The time derivative of J is given by: ∂ ∂ J ( X , t ) ∂ ∂ t = ∂ ∂ ∂ ∂ t ( det F ) = ( det F ) tr ⁡ ⁡ ( F − − 1 ∂ ∂ F ∂ ∂ t ) = ( det F ) tr ⁡ ⁡ ( ∂ ∂ X ∂ ∂ φ φ ∂ ∂ ∂ ∂ t ( ∂ ∂ φ φ ∂ ∂ X ) ) = ( det F ) tr ⁡ ⁡ ( ∂ ∂ X ∂ ∂ φ φ ∂ ∂ ∂ ∂ X ( ∂ ∂ φ φ ∂ ∂ t ) ) = ( det F ) tr ⁡ ⁡ ( ∂ ∂ ∂ ∂ x ( ∂ ∂ φ φ ∂ ∂ t ) ) = ( det F ) ( ∇ ∇ ⋅ ⋅ v ) = J ( X , t ) ∇ ∇ ⋅ ⋅ v ( φ φ ( X , t ) , t ) = J ( X , t ) ∇ ∇ ⋅ ⋅ v ( x , t ) .