How do I calculate the integral $\int_{S^{n-1}} x_1x_2 dS$ ,where $S^{n-1}$ is the $n$-dimensional sphere. The answer can have the expression $\text{Vol} (S^{n-1})$.

I know how to calculate the same integral when we are looking at the $2$-dimensional sphere or the $3$-dimensional sphere, since I can use the parametrization of those sphere and use the formula:

$$\int_{M} f = \int f \circ r(\theta, \alpha) \sqrt{\Gamma (r_{\theta}, r_{\alpha})}$$

Where $\Gamma (r_{\theta}, r_{\alpha})$ is the Gram determinant.

How can I do this in $n$-dimensions?

Help would be appreciated

  • $\begingroup$ You could parametrize the "upper half" of $S^{n-1}$ by $\phi : B_{n-1}\to S^{n-1}$, $\phi(y) = (y,1-|y|^2)$, where $B_{n-1}$ is the $(n-1)$-dimensional ball in $\mathbb R^{n-1}$. $\endgroup$ – amsmath Apr 16 at 17:55
  • $\begingroup$ I am not sure how that helps me $\endgroup$ – Gabi G Apr 16 at 20:53
  • $\begingroup$ The area element at $(x_1, x_2, \ldots, x_n)$ and $(-x_1, x_2, \ldots, x_n)$ is the same, and the sphere is symmetric wrt the hyperplane $x_1 = 0$. $\endgroup$ – Maxim Apr 18 at 10:05
  • $\begingroup$ Yes, this is how I eventually solved it $\endgroup$ – Gabi G Apr 18 at 10:43

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