Kenneth Kuttler

462 CHAPTER 17. THE AREA FORMULA

17.5 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 one to one Lipschitz mapping and m≥ n. The coarea formulais a statement about the Hausdorff measure of a set which involves the inverse image of f .It looks a little like the method of shells in Calculus. We will let f :Rn→Rm where m≤ nin what follows.

It is possible to obtain the coarea formula as a computation involving the area formulaand some simple linear algebra and this is the approach taken here. I found this formula in[17] which has a somewhat different proof. I find this material very hard, so I hope whatfollows doesn’t have grievous errors. I have never had occasion to use this coarea formula,but I think it is obviously of enormous significance and gives a very interesting geometricassertion. I will use the form of the chain rule in Theorem 17.3.5 as needed.

To begin with, here is the linear algebra identity. Recall that for a real matrix A∗ is justthe transpose of A. Thus AA∗ and A∗A are symmetric.

Theorem 17.5.1 Let A be an m×n matrix and let B be an n×m matrix for m≤ n.Then for I an appropriate size identity matrix, det(I +AB) = det(I +BA) .

Proof: Use block multiplication to write(I +AB 0

B I

)(I A0 I

(I +AB A+ABA

B BA+ I

)(

I A0 I

)(I 0B I +BA

(I +AB A+ABA

B I +BA

)Hence (

I +AB 0B I

)(I A0 I

(I A0 I

)(I 0B I +BA

)so (

I A0 I

)−1( I +AB 0B I

)(I A0 I

(I 0B I +BA

)which shows that the two matrices(

I +AB 0B I

(I 0B I +BA

)are similar and so they have the same determinant. Thus det(I +AB) = det(I +BA). Notethat the two matrices are different sizes. ■

With these lemmas it is now possible to establish the coarea formula. First we defineΛ(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 1.9.14,

det(Df (x)Df (x)∗

)= ∑

i∈Λ(n,m)

(detDxi

f (x))2

462 CHAPTER 17. THE AREA FORMULA17.5. The Coarea FormulaThe area formula was discussed above. This formula implies that for E a measurable setHO" ($(E)) = | Few) Jalawhere f : R” — R” for f a one to one Lipschitz mapping and m > n. The coarea formulais a statement about the Hausdorff measure of a set which involves the inverse image of f.It looks a little like the method of shells in Calculus. We will let f : R’ — R” where m <nin what follows.It is possible to obtain the coarea formula as a computation involving the area formulaand some simple linear algebra and this is the approach taken here. I found this formula in[17] which has a somewhat different proof. I find this material very hard, so I hope whatfollows doesn’t have grievous errors. I have never had occasion to use this coarea formula,but I think it is obviously of enormous significance and gives a very interesting geometricassertion. I will use the form of the chain rule in Theorem 17.3.5 as needed.To begin with, here is the linear algebra identity. Recall that for a real matrix A* is justthe transpose of A. Thus AA* and A*A are symmetric.Theorem 17.5.1 Let A be an m xn matrix and let B be ann xm matrix form <n.Then for I an appropriate size identity matrix, det (I+ AB) = det (I+ BA).Proof: Use block multiplication to writeCe rl Tt)5 4)(8 Sen) (9 5)I+AB A+ABABA+IHenceSOCot) CO TC oan )which shows that the two matricesI+AB 0 I 0Ce) Ca ras )are similar and so they have the same determinant. Thus det (J + AB) = det (I+ BA). Notethat the two matrices are different sizes.With these lemmas it is now possible to establish the coarea formula. First we defineA(n,m) as all possible ordered lists of m numbers taken from {1,2,...,2}. Recall 2 € R”and f (x) € R” where m <n. Recall that this was part of the Binet Cauchy theorem,Theorem 1.9.14,det (Df (x) Df (x)")= YL (detDe, f(x)’iE A(n,m)