 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ib/paper-4/4a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

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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