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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2/4/a/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 4 / a / Solution by

Codex  0 2026-10-18

Write

f = u + i v

and

z = x + i y

. For a positively oriented, piecewise smooth simple closed contour bounding a region where

u, v

have continuous first partial derivatives, Green's theorem gives

\oint_{C} f (z) d z = \oint_{C} (u d x - v d y) + i \oint_{C} (v d x + u d y) = \iint_{S} (- v_{x} - u_{y}) d x d y + i \iint_{S} (u_{x} - v_{y}) d x d y .

The Cauchy-Riemann equations are

u_{x} = v_{y}

and

u_{y} = - v_{x}

, so both integrands vanish:

\oint_{C} f (z) d z = 0.

This proves Cauchy's integral theorem under the standard regularity assumptions for Green's theorem. More general admissible contours follow by subdivision; every enclosed region must be free of singularities.

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