Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01
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The area form in polar coordinates

Example

On the polar chart domain {(r,θ):r>0}, the Euclidean area form pulls back as

F∗(dx∧dy)=r dr∧dθ,

where F(r,θ)=(rcos⁡θ,rsin⁡θ).

Facts & Assumptions

Given: The polar-coordinate map F(r,θ)=(rcos⁡θ,rsin⁡θ).

[F1]

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

[L1]

Pullback preserves wedge products (Pullback of forms is smooth functorial and preserves wedges).

Verification

technique · direct
1.1

By [F1], F∗dx=cos⁡θ dr−rsin⁡θ dθ,F∗dy=sin⁡θ dr+rcos⁡θ dθ.

F1givenalgebra
2.1

Using [L1] and bilinearity of the wedge product, F∗(dx∧dy)=F∗dx∧F∗dy=(cos⁡θ dr−rsin⁡θ dθ)∧(sin⁡θ dr+rcos⁡θ dθ)=r dr∧dθ.

L1step 1.1algebra
3.1

Thus the area form becomes r dr∧dθ in polar coordinates.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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