Dielectric wireless receiver

In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains.

Let $M,N$ be smooth Riemannian manifolds of respective dimensions $m\ge n$. Let $F:M⟶N$ be a smooth surjection such that the pushforward (differential) of $F$ is surjective almost everywhere. Let $\phi :M⟶[0,\mathrm{\infty }]$ a measurable function. Then, the following two equalities hold:

$\underset{x\in M}{\int }\phi (x)dM=\underset{y\in N}{\int }\underset{x\in F^{-1}(y)}{\int }\phi (x)\frac{1}{NJF(x)}d{F}^{-1}(y)dN$

$\underset{x\in M}{\int }\phi (x)NJF(x)dM=\underset{y\in N}{\int }\underset{x\in F^{-1}(y)}{\int }\phi (x)d{F}^{-1}(y)dN$

where $NJF(x)$ is the normal Jacobian of $F$, i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel.

Note that from Sard's lemma almost every point $y\in Y$ is a regular point of $F$ and hence the set $F^{-1}(y)$ is a Riemannian submanifold of $M$, so the integrals in the right-hand side of the formulas above make sense.

References

• Chavel, Isaac (2006) Riemannian Geometry. A Modern Introduction. Second Edition.

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