1040 CHAPTER 29. THE AREA FORMULA

Proof: This follows from Lemma 29.9.4 and Lemma 29.9.5. Since

y→H n−m (A∩ f−1 (y))

is measurable, ∫Rm

H n−m (A∩ f−1 (y))

dy =∫ ∗Rm

H n−m (A∩ f−1 (y))

dy.

Now let p = n, and s = n−m in Lemma 29.9.4.With these lemmas it is now possible to establish the coarea formula. First we define

Λ(n,m) as all possible ordered lists of m numbers taken from {1,2, ...,n} . Recall x ∈Rn and f(x) ∈ Rm where m ≤ n. Recall that this was part of the Binet Cauchy theorem,Theorem 30.2.1,

det(Df(x)Df(x)∗

)= ∑

i∈Λ(n,m)

(detDxi f(x)

)2

Now let ic ∈Λ(n,n−m) consist of the remaining indices taken in order where i∈Λ(n,m) .For i = (i1, · · · , im), define xi ≡ (xi1 , ...,xim) and xic to be the other components of x takenin order. Then let

fi (x)≡(

f(x)xic

)Thus there are C (n,n−m) = D(n,m) different fi.

Example 29.9.7 Say f : R4→ R2. Here are some examples for fi:f1 (x1,x2,x3,x4)f2 (x1,x2,x3,x4)

x2x4

 ,

f1 (x1,x2,x3,x4)f2 (x1,x2,x3,x4)

x1x2

 ,

f1 (x1,x2,x3,x4)f2 (x1,x2,x3,x4)

x3x4

Thus fi : Rn→ Rn. For example, if i consists of the first m of these indices, you have

Dfi (x) =(

Dxi f(x) ∗0 I

)and so

detDfi (x) = detDxi f(x) . (29.9.47)

It is the same with other i ∈ Λ(n,m), except you may have a minus sign. This will notmatter here.

Earlier with the area formula, we integrated J∗ (x) ≡ det(Df(x)∗Df(x)

)1/2. With the

coarea formula, we integrate J∗ (x)≡ det(Df(x)Df(x)∗

)1/2. This proof involves doing thisintegration and seeing what happens.

Theorem 29.9.8 Let A be a measurable set in Rn and let f : Rn→ Rm be a Lipschitz map.Then the following formula holds along with all measurability assertions needed for it tomake sense. ∫

RmH n−m (A∩ f−1 (y)

)dy =

∫A

J∗ (x)dx (29.9.48)

whereJ∗ (x)≡ det

(Df(x)Df(x)∗

)1/2.