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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 \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.
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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.
show/hide this revision's text 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.)

$$\int_C \frac{-y\,dx+x\,dy}{x^2+y^2}$$

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.
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