398 CHAPTER 14. INTEGRATION ON MANIFOLDS

Here du signifies dmp (u) and

Ji (u)≡(det(DR−1

i (u)∗DR−1i (u)

))1/2

I need to show that the same thing is obtained if another atlas and/or partition of unityis used.

Theorem 14.2.2 The functional L is well defined in the sense that if another atlasis used, then for f ∈Cc (Ω) , the same value is obtained for L f .

Proof: Let the other atlas be{(Vj,S j)

}sj=1 where v ∈ Vj and S j has the same prop-

erties as the Ri. Then(S j ◦R−1

i

)(u) = v so R−1

i (u) = S−1j (v) and so R−1

i (u) =

S−1j

((S j ◦R−1

i

)(u))

implying DR−1i (u) = DS−1

j (v)D(S j ◦R−1

i

)(u) . Therefore,

Ji (u) =(det(DR−1

i (u)∗DR−1i (u)

))1/2

=

det

p×p︷ ︸︸ ︷

D(S j ◦R−1

i)∗(u)

(p×q)(q×p)︷ ︸︸ ︷DS−1

j (v)∗DS−1j (v)

p×p︷ ︸︸ ︷D(S j ◦R−1

i)(u)



1/2

=[det(

D(S j ◦R−1

i)∗(u))

det(D(S j ◦R−1

i)(u))]1/2

J j (v)

=∣∣det

(D(S j ◦R−1

i)(u))∣∣J j (v) (14.5)

SimilarlyJ j (v) =

∣∣∣det(

D(Ri ◦S−1

j

)(v))∣∣∣Ji (u) . (14.6)

Let L̂ go with this new atlas. Thus

L̂( f )≡s

∑j=1

∫S j(V j)

f(S−1

j (v))

η j

(S−1

j (v))

J j (v)dv (14.7)

where η j is a partition of unity associated with the sets Vj as described above. Now lettingψ i be the partition of unity for the Ui, v= S j ◦R−1

i (u) for u ∈Ri (Vj ∩Ui) .

∫S j(V j)

f(S−1

j (v))

η j

(S−1

j (v))

J j (v)dv

=r

∑i=1

∫S j(V j∩Ui)

f(S−1

j (v))

ψ i

(S−1

j (v))

η j

(S−1

j (v))

J j (v)dv

By Lemma 11.8.1, the assumptions of differentiability imply that the boundary points ofΩ are always mapped to a set of measure zero so these can be neglected if desired. NowS j (Vj ∩Ui) = S j ◦R−1

i (Ri (Vj ∩Ui)) and so using 14.6, the above expression equals

r

∑i=1

∫Ri(V j∩Ui)

f(R−1

i (u))

ψ i(R−1

i (u))

η j(R−1

i (u))·∣∣∣det

(D(Ri ◦S−1

j

)(v))∣∣∣Ji (u)

∣∣detD(S j ◦R−1

i)(u)∣∣du