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Let $q \in R^3$ be some vector. Let $f: R^3 \to R$ be the function $f(x) = |x-q| ^2$. I want to solve the integral $\int_{S^2} fdS$ , where $S^2$ is the $2$-dimensional sphere.

I tried using the parametrization $S^2 = r(\theta, \alpha) = (cos(\alpha )sin(\theta), sin(\alpha )sin(\theta), cos(\theta ))$, $\theta \in (0,\pi ), \alpha \in (0, 2\pi ) $.

Then I use the formula $$\int_{M} f = \int f \circ r(\theta, \alpha) \sqrt{\Gamma (r_{\theta}, r_{\alpha})}$$

By calculation $\sqrt{ \Gamma (r_{\theta}, r_{\alpha})} = sin(\theta)$, so what i get is $$\int_0^{\pi} \int_0^{2\pi} [(cos(\alpha )sin(\theta)-q_1)^2 + (sin(\alpha )sin(\theta) -q_2)^2 + (cos(\theta )-q_3)^2]sin(\theta)d\alpha d\theta = $$

$$\int_0^{\pi} \int_0^{2\pi} [1 + q_1^2 + q_2^2 + q_3^2 -2q_1cos(\alpha )sin(\theta) -2q_2sin(\alpha )sin(\theta) -2q_3cos(\theta )]sin(\theta)d\alpha d\theta$$

Then answer to which I got is $4\pi (1+ ||q||)$, am I correct?

Help would be appreciated.

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  • $\begingroup$ For $q=0$ you get $f=1$ over the sphere and the integral is just the surface area $4\pi$, so your calculation is incorrect. $\endgroup$ – eranreches Apr 16 at 22:15
  • $\begingroup$ Then could you show me how to do it. I just simply inserted the composition. $\endgroup$ – Gabi G Apr 16 at 23:04
  • $\begingroup$ It would be better if you can elaborate more on your calculation, so we can spot the exact point where you have a problem. $\endgroup$ – eranreches Apr 16 at 23:10
  • $\begingroup$ Okay, so I edited my answer. I have also found a mistake in my calculations, so I also updated my result. $\endgroup$ – Gabi G Apr 17 at 0:19
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Note that $\left\|q\right\|^{2}=q_{1}^{2}+q_{2}^{2}+q_{3}^{2}$ so you should have this instead of $\left\|q\right\|$ (thanks to @kimchilover for the correction). As you can see now, for $q=0$ you indeed get

$$\int f{\rm dS}=\int{\rm dS}=4\pi\left(1+\left\|q\right\|^{2}\right)=4\pi$$

which is just the surface area of the unit sphere.

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  • $\begingroup$ Okay, thanks. I have indeed forgotten the square $\endgroup$ – Gabi G Apr 17 at 12:08

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