

I know that there are several generalizations of this theorem, but since I'm not familiar with general differential geometry, I'd like to find a reference which particularly considers the case of the manifold boundary $\partial M$ of a "$d$-dimensional embedded $C^1$submanifold of $\mathbb R^d$ with boundary"$^2$.

Is there a nice reference which presents and proofs this case in a rigorous way?

Please note that my main background is abstract measure/probability theory.

$^1$ i.e. for each $p\in\partial K$, there is an open neighborhood $U$ of $p$ and a $\psi\in C^1(U)$ with $K\cap U=\{\psi\le0\}$ and $\psi'(x)\ne0$ for all $x\in U$. $\partial K$ is denoting the topological boundary of $K$ and it is a $(d-1)$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$.

$^2$ i.e. each point of $M$ is locally $C^1$-diffeomorphic to $\mathbb H^d:=\mathbb R^{d-1}\times[0,\infty)$ and the manifold boundary $\partial M$ is a $(d-1)$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ (without boundary).