I'm having major trouble every time I need to parametrise a surface in order to take a surface integral, I just have no idea where to even start half of the time. Is there some kind of method that can be used or is it just about thinking how we should best define the surface?
Example of one I'm trying to parametrise.
$$x^2+y^2-2x+z^2=0$$
How could I do this one, I was thinking I should complete the square in $x$ then say rearrange to get $z=f(x,y)$ then say $x=\alpha, y=\beta$ then $z=f(\alpha,\beta)$ would be a parametrisation but according to the solution there are trigonometric functions involved and angles etc. And I just don't see where they come from and the solution doesn't explain it at all.
Any help?
You have $$ x^2+y^2-2x+z^2=0\Longrightarrow (x-1)^2 + y^2+z^2=1. $$ We know that $$ \cos^2 a + \sin^2 a = 1. $$ Let's use it! Denote $$ (x-1)^2 + y^2 = \sin^2\theta,\\ z=\cos\theta $$ Rewrite first: $$ (x-1)^2 + y^2 = \sin^2\theta\Longrightarrow \frac{(x-1)^2}{\sin^2\theta}+\frac{y^2}{\sin^2\theta}=1, $$ and use previous trick again: $$ x-1=\sin\theta\cos\phi,\\ y=\sin\theta\sin\phi. $$