1262 CHAPTER 36. INTEGRATION ON MANIFOLDS

This establishes 36.2.7.To establish 36.2.8, let f ∈ L1 (Γ,µ) and let {(Γi,gi)} be an atlas and {ψ i} be a partition

of unity. Then f ψ i ∈ L1 (Γ,µ) and is zero off Γi. Therefore, from what was just shown,

∫Γ

f dµ =∞

∑i=1

∫Γi

f ψ idµ

=∞

∑r=1

∫gr(Γr)

ψr f (hr (u))Jr (u)du

36.3 Comparison With H n

The above gives a measure on a manifold, Γ. I will now show that the measure obtained isnothing more than H n, the n dimensional Hausdorff measure. Recall Λ(p,n) was the setof all increasing lists of n indices taken from {1,2, · · · , p}

Recall

Ji (u)≡

 ∑I∈Λ(p,n)

(∂(xi1 · · ·xin

)∂ (u1 · · ·un)

)21/2

where here the sum is taken over all possible increasing lists of n indices, I, from {1, · · · , p}and x = hiu and the functional was given as

Λ f ≡∞

∑i=1

∫giΓi

f ψ i (hi (u))Ji (u)du (36.3.10)

where the {ψ i}∞

i=1 was a partition of unity subordinate to the open sets, {Wi}∞

i=1 as de-scribed above. I will show

Ji (u) = det(Dh(u)∗Dh(u)

)1/2

and then use the area formula. The key result is really a special case of the Binet Cauchytheorem and this special case is presented in the next lemma.

Lemma 36.3.1 Let A = (ai j) be a real p×n matrix in which p≥ n. For I ∈Λ(p,n) denoteby AI the n×n matrix obtained by deleting from A all rows except for those correspondingto an element of I. Then

∑I∈Λ(p,n)

det(AI)2 = det(A∗A)

Proof: For ( j1, · · · , jn) ∈ Λ(p,n) , define θ ( jk)≡ k. Then for

{k1, · · · ,kn}= { j1, · · · , jn}

definesgn(k1, · · · ,kn)≡ sgn(θ (k1) , · · · ,θ (kn)) .