Let $n,k$ be positive integers satisfying $k\le n$, $\varphi\colon\mathbb{R}^n\to\mathbb{R}$ be a continuous function whose support is compact and $f\colon\mathbb{R} ^k \to \mathbb{R}^n$ be an injective Lipschitz continuous map.
In the book "Rectifiable Sets, Densities and Tangent Measures" of Camillo De Lellis, he uses the following equation;
\begin{equation} \int_{f(\mathbb{R}^k)} \varphi (x) d\mathcal{H}^k (x) = \int_{\mathbb{R}^k} \varphi(f(y))Jf(y)d\mathcal{L}^k (y) \end{equation} where $\mathcal{H}^k$ and $\mathcal{L}^k$ are the $k$-dimensional Hausdorff measure and the $k$-dimensional Lebesgue measure, respectively.
How to prove this equation? (We can use the Area Formula.)