Therefore, d d t ( ∫ ∫ Ω Ω ( t ) f ( x , t ) d V ) = ∫ ∫ Ω Ω ( t ) ( ∂ ∂ f ( x , t ) ∂ ∂ t + ( ∇ ∇ f ( x , t ) ) ⋅ ⋅ v ( x , t ) + f ( x , t ) ∇ ∇ ⋅ ⋅ v ( x , t ) ) d V , or, d d t ( ∫ ∫ Ω Ω ( t ) f d V ) = ∫ ∫ Ω Ω ( t ) ( ∂ ∂ f ∂ ∂ t + ∇ ∇ f ⋅ ⋅ v + f ∇ ∇ ⋅ ⋅ v ) d V .