Dist Curl dist Grad eq zero
Space.distCurl_distGrad_eq_zero
Plain-language statement
The curl of a grad is equal to zero.
Exact Lean statement
@[simp]
lemma distCurl_distGrad_eq_zero (f : (Space) →d[ℝ] ℝ) :
∇ᵈ ⨯ (∇ᵈ f) = 0Formal artifact
Lean source
@[simp]lemma distCurl_distGrad_eq_zero (f : (Space) →d[ℝ] ℝ) : ∇ᵈ ⨯ (∇ᵈ f) = 0 := by ext η i simp only [distCurl_coord_apply, Fin.isValue, _root_.zero_apply, PiLp.zero_apply] rw [distGrad_eq_sum_basis, distGrad_eq_sum_basis] simp only [Fin.isValue, neg_smul, Finset.sum_neg_distrib, PiLp.neg_apply, WithLp.ofLp_sum, WithLp.ofLp_smul, PiLp.ofLp_single, Finset.sum_apply, Pi.smul_apply, Pi.single_apply, smul_eq_mul, mul_ite, mul_one, mul_zero, Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte, neg_neg] rw [← map_neg, ← map_add, ← ContinuousLinearMap.map_zero f] congr ext x simp only [Fin.isValue, _root_.add_apply, _root_.neg_apply, _root_.zero_apply] rw [schwartMap_fderiv_comm] simp- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/Space/Derivatives/Curl.lean:734-749
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