CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-11

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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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 $(r, θ) \mapsto (r \cos θ, r \sin θ)$ 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 $[0, 1] \times [0, 2 π]$ .

[L1]

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

[L2]

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

Counterexample

technique · direct

1.1

For every $r > 0$ , periodicity [L1] gives $P (r, 0) = P (r, 2 π)$ 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 $\det D P (r, θ) = r$ . Thus $\det D P (0, θ) = 0$ , 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

Nothing in the library uses this result yet.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.