494 CHAPTER 18. DIFFERENTIAL FORMS
=∫
V0
∂
∂ t(f (t,h(t,x)))det(F)dmp (x)+
∫V0
f (t,h(t,x)) trace(F ′F−1)det(F)dmp (x)
=∫
V0
(∂
∂ tf (t,h(t,x))+∑
i
∂f
∂yi
∂yi
∂ t
)det(F)dmp (x)
+∫
V0
f (t,h(t,x)) trace(F ′F−1)det(F)dmp (x)
=∫
Vt
∂
∂ tf (t,y)dmp (y)+
∫Vt
∑i
∂f
∂yi
∂yi
∂ t+f (t,y) trace
(F ′F−1)dmp (y)
Now v ≡ ∂
∂ th(t,x) and also, as noted above, y ≡ h(t,x) defines y as a function of x andso trace
(F ′F−1
)= ∑α
∂vi∂xα
∂xα
∂yi. Hence the double sum ∑α,i
∂vi∂xα
∂xα
∂yiis ∂vi
∂yi= ∇y · v. The
above then gives
∫Vt
∂
∂ tf (t,y)dmp (y)+
∫Vt
(∑
i
∂f
∂yi
∂yi
∂ t+f (t,y)∇y ·v
)dmp (y)
=∫
Vt
∂
∂ tf (t,y)dmp (y)+
∫Vt
(D2f (t,y)v+f (t,y)∇y ·v)dmp (y) (18.13)
Now consider the ith component of the second integral in the above. It is∫Vt
∇y fi (t,y) ·v+ fi (t,y)∇y ·vdmp (y) =∫
Vt
∇y · ( fi (t,y)v)dmp (y)
At this point, use the divergence theorem to get this equals∫
∂Vtfi (t,y)v ·ndH p−1. There-
fore, from 18.13 and 18.12,
ddt
∫Vt
f (t,y)dmp (y) =∫
Vt
∂
∂ tf (t,y)dmp (y)+
∫∂Vt
f (t,y)v ·ndH p−1 (18.14)
this is the Reynolds transport formula.
Proposition 18.9.4 Let y= h(t,x) where h is C1 and let f be Lipschitz continuousand let Vt ≡ h(t,V0) where V0 is a bounded open set which is on one side of a Lipschitzboundary so that the divergence theorem holds for V0. Then 18.14 is obtained.
As with the divergence theorem, Some generalization should be possible to the casewhere h giving the motion is only Lipschitz by using the version of the chain rule inTheorem 17.3.5 in the above argument when needed.
18.10 Exercises
1. Let f :Rn→R be given by f (x)≡(
∑ni=1
x2i
a2i
)1/2where each ai > 0. Thus for y > 0
f−1 (y) is the boundary of an n dimensional ellipsoid. Using change of variablesformula and the coarea formula, find the area of f−1 (r).