22.4. SOME APPLICATIONS OF THE DIVERGENCE THEOREM 419
That last term follows from observing that you have some Ukn times terms which have atleast one other factor involving some Un j. Simply expand the resulting cofactors along thebottom row. Therefore, multiplying this out gives 1+ trace(U)+o(U) . ■
With this lemma, it is easy to find Ddet(F) whenever F is invertible.
det(F +U) = det(F(I +F−1U
))= det(F)det
(I +F−1U
)= det(F)
(1+ trace
(F−1U
)+o(U)
)= det(F)+det(F) trace
(F−1U
)+o(U)
Therefore,det(F +U)−det(F) = det(F) trace
(F−1U
)+o(U)
This proves the following.
Proposition 22.4.3 Let F−1 exist. Then Ddet(F)(U) = det(F) trace(F−1U
).
From this, suppose F (t) is a p× p matrix and all entries are differentiable. Then thefollowing describes d
dt det(F)(t) .
Proposition 22.4.4 Let F (t) be a p× p matrix and all entries are differentiable. Then
ddt
det(F)(t) = det(F (t)) trace(F−1 (t)F ′ (t)
)= det(F (t)) trace
(F ′ (t)F−1 (t)
)(22.15)
The situation of interest is where x is the material coordinates and y the spacial co-ordinates and y = h(t,x) with F = F (t,x) = D2h(t,x) . I will write ∇y to indicate thegradient with respect to the y variables and F ′ to indicate ∂
∂ t F (t,x). Note that h(t,x) = yand so by the inverse function theorem, this defines x as a function of y, also as smooth ash because it is always assumed detF > 0.
Now let Vt be h(t,V0) where V0 is an open set whose boundary is sufficient for usingthe divergence theorem. Let f (y,t) be differentiable with as many derivatives as neededto make the computations valid. The idea is to simplify
ddt
∫Vt
f (t,y)dV (y)
This will involve the change of variables in which the Jacobian will be det(F) . It will notbe necessary to take the absolute value because det(F)≤ 0 is not physically possible. Then,it is fairly routine to justify the interchange of the derivative and the integral under suitableassumptions. The best would be to use the dominated convergence theorem, but formally,it is like saying the derivative of a sum is the sum of the derivatives. There is of coursethe question whether the divergence theorem will continue to hold for Vt . This will endup holding under typical assumptions normally used for assumptions that the divergencetheorem will hold for V0. For example, if h(t, ·) is smooth and the boundary of V0 isLipschitz, all will be well, but this is an application of things like Rademacher’s theoremand the area formula.
ddt
∫Vt
f (t,y)dV (y) =ddt
∫V0
f (t,h(t,x))det(F)dV (x) (22.16)