extDerivWithin_univ — Mathlib

English

The exterior derivative of a 0-form given by a constant map equals the corresponding 1-form formed by the derivative of the underlying function.

Русский

Экст_deriv нулевой формы, заданной константой, равен соответствующей 1-форме, полученной из производной функции.

LaTeX

$$$extDeriv (fun x \mapsto ContinuousAlternatingMap.constOfIsEmpty 𝕜 E (Fin 0) (f x)) x = .ofSubsingleton _ _ _ (0 : Fin 1) (fderiv 𝕜 f x)$$$

Lean4

@[simp]
theorem extDerivWithin_univ (ω : E → E [⋀^Fin n]→L[𝕜] F) : extDerivWithin ω univ = extDeriv ω :=
  by
  ext1 x
  rw [extDerivWithin, extDeriv, fderivWithin_univ]

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