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My question is, essentially, suppose I have two simply connected subset of $\mathbb{R}^n$, if I know that the boundaries of both are very close, how can I bound the determinant of the Jacobian between ...

Suppose we are dealing with diffusion over a boundaryless manifold $M$ (for simplicity let's say it's a surface). In that case, we have \begin{align} \int_M W \Delta U \mathrm{d} x & = -\int_M \...