Measure on sphere

Considering an n-sphere, which is the unit sphere in \mathbb{R}^n, we usually encounter this kind of integration

\displaystyle \int_{\mathbb{S}^n}f(x)dS_x

here dS_x is the sphere measure. For example find the value of \displaystyle \int_{\mathbb{S}^n}x^+_ndS_x.

To calculate this integral, we have to understand dS_x. What is the relation to Lebesgue measure in Euclidean space?

The key point is viewing \mathbb{S}^n as a manifold, then dS_x is the volume element of \mathbb{S}^n, which can be written as dS_x=\sqrt{g}du_1\wedge du_2\wedge \cdots \wedge du_{n-1} if u_1,u_2,\cdots,u_{n-1} is the local coordinates of \mathbb{S}^n.

\displaystyle \psi: B_1(0)\mapsto \mathbb{S}_+^n

\displaystyle (u_1,u_2,\cdots,u_{n-1})\to (u_1,u_2,\cdots,u_{n-1}, \gamma)

where B_1(0)\subset \mathbb{R}^{n-1} and \gamma=\sqrt{1-u^2_1-u^2_2-\cdots-u^2_{n-1}}.

Suppose the metric of \mathbb{R}^n is g_0, then it induces a metric \psi^*g_0 on \mathbb{S}^n.

\displaystyle \psi^*g_0\left(\frac{\partial }{\partial u_i}, \frac{\partial }{\partial u_j}\right)=\delta_{ij}+\frac{\partial \gamma}{\partial u_i}\frac{\partial \gamma}{\partial u_j}.

Since (u_1,u_2,\cdots,u_{n-1}) can serve as a local coordinates of \mathbb{S}_+^n, then

\displaystyle dS_x=\sqrt{\psi^*g_0}du_1\wedge du_2\wedge \cdots \wedge du_{n-1}=\sqrt{1+|D\gamma|^2}du_1\wedge du_2\wedge \cdots \wedge du_{n-1}.

Or we can get from the volume form

\displaystyle \omega = \sum_{j=1}^{n} (-1)^{j-1} x_j \,dx_1 \wedge \cdots \wedge dx_{j-1} \wedge dx_{j+1}\wedge \cdots \wedge dx_{n}

and substitute x_{n} by \gamma.

So

\displaystyle \int_{\mathbb{S}^n}x^+_ndS_x=\int_{B_1(0)}\gamma\sqrt{1+|D\gamma|^2}du_1\wedge du_2\wedge \cdots \wedge du_{n-1}

which can be calculated by the polar coordinates.

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