Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • 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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Pullback of the circle angular form along a parametrized curve

Example

Let c:R→S1⊆R2 be the parametrized unit circle c(t)=(cos⁡t,sin⁡t), and let

η=−y dx+x dy.

Then

c∗η=dt.

Facts & Assumptions

Given: The curve c(t)=(cos⁡t,sin⁡t) and the 1-form η=−y dx+x dy.

[F1]

Pullback of a differential form is defined by composing with the differential (The pullback of a differential form).

Verification

technique · direct
1.1F1givenalgebra

Along the curve, x∘c=cos⁡t, y∘c=sin⁡t, so c∗x=cos⁡t,c∗y=sin⁡t,c∗dx=−sin⁡t dt,c∗dy=cos⁡t dt.

2.1step 1.1algebra

Therefore c∗η=−(sin⁡t)(−sin⁡t dt)+(cos⁡t)(cos⁡t dt)=(sin⁡2t+cos⁡2t) dt=dt.

3.1step 2.1∎

Thus the angular form pulls back to the standard parameter form on the circle.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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.

Sources