Let J (X, t) = det F (X, t). Define f ^ ^ ( X , t ) = f ( φ φ ( X , t ) , t ) . Then the integrals in the current and the reference configurations are related by ∫ ∫ Ω Ω ( t ) f ( x , t ) d V = ∫ ∫ Ω Ω 0 f ( φ φ ( X , t ) , t ) J ( X , t ) d V 0 = ∫ ∫ Ω Ω 0 f ^ ^ ( X , t ) J ( X , t ) d V 0 .