curveIntegral_symm — Mathlib

English

For t < 1/2, the integral along γab.trans γbc at t equals 2 times the integral along γab at 2t.

Русский

При t < 1/2 интеграл по γab.trans γbc в точке t равен 2× интегралу по γab в точке 2t.

LaTeX

$$$ \\operatorname{curveIntegralFun}(\\omega, γ_{ab}.trans γ_{bc}, t) = 2 \\cdot \\operatorname{curveIntegralFun}(\\omega, γ_{ab}, 2t) $$$

Lean4

@[simp]
theorem curveIntegral_symm (ω : E → E →L[𝕜] F) (γ : Path a b) : ∫ᶜ x in γ.symm, ω x = -∫ᶜ x in γ, ω x := by
  simp [curveIntegral, curveIntegralFun_symm]

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