Hausdorff dimension and surface measure

Question

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" (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!

Answer 1

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.