Could someone please indicate me some reference that contains the proof of the following theorem?
Below $mathcal{H}^n$ denotes the $n$-dimensional Hausdorff outer measure in $mathbb{R}^n$.
Theorem: Let $Msubset mathbb{R}^N$ be a $k$-dimensional manifold of class $C^1$, $1leq kleq N$.
- Let $varphi$ be a local chart, that is, $varphi:Ato M$ is a function of class $C^1$ for some open set $A subset mathbb{R}^k$ such that $nabla varphi $ has maximum rank $k$ in $A$. Define $g_{ij}:=frac{partial varphi }{partial y_i}!cdot !frac{partial varphi }{partial y_j}$, where $cdot$ is the inner product in $mathbb{R}^N$. Then $varphi (A)$ has Hausdorff dimension $k$ and
$$mathcal{H}^k(varphi(A))=int_A sqrt{det g_{ij}(y)}dy$$
- $M$ has Hausdorff dimension $k$ and that $mathcal{H}^k(M)$ is the standard surface measure of $M$.
I found this theorem in the file “Measure and Integration” (pg 9).
I searched for some reference that contains the proof of the above theorem but couldn’t find it.
I am posting this request here since a similar question on Mathematics Stack Exchange didn’t receive an answer!
