Theorems · Theorem · functional analysis

Orientation.volumeForm_def

∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {n : ℕ}
  [_i : Fact (Module.finrank ℝ E = n)] (o : Orientation ℝ E (Fin n)),
  o.volumeForm =
    Nat.casesAuxOn (motive := fun a => n = a → E [⋀^Fin n]→ₗ[ℝ] ℝ) n
      (fun h =>
        Eq.ndrec (motive := fun {n} =>
          [_i : Fact (Module.finrank ℝ E = n)] → Orientation ℝ E (Fin n) → E [⋀^Fin n]→ₗ[ℝ] ℝ)
          (fun [Fact (Module.finrank ℝ E = 0)] o =>
            have opos := AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1;
            ⋯.by_cases (fun x => opos) fun x => -opos)
          ⋯ o)
      (fun n_1 h =>
        Eq.ndrec (motive := fun {n} =>
          [_i : Fact (Module.finrank ℝ E = n)] → Orientation ℝ E (Fin n) → E [⋀^Fin n]→ₗ[ℝ] ℝ)
          (fun [Fact (Module.finrank ℝ E = n_1 + 1)] o => (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det) ⋯ o)
      ⋯

Defined in

Mathlib.Analysis.InnerProductSpace.Orientation

Cited by

4 results in Mathlib

Foundations

Depth 246 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceFact