I have some smooth function $g(x) \colon \mathbb{R}^{n}_{+} \to \mathbb{R}_+$ such that $G_{t} = \{ x \in \mathbb{R}^n_+ \mid g(x) \leqslant t \}$ is compact. I consider a function $ f(t) = \int\limits_{G_t}a(x)dx_1 \wedge ... \wedge dx_n $ I want to find its derivative.
In this article http://amath.colorado.edu/pub/wavelets/papers/BEYLKI-1984.pdf author uses the represenation of the form $dx_1 \wedge ... \wedge dx_n = dg(x) \wedge \Omega$ to reduce an integral of the form $dx$ to an iterated integral. Is it possible to do something similar here? I think the answer is f'(t) = \int\limits_{ \{ x\mid g(x)=t \}} a(x) \Omega