An old exam question: Let $M$ be oriented submanifold of $\mathbb{R}^{n}$ of dimension $k$, let $\omega$ be a $k$-form on $M$.
1) Define $\displaystyle\int_{M}\omega$
2) let $M:=\{x\in\mathbb{R}^{3}\;|x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\;\;\text{and}\;x_{i}>0,\;i=1,2,3\}$. Compute:
$\displaystyle\int_{M}dx_{3}\wedge dx_{2}$
Solution 2)
In polar coordinates $\begin{cases}x_{2}=\sin\theta\sin\varphi\\x_{3}=\cos\theta\end{cases}\;$ where $\;\theta\in[0,\pi]$ and $\varphi\in[0,2\pi]$ and $M$ is the unit sphere $S^{2}$ in $\mathbb{R}^{3}$. The differentials are
$dx_{2}=\sin\theta\sin\varphi d\theta+\sin\theta\cos\varphi d\varphi$
$dx_{3}=-\sin\theta d\theta$
$dx_{3}\wedge dx_{2}=-sin^2 \theta\cos \varphi d\theta \wedge d\varphi$
hence
$\displaystyle\int_{S^{2}}dx_{3}\wedge dx_{2}=-\displaystyle\int_{S^{2}}sin^2 \theta\cos \varphi d\theta \wedge d\varphi=-\displaystyle\int_{0}^{\pi}\displaystyle\int_{0}^{2\pi}sin^2 \theta\cos \varphi d\theta d\varphi=-\displaystyle\int_{0}^{\pi}\sin^2 \theta d\theta \underbrace{\displaystyle\int_{0}^{2\pi}cos\varphi d\varphi}_{=0}=0$
Is this correct?
How do we define formally the integral in 1)?
Thank you.