CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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Polar coordinates on a full closed angular period are not injective and are singular at radius zero

Statement refuted

False claim. The polar map $(r,\theta)\mapsto(r\cos\theta,r\sin\theta)$ is injective with invertible derivative on every closed rectangle of nonnegative radii and one full angular period.

Facts & Assumptions

Given: The polar map $P$ on $[0,1]\times[0,2\pi]$ .

[L1]

Sine and cosine have common period $2\pi$ (The zero sets of sine and cosine and the least positive common period 2 pi).

[L2]

Sine and cosine satisfy $\sin^2\theta+\cos^2\theta=1$ (Parity and the Pythagorean identity for sine and cosine).

Counterexample

technique · direct

1.1

For every $r>0$ , periodicity [L1] gives $P(r,0)=P(r,2\pi)$ although the two parameter points differ. Thus the two closed seam faces already destroy injectivity.

L1given

2.1

At $r=0$ , every angle maps to the origin, providing infinitely many preimages even away from comparing the seam endpoints.

givenstep 1.1

3.1

Direct differentiation and [L2] give $\det DP(r,\theta)=r$ . Thus $\det DP(0,\theta)=0$ , so the derivative is singular along the entire zero-radius edge and the map violates both claimed hypotheses.

L2step 2.1∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 75 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.