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Let $\sigma$ be the "normalized Lebesgue" (Haar, really...) measure on the unit sphere $S=S^{n-1} \subset \mathbb R^n$. That is, $\sigma$ has support $S$, it is uniformly distributed, and $\int_S d\sigma = 1$.

I'm having a hard time figuring out how to understand and evaluate integrals $\int_S f\left(x\right) d\sigma\left(x\right)$ of non-trivial (let's say, even and non-constant) functions $f$. This is not really "the" $\mathbb{R}^n$ Lebesgue measure or a restriction thereof, so I can't easily convert it to a Riemann integral (unless there's something I haven't noticed).

As a simple example, how do I evaluate $\int_S (x_1)^2 d\sigma(x)$ when $n=2$? Please explain in a way that I can generalize.

Note: What I'm eventually really interested in is proving that the Fourier transform of $\sigma$, that is, $\hat\sigma(y) := \int_S \exp \left(2\pi i x\cdot y \right) d\sigma(x)$, which is always real (as $\sigma$ is symmetric), is negative on some sphere (in the Fourier-space) centered at zero. This is not supposed to be hard. I'm thinking it's going to look like a "sinc" function which is negative infinitely many times when you move away from the origin, but I don't know how to actually show that. Some pointers (but not a full proof please) would be welcome :)

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The short answer is: polar coordinates. More generally, to integrate over a manifold, you parametrize it, and compute the volume form in coordinates (which corresponds to a Jacobian). –  Nate Eldredge Dec 10 '12 at 20:06
    
That sounds like something I'd like to know how to do :) Do you have a reference where I can find a simple example of integrating over a manifold? –  Yoni Rozenshein Dec 11 '12 at 8:32
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It sounds like you're on the wrong track to answer your question about the Fourier transform--these integrals can't be evaluated in any explicit sense (though one can derive formulas for them involving Bessel functions). The shortest solution to your problem that I can see would be to compute the integral $$\int_{{\mathbb R}^n} e^{-\beta \pi |y|^2} \hat \sigma(y) dy$$ exactly as a function of $\beta > 0$ using Fubini, then argue that the limiting behavior as $\beta \rightarrow 0^{+}$ of this integral contradicts the assumption that $\hat \sigma(y)$ is nonnegative.

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Interesting, I'll try that! Regarding the fact that the integral can't be evaluated in an explicit sense, well, okay, but what I was thinking of doing is using the fact that the Taylor series for $exp$ converges compactly uniformly, so if I know how to compute the integral of a polynomial, I may be able to work out a good enough approximation of the Fourier transform. Sort of the way you use the Dirichlet kernel to approach the delta function. –  Yoni Rozenshein Dec 11 '12 at 8:30
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