Using the divergence theorem and the identity (a ⊗ b) · n = (b · n) a, we have d d t ( ∫ ∫ Ω Ω ( t ) f d V ) = ∫ ∫ Ω Ω ( t ) ∂ ∂ f ∂ ∂ t d V + ∫ ∫ ∂ ∂ Ω Ω ( t ) ( f ⊗ ⊗ v ) ⋅ ⋅ n d A = ∫ ∫ Ω Ω ( t ) ∂ ∂ f ∂ ∂ t d V + ∫ ∫ ∂ ∂ Ω Ω ( t ) ( v ⋅ ⋅ n ) f d A . Q.E.D.