TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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For a differentiable scalar field, $D_{v} f (a) = ⟨ \nabla f (a), v ⟩$ and the unit direction of steepest ascent is the normalized gradient

Statement

If a scalar-valued $f : U \to R$ is totally differentiable at $a$ , then $D_{v} f (a) = ⟨ \nabla f (a), v ⟩$ for every $v$ . Among unit vectors $v$ , this is at most $∥ \nabla f (a) ∥_{2}$ ; if the gradient is nonzero, equality holds exactly in the direction $\nabla f (a) / ∥ \nabla f (a) ∥_{2}$ . If the gradient is zero, every unit direction has directional derivative zero.

Facts & Assumptions

Given: A scalar-valued totally differentiable $f$ at $a$ and a direction $v$ .

[L1]

A total derivative computes every directional derivative, and its matrix is the Jacobian (A total derivative computes every directional derivative, and its matrix is the Jacobian).

[L2]

Cauchy--Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥_{2} ∥ v ∥_{2}$ (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

Proof

technique · cases

1.1

By [L1], $D_{v} f (a)$ is the Jacobian row applied to $v$ , namely $\sum_{j} \partial_{j} f (a) v_{j} = ⟨ \nabla f (a), v ⟩$ .

L1L2

2.1

If $\nabla f (a) \neq 0$ and $∥ v ∥_{2} = 1$ , [L2] gives $D_{v} f (a) \leq ∥ \nabla f (a) ∥_{2}$ , with equality at $v = \nabla f (a) / ∥ \nabla f (a) ∥_{2}$ .

assume-case nonzerostep 1.1L2algebra

3.1

If $\nabla f (a) = 0$ , step 1.1 makes every directional derivative zero; together with step 2.1 this proves the stated alternatives.

assume-case zerostep 1.1step 2.1cases-exhaustive∎

Depends on

Used by

Dependency tree · two levels

19 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. Lebl, Basic Analysis I, §8.3 (standard reference, not scraped)

For a differentiable scalar field, Dvf(a)=⟨∇f(a),v⟩Dv​f(a)=⟨∇f(a),v⟩ and the unit direction of steepest ascent is the normalized gradient