$\int_C \frac{-y \ dx + x$\int_C \dy}{x^2+y^2}$frac{-y\,dx+x\,dy}{x^2+y^2}$$ However, this integral turns out to be too complicated to calculate. What should we do now? Are there any efficient and strong methods to quickly and calculate the winding number? Thanks. 3 formatting, latex We have a closed curve C on the plane given by parametric equations: x=x(t), y=y(t), t changes between a and b, x and y are smooth functions. We want to calculate the winding number of this curve around the origin. The most natural way to do it is to calculate the path integral:$$\int_C \int_C \frac{-y\,dx+x\,dy}{x^2+y^2}$$frac{-y \ dx + x \ dy}{x^2+y^2} However, this integral turns out to be too complicated to calculate. What should we do now? Are there any efficient and strong methods to quickly and calculate the winding number? Thanks. 2 TeXified We have a closed curve C on the plane given by parametric equations: x=x(t), y=y(t), t changes between a and b, x and y are smooth functions. We want to calculate the winding number of this curve around the origin. The most natural way to do it is to calculate the path integral: Int_C -ydx/(x^2+y^2) + xdy/(x^2+y^2) (I'm sorry, I'm having trouble with typing fomulas, but I hope it's clear enough.) $$\int_C \frac{-y\,dx+x\,dy}{x^2+y^2}$\$