## 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.