I am currently trying to work out the n-dimensional volume of
$$ \mathbf{S}^n_+ := \{x \in \mathbb{R}^{n+1}:||x||=1 , x_{n+1}>0\}\subset \mathbb{R}^{n+1} $$
Using the fact that $\mathbf{S}^n_+$ is an $n$-manifold, I used:
$$\psi: B^n_1 \longrightarrow \mathbf{S}^n_+ $$
$$x \mapsto (x_1,\dots,x_n,\sqrt{1-||x||^2} ) $$
as a parameterization, with $B^n_1$ being a $n$-ball with radius $1$ . And the fact that
$$Vol_n(\mathbf{S}^n_+)=\int_ {B^n_1} \sqrt{det(\psi^{’T}\psi’)}d\lambda$$
But I have problems evaluating the integral, I have proceed until this step, but don‘t know how to get further:
$$\int_ {B^n_1} \sqrt{det(\psi^{’T}\psi’)}d\lambda= \int_ {B^n_1} \sqrt{1+||grad(\psi)||^2}d\lambda=$$
$$\int_ {B^n_1} \sqrt{1+\sum_{i=1}^n \frac{x_i}{1-||x||}}d\lambda= \int_ {B^n_1} \frac{1}{\sqrt{1-||x||}}d\lambda= \dots $$
I would be glad to get some help and/or pointers.
