 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ib/paper-1/3c/b/solution

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

Write

f = u + i v

and

z = x + i y

u_{x} = v_{y}

and

u_{y} = - v_{x}

\frac{\partial ( u , v )}{\partial ( x , y )} = u_{x} v_{y} - u_{y} v_{x} = u_{x}^{2} + v_{x}^{2} = ∣ u_{x} + i v_{x} ∣^{2} = ∣ f^{'} (z) ∣^{2} .

This identity also holds at points where

f^{'} (z) = 0

; there the Jacobian is zero and local angle preservation is not guaranteed.

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