Let $M$ be an ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\leq1$. I explicitly computed $$\int_{\partial M} z^2\mathrm{d}x\wedge\mathrm{d}y=0$$ Can one use Stokes' theorem here? I mean $$\int_{\partial M} z^2\mathrm{d}x\wedge\mathrm{d}y=\int_M 2z\mathrm{d}z\wedge\mathrm{d}x\wedge\mathrm{d}y$$ so, can we conclude by symmetry of the domain, that the last integral is $0$? If yes, what kind of formal verifications are needed? I am sorry if this is completely trivial, just trying to find my way around this stuff.
Integral over an ellipsoid of $z^2\mathrm{d}x\wedge\mathrm{d}y$, Stokes' theorem
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integration
differential-geometry
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1What about considering the behavior of the original integral under $(x, y, z) \to (-x, y, -z)$? – 2017-02-23
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0I see. Is it not a coincidence, that there are such transformations on integrals of both sides of Stokes' formula? – 2017-02-23