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Results tagged with integration
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user 91442
Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...
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Fubini's theorem on arbitrary foliations
It must be, because it is so natural to desire an integration over the original foliations.
This is a paraphrase of the question asked previously here. … The correcting factor is can be memorized as
$$ \frac{\text{ Jacobian along fibers} \times \text {Jacobian along base}}{\text{full Jacobian}} \ .$$
Corollary: Integration in spherical coordinates. …
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A Curved/Warped Version of Fubini's Theorem
I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$.
Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed …
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A Curved/Warped Version of Fubini's Theorem
I have the answer here: Fubini's Theorem on Arbitrary Foliations
$$\int_U f = \int_{U_{\eta_0}} \left(\int_{U_\xi} f(\xi,\eta) \frac{|\det DG_{U_\xi} (\xi,\eta)| \cdot |\det DG_{U_{\eta_0}} (\xi,\eta …
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