29.9. THE COAREA FORMULA 1035

of Fubini’s theorem. Recall that if n > m and Rn = Rm×Rn−m, we may take a productmeasurable set, E ⊆ Rn, and obtain its Lebesgue measure by the formula

mn (E) =∫Rm

∫Rn−m

XE (y,x)dmn−mdmm

=∫Rm

mn−m (Ey)dmm =∫Rm

H n−m (Ey)dmm.

Let π1 and π2 be defined by π2 (y,x) = x,π1 (y,x) = y. Then Ey = π2(π−11 (y)∩E

)and

so

mn (E) =∫Rm

H n−m (π2(π−11 (y)∩E

))dmm

=∫Rm

H n−m (π−11 (y)∩E

)dmm. (29.9.45)

Thus, the notion of product measure yields a formula for the measure of a set in termsof the inverse image of one of the projection maps onto a smaller dimensional subspace.The coarea formula gives a generalization of 29.9.45 in the case where π1 is replaced byan arbitrary Lipschitz function mapping Rn to Rm. In general, we will take m < n in thispresentation. Whereas in the area formula the Lipschitz function has m≥ n.

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 formulain [47]. This is a good place to obtain a slightly different proof. This argument follows[84] which came from [47]. I find this material very hard, so I hope what follows doesn’thave grievous errors. I have never had occasion to use this coarea formula, but I think it isobviously of enormous significance and gives a very interesting geometric assertion. I willuse the form of the chain rule in Theorem 29.8.2 as needed.

To begin with we give the linear algebra identity which will be used. Recall that for areal matrix A∗ is just the transpose of A. Thus AA∗ and A∗A are symmetric.

Theorem 29.9.1 Let A be an m× n matrix and let B be an n×m matrix for m ≤ n. Thenfor 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

)