18.9. THE REYNOLDS TRANSPORT FORMULA 493
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 follow-
ing.
Proposition 18.9.2 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 18.9.3 Let F (t) be a p× p matrix and all entries are at least Lipschitz.Then for a.e. t
ddt
det(F)(t) = det(F (t)) trace(F−1 (t)F ′ (t)
)= det(F (t)) trace
(F ′ (t)F−1 (t)
)(18.11)
Proof: From the above,
det(F (t +h))−det(F (t))
= det(F (t)) trace(F−1 (F (t +h)−F (t))
)+
=o(h) since F ′ existso(F (t +h)−F (t))
Dividing by h and taking a limit yields 18.11. â– Let y= h(t,x) with F = F (t,x) = D2h(t,x) . I will write ∇y to indicate the gradient
with respect to the y variables and F ′ to indicate ∂
∂ t F (t,x). I will be assuming what isneeded to use the various theorems. In particular let h be differentiable and one to one in x.Note that h(t,x) = y and so by the inverse function theorem, or actually Corollary 8.10.6,this defines x as a function of y, also differentiable as h because it is always assumeddetF > 0.
Now let Vt be h(t,V0) where V0 is an open bounded set. Let V0 have a Lipschitz bound-ary so one can use the divergence theorem on V0. Thus this is concerned with smooth mo-tion of a bounded open set with Lipschitz boundary. Let (t,y)→ f (t,y) be Lipschitz. Theidea is to simplify d
dt∫
Vtf (t,y)dmp (y). This will involve the change of variables in which
the Jacobian will be det(F) which is assumed positive thus preserving the orientation ofthe normal vector for V0 and Vt . In applications of this theory, det(F)≤ 0 is not physicallypossible. Since h(t, ·) is better than Lipschitz and the boundary of V0 is Lipschitz, Vt willbe such that one can use the divergence theorem because the composition of Lipschitz func-tions is Lipschitz. See Corollary 14.3.6. Then, using the dominated convergence theoremas needed along with the area formula,
ddt
∫Vt
f (t,y)dmp (y) =ddt
∫V0
f (t,h(t,x))det(F)dmp (x) (18.12)
=∫
V0
∂
∂ tf (·,h(·,x))det(F)dmp (x)+
∫V0
f (t,h(t,x))∂
∂ t(det(F))dmp (x)