Therefore, d d t ( ∫ ∫ Ω Ω ( t ) f ( x , t ) d V ) = ∫ ∫ Ω Ω 0 ( ∂ ∂ ∂ ∂ t ( f ^ ^ ( X , t ) ) J ( X , t ) + f ^ ^ ( X , t ) J ( X , t ) ∇ ∇ ⋅ ⋅ v ( x , t ) ) d V 0 = ∫ ∫ Ω Ω 0 ( ∂ ∂ ∂ ∂ t ( f ^ ^ ( X , t ) ) + f ^ ^ ( X , t ) ∇ ∇ ⋅ ⋅ v ( x , t ) ) J ( X , t ) d V 0 = ∫ ∫ Ω Ω ( t ) ( f ˙ ˙ ( x , t ) + f ( x , t ) ∇ ∇ ⋅ ⋅ v ( x , t ) ) d V . where f ˙ ˙ is the material time derivative of f. The material derivative is given by f ˙ ˙ ( x , t ) = ∂ ∂ f ( x , t ) ∂ ∂ t + ( ∇ ∇ f ( x , t ) ) ⋅ ⋅ v ( x , t ) .