For a problem on one of my homework sheets I have encountered the following kind of problem we did not discuss in class yet (or at least I don't think so).
Let's say we have differential form $\omega$ and a manifold $M$ given like $$M := \{(x,y,z)\in\mathbb{R}^3 |\; x^2+y^2 = 9, -1\leq z \leq 1\} $$ Then $\partial M$ consists of two circles (one at $z=1$ and one at $z = -1$). How do I then calculate $$\int_{\partial M} \omega\enspace? $$
Normally I would parameterize $\partial M$ with a mapping $\varphi: U \rightarrow \mathbb{R}^3$. Then calculate the pullback $\varphi^{*}\omega$ by substituting $\varphi$ into $\omega$. The integral is then calculated as $$\int_{\partial M}\omega = \int_{U}\varphi^{*}\omega\;. $$
But in this case $\partial M$ consists of two different manifolds (the two circles) which each would require a separate parameterization. How can a integral like this be calculated?
You can do essentially the same as in the connected case. You need two charts for the two components of the boundary and each of them becomes an integral over a circle. The most tricky bit is to get the orientation (ie the signs) correct.
Edit: For the orientations, what you want is that the orientation of the boundary plus an inward pointing normal (normal to the boundary and inward towards $M$) is the same as the orientation of $M$. In your example this means that projected to the $x,y$-plane one boundary circle is clockwise and the other one is counter-clockwise.