Topics In Analysis

35.1 The Area Formula Again

Recall the area formula presented earlier. For convenience, here it is.

Obviously, one can obtain improved versions of this important theorem by using Rademacher’s theorem and condition 35.1.3. As mentioned earlier, a function which satisfies 35.1.3 is called locally Lipschitz at x. Here is a simple lemma which is in the spirit of similar lemmas presented in the chapter on Hausdorff measures.

Proof: Let N_k be defined as

Nk ≡ {x ∈ N : for some Rx > 0,|h(z)− h(y)| ≤ k |z− y | for all y,z ∈ B (x,Rx )}

Thus N_k ↑ N. Let ε > 0 be given and let U ⊇ V _k ⊇ N be open and m_n

< . Now fix δ > 0. For x ∈ N_k let B ⊆ V _k such that r_x < min. By the Vitali covering theorem, there exists a disjoint sequence of these balls, _i=1^∞ such that _i=1^∞, the corresponding sequence of balls having the same centers but five times the radius covers N_k. Then diam < 2δ∕k. Hence _i=1^∞ covers h and diam < 2δ. It follows

Since δ was arbitrary, this shows ℋⁿ ≤ ε. Since k was arbitrary, this shows ℋⁿ = lim_k→∞ℋⁿ ≤ ε. Since ε is arbitrary, this shows ℋⁿ = 0. This proves the lemma.

Now with this lemma, here is one of many possible generalizations of the area formula.

Proof: For x ∈ A, there exists a ball, B_x on which h is Lipschitz. By Rademacher’s theorem, h is differentiable a.e. on B_x. There is a countable cover of A consisting of such balls on which h is Lipschitz. Therefore, h is differentiable on A₀ ⊆ A where m_n

= 0. Then by the earlier area formula,

∫ ∫ n h(A0)g (y )dℋ = A0 g(h (x ))J (x)dmn

By Lemma 35.1.2

This proves the theorem.

Note how a special case of this occurs when h is one to one and C¹. Of course this yields the earlier change of variables formula as a still more special case.

In addition to this, recall the divergence theorem, Theorem 27.2.14 on Page 3372. This theorem was stated for bounded open sets which have a Lipschitz boundary. This definition of Lipschitz boundary involved an assumption that certain Lipschitz mappings had a derivative a.e. Rademacher’s theorem makes this assumption redundant. Therefore, the statement of Theorem 27.2.14 remains valid with the following definition of a Lipschitz boundary.