coordinateDivergence_eq_sum_fderiv_apply_of_hasFDerivAt

Plain-English statement

- If a vector field has Frechet derivative `F'` at `x`, then the ASTIS coordinate divergence is the coordinate trace-style sum `∑ᵢ (F' eᵢ)ᵢ`. This matches the pointwise divergence summand shape used by Mathlib's box-integral divergence theorem. It is still not an integration theorem or an integration-by-parts result.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Syntax used on this page

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Lean statement

theorem coordinateDivergence_eq_sum_fderiv_apply_of_hasFDerivAt
    {ι : Type*} [Fintype ι] [DecidableEq ι]
    {F : EuclideanSpace ℝ ι → EuclideanSpace ℝ ι}
    {F' : EuclideanSpace ℝ ι →L[ℝ] EuclideanSpace ℝ ι}
    {x : EuclideanSpace ℝ ι}
    (hF : HasFDerivAt F F' x) :
    coordinateDivergence F x =
      ∑ i, F' (EuclideanSpace.single i (1 : ℝ)) i := by
  dsimp [coordinateDivergence]
  refine Finset.sum_congr rfl ?_
  intro i _hi
  let pr : EuclideanSpace ℝ ι →L[ℝ] ℝ :=
    PiLp.proj (p := 2) (𝕜 := ℝ) (fun _ : ι => ℝ) i
  have hcomp : HasFDerivAt (fun y : EuclideanSpace ℝ ι => F y i)
      (pr.comp F') x := by
    simpa [pr, Function.comp_def] using (pr.hasFDerivAt.comp x hF)
  have hline := hcomp.hasLineDerivAt (EuclideanSpace.single i (1 : ℝ))
  simpa [pr] using hline.lineDeriv

/-- If a vector field is differentiable at `x`, then the ASTIS coordinate
divergence is the Mathlib divergence-theorem summand with `fderiv ℝ F x`.

This is the pointwise bridge needed before instantiating Mathlib's integral
divergence theorem.  It does not assert integrability, face terms, boundary
decay, integration by parts, generator domains, or invariant-law consequences. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:60published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 calculus root: align ASTIS coordinate divergence with the Mathlib divergence-theorem summand shape `sum_i F' e_i i` under `HasFDerivAt F F' x`

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
`have` creates a named intermediate mathematical fact.
`refine` instantiates a reusable theorem while leaving explicit subgoals.
`simpa` closes the goal after a controlled simplification of a typed result.

Why the statement has this shape

The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.

Hidden assumptions and non-claims

Integrability is an input or proved output; a displayed integral alone does not supply it.
Totalized `fderiv` values must not be read as a differentiability theorem.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.
A formal generator display does not establish a closed operator domain.