Converting into integrals over the reference configuration, we get d d t ( ∫ ∫ Ω Ω ( t ) f ( x , t ) d V ) = lim Δ Δ t → → 0 1 Δ Δ t ( ∫ ∫ Ω Ω 0 f ^ ^ ( X , t + Δ Δ t ) J ( X , t + Δ Δ t ) d V 0 − − ∫ ∫ Ω Ω 0 f ^ ^ ( X , t ) J ( X , t ) d V 0 ) .