Differential dot gradient

Proved

BirkhoffGlobalSection.differential_dot_gradient

by Mazecto · Oct 3, 2026 · Mathlib 0df444a (Lean v4.33.1)

differential-geometryreal-analysis

A phase-space function's Fréchet differential equals the coordinate pairing with its designated gradient.

All displayed regularity, tangency, normalization and supplied-data hypotheses of the exact formal statement are retained.

Preamble

import Theorems.Thm_BirkhoffGlobalSection_contact_frame_basis_bundle
open BirkhoffGlobalSection Set Filter SignType
open scoped ContDiff Topology
set_option linter.unusedVariables false


set_option autoImplicit false
set_option maxHeartbeats 800000

Formal statement

theorem BirkhoffGlobalSection.differential_dot_gradient :
∀ (F : Phase → ℝ) (y v : Phase),
    fderiv ℝ F y v = v ⬝ᵥ TangentialHessian.grad F y := by sorry

Source

Independent extraction of component 1 of the Proved theorem BirkhoffGlobalSection.contact_frame_basis_bundle. Original verified source: Solutions/Sol_BirkhoffGlobalSection_contact_frame_basis_bundle.lean. Exact original component retained.

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