1034 CHAPTER 29. THE AREA FORMULA
=∫
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≡ ∂
∂ t h(t,x) and also, as noted above, y≡ h(t,x) defines y as a function of x and sotrace
(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) (29.8.43)
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.
Therefore, from 29.8.43 and 29.8.42,
ddt
∫Vt
f(t,y)dmp (y) =∫
Vt
∂
∂ tf(t,y)dmp (y)+
∫∂Vt
f(t,y)v ·ndA (29.8.44)
this is the Reynolds transport formula.
Proposition 29.8.7 Let y = h(t,x) where h is Lipschitz continuous and let f also be Lip-schitz continuous and let Vt ≡ h(t,V0) where V0 is a bounded open set which is on oneside of a Lipschitz boundary so that the divergence theorem holds for V0. Then 29.8.44 isobtained.
29.9 The Coarea FormulaThe area formula was discussed above. This formula implies that for E a measurable set
H n (f(E)) =∫
XE (x)J∗ (x)dm
where f : Rn → Rm for f a Lipschitz mapping and m ≥ n. It is a version of the change ofvariables formula for multiple integrals. The coarea formula is a statement about the Haus-dorff measure of a set which involves the inverse image of f. It is somewhat reminiscent