I am reading this paper.

Let $\Gamma(t)$ be a smooth closed connected oriented hypersurface for each $t \in [0,T]$. Define the set $$S = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$

On page 5 of the document, the authors say

the identity $$\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)(1+(\mathbf w \cdot \mathbf n)^2)^{-\frac{1}{2}}\;d\sigma$$ holds

where $\mathbf n$ is the unit normal on $\Gamma(t)$ and $\mathbf w$ is a velocity field that advects $\Gamma(t)$ (these definitions found in page 2).

**QUESTION** How does one prove this identity? In fact, I thought that
$$\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)\;d\sigma$$
is supposed to be true.

The square rooted term reminds me of using a parametrisation, but not sure what or how. I wonder what it even means to write down a product integral as on the RHS...

I also posted this on MSE.