TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Change-of-coordinate formula for tangent bases

Statement

Let $(U, x)$ and $(V, y)$ be smooth charts on $M$ with $p \in U \cap V$ . Then $\partial_{x^{i}} ∣_{p} = \sum_{j = 1}^{n} \frac{\partial (y^{j} \circ x^{- 1})}{\partial x^{i}} (x (p)) \partial_{y^{j}} ∣_{p}$ for each $i$ .

Facts & Assumptions

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

[L1]

The coordinate derivations form a basis of the tangent space (Coordinate derivations form a basis of the tangent space).

Proof

technique · direct

1.1

Both sides are derivations by [L1], so it is enough to compare their values on the coordinate germs $[y^{k}]$ , which form a separating family in the chart $y$ .

L1given

2.1

Applying the left-hand side to $[y^{k}]$ gives $\partial (y^{k} \circ x^{- 1}) / \partial x^{i}$ at $x (p)$ by definition. Applying the right-hand side to $[y^{k}]$ gives $\sum_{j} (\partial (y^{j} \circ x^{- 1}) / \partial x^{i}) (x (p)) \partial_{y^{j}} ∣_{p} ([y^{k}])$ , and only the $j = k$ term survives because $\partial_{y^{j}} ∣_{p} ([y^{k}]) = δ_{j k}$ . So the two sides agree on every coordinate germ $[y^{k}]$ .

L1step 1.1

3.1

Therefore the two derivations agree on a basis of germs and hence are equal.

step 2.1∎

Depends on

Coordinate derivations form a basis of the tangent space

Used by

A coordinate tuple is not an intrinsic tangent vector Counterexample
Tangent basis change between Cartesian and polar coordinates Example
Curve velocity coordinates depend on the chart False statement
Cotangent coordinate changes use the inverse transpose Jacobian Lemma
Tangent-bundle chart transitions are smooth with smooth inverses Lemma

Dependency tree · two levels

5 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)