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I came across the term in Tom Apostol Calculus volume 2 (page 390) as a line integral , which seemed to be equivalent to the wikipedia definition in terms of polar coordinate.

The line Integral in Apostol text simplifies to somewhat :-

$$\tan^{-1}(\frac {Y(a')-y_0}{X(a')-x_0})-\tan^{-1}(\frac {Y(a)-y_0}{X(a)-x_0})$$ but this will be identically zero , and so will be the definition in the wikipedia article as range of $\theta$ in polar coordinate is $[0,2\pi]$.

So where am I wrong , how to compute the Winding Number , what is the actual definition of Winding Number?

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