How can I find the winding number of a curve?

Question

I need to find the winding number of the closed curve $c(t)=(a \cos(t),b \sin(t))^T $, where $a,b > 0$ and $c:[0,2\pi) \to\mathbb{R}^2\setminus\{0\}$.

I don't understand how to do this.

Answer 1

Integrate $d\theta = d(tan^{-1}(y/x)) = \frac{-y \ dx+x \ dy}{x^2+y^2}$. You have $x = a \cos(t)$ and $y=b\sin(t)$ hence $dx=-a\sin(t) dt$ and $dy=b\cos(t)dt$. We find: $$ d \theta = \frac{abdt}{a^2\cos^2(t)+b^2\sin^2(t)} $$ now integrate... or perhaps there is a lazier argument using homotopy.