That this derivation is for a material element is implicit in the time constancy of the reference configuration: it is constant in material coordinates. The time derivative of an integral over a volume is defined as d d t ( ∫ ∫ Ω Ω ( t ) f ( x , t ) d V ) = lim Δ Δ t → → 0 1 Δ Δ t ( ∫ ∫ Ω Ω ( t + Δ Δ t ) f ( x , t + Δ Δ t ) d V − − ∫ ∫ Ω Ω ( t ) f ( x , t ) d V ) .