Past exam of the mathematics course of the University of Cambridge / 2005 / ib / Paper 4 / 4A / a

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- Solution a

Solution

Where both derivatives exist, the ordinary product rule gives

\frac{( γ _{1} γ _{2} ) ^{'}}{γ _{1} γ _{2}} = \frac{γ _{1}^{'}}{γ _{1}} + \frac{γ _{2}^{'}}{γ _{2}} .

Neither denominator vanishes. Integrate over a common piecewise-smooth subdivision and use the contour characterization of the winding number:

n (γ_{1} γ_{2}, 0) = n (γ_{1}, 0) + n (γ_{2}, 0) .

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