LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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Cotangent coordinate changes use the inverse transpose Jacobian

Statement

If $(U, x)$ and $(V, y)$ are smooth charts on $M$ with $p \in U \cap V$ , then the coordinate change on cotangent-fiber coordinates is given by the inverse transpose Jacobian of $y \circ x^{- 1}$ at $x (p)$ .

Facts & Assumptions

Given: Smooth charts $(U, x)$ and $(V, y)$ containing $p$ .

[L1]

Tangent bases transform by the Jacobian of the coordinate change (Change-of-coordinate formula for tangent bases).

[L2]

Coordinate differentials are dual to the coordinate tangent bases (Coordinate differentials form the dual cotangent basis).

[L3]

Matrix inversion preserves $C^{k}$ regularity (Matrix inversion preserves $C^{k}$ regularity where the determinant is nonzero).

Proof

technique · direct

1.1

Let $J : = D (y \circ x^{- 1}) (x (p))$ . By [L1], the $x$ -basis of tangent vectors is obtained from the $y$ -basis by multiplication with $J$ . Dual bases therefore transform by $J^{- T}$ , so if a covector has coordinate column $ξ$ in the $x$ -basis and $η$ in the $y$ -basis, then $η = J^{- T} ξ$ .

L1L2given

2.1

The base-point dependence of $J^{- T}$ is smooth because $J$ varies smoothly with the chart change and [L3] gives smooth inversion on invertible matrices.

L3step 1.1

3.1

Hence cotangent coordinate changes use the inverse transpose Jacobian.

step 1.1step 2.1∎

Depends on

Used by

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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)