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 $M\subset \mathbb{R}^N$ be a $k$-dimensional manifold of class $C^1$, $1\leq k\leq N$.


1. Let $\varphi$ be a local chart, that is, $\varphi:A\to M$ 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


$$\mathcal{H}^k(\varphi(A))=\int\_A \sqrt{\det g\_{ij}(y)}dy$$


2. $M$ has Hausdorff dimension $k$ and that $\mathcal{H}^k(M)$ is the standard surface measure of $M$.




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I found this theorem in the file ["Measure and Integration"](http://giovannileoni.weebly.com/uploads/3/1/0/5/31054371/mi-lectures-2011-12-09-revised-signed.pdf) (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](https://math.stackexchange.com) didn't receive an answer!



Hausdorff dimension and surface measure
Section 3.3.4D in Evans and Gariepy "Measure Theory and Fine Properties of Functions". Chapter 3 of Federer's "Geometric Measure Theory" is also a canonical reference.


