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For M a connected compact manifold, $T$ is in $C^{1+\nu}(M,M)$ with $\nu\in(0,1)$, i.e., $DT$ is some Hölder continuous function with Hölder exponent $\nu$. Denote by $m$ the Lebesgue measure on $M$ ...

This is not the most theoretically challenging question; rather it is more of a reference request for a simple formula (which must be known). Let $\left\{B_{\epsilon_n}(x_n)\right\}_{n=0}^N$ be a set ...

Let me motivate my question with this example. The volume integral of a ball $\int_{B(0,R)} dx$ can be written as an integral over the surface of balls, i.e. $$\int_{B(0,R)} dx = \int_0^R \int_{\...