Since Ω 0 is independent of time, we have d d t ( ∫ ∫ Ω Ω ( t ) f ( x , t ) d V ) = ∫ ∫ Ω Ω 0 ( lim Δ Δ t → → 0 f ^ ^ ( X , t + Δ Δ t ) J ( X , t + Δ Δ t ) − − f ^ ^ ( X , t ) J ( X , t ) Δ Δ t ) d V 0 = ∫ ∫ Ω Ω 0 ∂ ∂ ∂ ∂ t ( f ^ ^ ( X , t ) J ( X , t ) ) d V 0 = ∫ ∫ Ω Ω 0 ( ∂ ∂ ∂ ∂ t ( f ^ ^ ( X , t ) ) J ( X , t ) + f ^ ^ ( X , t ) ∂ ∂ ∂ ∂ t ( J ( X , t ) ) ) d V 0 .