<Li> (A ⋅ (B × C)) D = (A ⋅ D) (B × C) + (B ⋅ D) (C × A) + (C ⋅ D) (A × B) (\ displaystyle \ left (\ mathbf (A) \ cdot \ left (\ mathbf (B) \ times \ mathbf (C) \ right) \ right) \ mathbf (D) = \ left (\ mathbf (A) \ cdot \ mathbf (D) \ right) \ left (\ mathbf (B) \ times \ mathbf (C) \ right) + \ left (\ mathbf (B) \ cdot \ mathbf (D) \ right) \ left (\ mathbf (C) \ times \ mathbf (A) \ right) + \ left (\ mathbf (C) \ cdot \ mathbf (D) \ right) \ left (\ mathbf (A) \ times \ mathbf (B) \ right)) </Li> <Li> (A × B) × (C × D) = (A ⋅ (B × D)) C − (A ⋅ (B × C)) D (\ displaystyle \ left (\ mathbf (A) \ times \ mathbf (B) \ right) \ times \ left (\ mathbf (C) \ times \ mathbf (D) \ right) = \ left (\ mathbf (A) \ cdot \ left (\ mathbf (B) \ times \ mathbf (D) \ right) \ right) \ mathbf (C) - \ left (\ mathbf (A) \ cdot \ left (\ mathbf (B) \ times \ mathbf (C) \ right) \ right) \ mathbf (D)) </Li> <Ul> <Li> ∇ (ψ + φ) = ∇ ψ + ∇ φ (\ displaystyle \ nabla (\ psi + \ phi) = \ nabla \ psi + \ nabla \ phi) </Li> <Li> ∇ (ψ φ) = φ ∇ ψ + ψ ∇ φ (\ displaystyle \ nabla (\ psi \, \ phi) = \ phi \, \ nabla \ psi + \ psi \, \ nabla \ phi) </Li> <Li> ∇ (A ⋅ B) = (A ⋅ ∇) B + (B ⋅ ∇) A + A × (∇ × B) + B × (∇ × A) (\ displaystyle \ nabla \ left (\ mathbf (A) \ cdot \ mathbf (B) \ right) = \ left (\ mathbf (A) \ cdot \ nabla \ right) \ mathbf (B) + \ left (\ mathbf (B) \ cdot \ nabla \ right) \ mathbf (A) + \ mathbf (A) \ times \ left (\ nabla \ times \ mathbf (B) \ right) + \ mathbf (B) \ times \ left (\ nabla \ times \ mathbf (A) \ right)) </Li> </Ul> <Li> ∇ (ψ + φ) = ∇ ψ + ∇ φ (\ displaystyle \ nabla (\ psi + \ phi) = \ nabla \ psi + \ nabla \ phi) </Li>

Curl of a scalar field times a vector field