I am looking for a reference for a 'well-known' formula in $\mathbb{R}^d$: $$ \frac{d}{dr} \int_{\lVert x\rVert\leq r} f(x)dx= \int_{\lVert y\rVert=r} f(y)dS(y), $$ where $dS$ is the Lebesgue surface measure on $\mathbb{R}^{d-1}$. For $d=3$, it is a particular case of the Reynolds transport theorem, however, how is it called or where was it formulated for other dimensions?

Clearly, the question is not about the proof, it is more or less straightforward, but about any reference only.

By the way, what is about the 'maximum' class of functions, for which this formula does hold? $L^1_{\mathrm{loc}}(\mathbb{R}^d)$ ?


It is a consequence of the co-area formula. In particular, I studied it in the books by Giaquinta and Modica, volumes 4 and 5.


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