1046 CHAPTER 29. THE AREA FORMULA

Now let s(x) = ∑pi=1 ciXEi (x) where Ei is measurable and ci ≥ 0. Then∫

Rns(x)J∗f(x)dx =

p

∑i=1

ci

∫Ei

J∗f(x)dx =p

∑i=1

ci

∫f(Ei)

H n−m (Ei∩ f−1(y))

dy

=∫

f(Rn)

p

∑i=1

ciHn−m (Ei∩ f−1(y)

)dy =

∫f(Rn)

[∫f−1(y)

s dH n−m]

dy

=∫

f(Rn)

[∫f−1(y)

s dH n−m]

dy. (29.10.61)

Theorem 29.10.1 Let g≥ 0 be Lebesgue measurable and let

f : Rn→ Rm, n≥ m

satisfy the Coarea formula. Then∫Rn

g(x)J∗f(x)dx =∫

f(Rn)

[∫f−1(y)

g dH n−m]

dy.

Proof: Let si ↑ g where si is a simple function satisfying 29.10.61. Then let i→ ∞

and use the monotone convergence theorem to replace si with g. This proves the change ofvariables formula.

Note that this formula is a nonlinear version of Fubini’s theorem. The “n−m di-mensional surface”, f−1 (y), plays the role of Rn−m and H n−m is like n−m dimensionalLebesgue measure. The term, J∗f(x), corrects for the error occurring because of the lackof flatness of f−1 (y).

The following is an easy example of the use of the coarea formula to give a familiarrelation.

Example 29.10.2 Let f : Rn → R be given by f (x) ≡ |x| . Then J∗ (x) ends up being 1.Then by the coarea formula,∫

B(0,r)dmn =

∫ r

0H n−1 (B(0,r)∩ f−1 (y)

)dy =

∫ r

0H n−1 (∂B(0,y))dy

Then mn (B(0,r))≡ αnrn =∫ r

0 H n−1 (∂B(0,y))dy. Then differentiate both sides to obtainnαnrn−1 =H n−1 (∂B(0,r)) . In particular H 2 (∂B(0,r)) = 3 4

3 πr2 = 4πr2. Of course αnwas computed earlier. Recall from Theorem 28.4.2 on Page 1005

αn = πn/2(Γ(n/2+1))−1

Therefore, the n−1 dimensional Hausdorf measure of the boundary of the ball of radius rin Rn is nπ p/2(Γ(n/2+1))−1rn−1.

I think it is clear that you could generalize this to other more complicated situations.The above is nice because J∗ (x) = 1. This won’t be so in general when considering otherlevel surfaces.