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Winding number
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===Differential geometry=== In [[differential geometry]], parametric equations are usually assumed to be [[Differentiable function|differentiable]] (or at least piecewise differentiable). In this case, the polar coordinate ''θ'' is related to the rectangular coordinates ''x'' and ''y'' by the equation: :<math>d\theta = \frac{1}{r^2} \left( x\,dy - y\,dx \right)\quad\text{where }r^2 = x^2 + y^2.</math> Which is found by differentiating the following definition for ΞΈ: :<math> \theta(t)=\arctan\bigg(\frac{y(t)}{x(t)}\bigg)</math> By the [[fundamental theorem of calculus]], the total change in ''θ'' is equal to the [[integral]] of ''dθ''. We can therefore express the winding number of a differentiable curve as a [[line integral]]: :<math>\text{wind}(\gamma,0) = \frac{1}{2\pi} \oint_{\gamma} \,\left(\frac{x}{r^2}\,dy - \frac{y}{r^2}\,dx\right).</math> The [[one-form]] ''dθ'' (defined on the complement of the origin) is [[closed and exact differential forms|closed]] but not exact, and it generates the first [[de Rham cohomology]] group of the [[punctured plane]]. In particular, if ''ω'' is any closed differentiable one-form defined on the complement of the origin, then the integral of ''ω'' along closed loops gives a multiple of the winding number.
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