TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Tangent vectors to a regular level set are exactly its curve velocities

Statement

Let $f : U \subseteq R^{m} \to R^{n}$ be $C^{k}$ , $k \geq 1$ , let $c$ be a regular value, and let $a \in f^{- 1} (c)$ . A vector $v$ lies in $T_{a} (f^{- 1} (c))$ if and only if it is the velocity at zero of a $C^{1}$ curve $γ : (- ε, ε) \to f^{- 1} (c)$ with $γ (0) = a$ .

Facts & Assumptions

Given: The map, regular value, point, and vector $v \in R^{m}$ .

[L1]

The tangent space is $T_{a} (f^{- 1} (c)) = \ker D f (a)$ (The tangent space to a regular level set), and the chain rule gives $(f \circ γ)^{'} (0) = D f (a) γ^{'} (0)$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

[L2]

Locally the fibre is $a + u + g (u)$ over $K = \ker D f (a)$ , with $g (0) = 0$ and $D g (0) = 0$ (A regular level set is locally a $C^{k}$ graph of dimension $m - n$ ).

Proof

technique · direct

1.1givenL1

For the forward direction, suppose $γ$ lies in the fibre and $γ (0) = a$ . Then $f \circ γ$ is constant, so [L1] gives $D f (a) γ^{'} (0) = 0$ and hence $γ^{'} (0) \in T_{a} (f^{- 1} (c))$ .

1.2givenL1L2construct

For the reverse direction, suppose $v \in T_{a} (f^{- 1} (c)) = K$ . Using [L2], define $γ (t) = a + t v + g (t v)$ for sufficiently small $∣ t ∣$ . This curve lies in the fibre, satisfies $γ (0) = a$ , and has $γ^{'} (0) = v + D g (0) v = v$ .

2.1step 1.1step 1.2∎

The two implications are independent and exhaustive. In particular $v = 0$ is realized by the same construction, or by the constant curve.

Depends on

Used by

Dependency tree · two levels

37 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

J. M. Lee, Introduction to Smooth Manifolds, tangent-space discussion after Theorem 8.8 (standard reference, not scraped)
L. W. Tu, An Introduction to Manifolds, Section 11.2 (standard reference, not scraped)