Theorems · Theorem · real analysis

second_derivative_symmetric

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} {F : Type u_3} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
  [IsRCLikeNormedField 𝕜] {f' : E → E →L[𝕜] F} {f'' : E →L[𝕜] E →L[𝕜] F} {x : E},
  (∀ (y : E), HasFDerivAt f (f' y) y) → HasFDerivAt f' f'' x → ∀ (v w : E), (f'' v) w = (f'' w) v

If a function is differentiable, and has two derivatives at x, then the second derivative is symmetric.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Symmetric

Cited by

0 results in Mathlib

Foundations

Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceIsRCLikeNormedField