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

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Coordinate derivations form a basis of the tangent space

Statement

If $(U, x)$ is a smooth chart on an $n$ -manifold $M$ with $p \in U$ , then the coordinate derivations $\partial_{1} ∣_{p}, \dots, \partial_{n} ∣_{p}$ form a basis of $T_{p} M$ .

Facts & Assumptions

Given: A smooth chart $(U, x)$ with $p \in U$ .

[L1]

Each coordinate operator $\partial_{i} ∣_{p}$ is a derivation at $p$ (Coordinate derivations are well-defined derivations).

[L2]

Every derivation annihilates constant germs (A derivation annihilates constant germs).

[L3]

Smooth functions on a Euclidean neighbourhood admit a first-order Hadamard factorization (First-order Hadamard factorization near a point).

Proof

technique · direct

1.1

By [L1], the coordinate operators belong to $T_{p} M$ . If $v \in T_{p} M$ and $f$ is represented in the chart by $\tilde{f}$ , then [L3] gives $\tilde{f} (u) - \tilde{f} (a) = \sum_{i} (u^{i} - a^{i}) g_{i} (u)$ near $a = x (p)$ ; applying $v$ to the corresponding germ and using [L2], one obtains $v ([f]) = \sum_{i} v ([x^{i}]) g_{i} (a) = \sum_{i} v ([x^{i}]) \partial_{i} ∣_{p} ([f])$ .

L2L3given

2.1

Step 1.1 shows $v = \sum_{i} v ([x^{i}]) \partial_{i} ∣_{p}$ , so the coordinate derivations span $T_{p} M$ .

step 1.1

2.2

If $\sum_{i} c_{i} \partial_{i} ∣_{p} = 0$ , apply this derivation to the coordinate germ $[x^{j}]$ ; only the $j$ th term survives, so $c_{j} = 0$ . Hence the coordinate derivations are linearly independent.

step 1.1

3.1

Therefore $\partial_{1} ∣_{p}, \dots, \partial_{n} ∣_{p}$ form a basis of $T_{p} M$ .

step 2.1step 2.2∎

Depends on

Used by

The tangent space of an n-manifold has dimension n Corollary
The induced tangent bundle chart Definition
The tangent space of Euclidean space Example
Canonical tangent and cotangent splittings for products Theorem
Change-of-coordinate formula for tangent bases Theorem
Coordinate differentials form the dual cotangent basis Theorem
Coordinate formula for the differential Theorem
Coordinate formula for the differential of a function Theorem
Curve contact classes are canonically isomorphic to derivation tangent vectors Theorem

Dependency tree · two levels

9 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)