868 APPENDIX A. THE THEORY OF THE RIEMANNN INTEGRAL∗

which implies the inequality,∫XU (x) f (h(x)) |detDh(x)|dVn ≤

∫Xh(U) (z) f (z)dVn

But now you can use the same information just derived to obtain equality.

x= h−1 (z)

and so from what was just done,∫XU (x) f (h(x)) |detDh(x)|dVn

=∫

Xh−1(h(U)) (x) f (h(x)) |detDh(x)|dVn

≥∫

Xh(U) (z) f (z)∣∣detDh

(h−1 (z)

)∣∣ ∣∣detDh−1 (z)∣∣dVn

=∫

Xh(U) (z) f (z)dVn

from the chain rule. In fact,

I = Dh(h−1 (z)

)Dh−1 (z) ,

so

1 =∣∣detDh

(h−1 (z)

)∣∣ ∣∣detDh−1 (z)∣∣ . ■

The change of variables theorem follows.

Theorem A.6.15 Let U be a bounded open set with ∂U having content 0. Also let h ∈C1(U ;Rn

)be one to one on U and Dh(x)−1 exists for all x ∈U. Let f ∈C

(U). Then∫

Xh(U) (z) f (z)dz =∫

XU (x) f (h(x)) |detDh(x)|dx

Proof: You note that the formula holds for f+ ≡ | f |+ f2 and f− ≡ | f |− f

2 . Now f =f+− f− and so ∫

Xh(U) (z) f (z)dz

=∫

Xh(U) (z) f+ (z)dz−∫

Xh(U) (z) f− (z)dz

=∫

XU (x) f+ (h(x)) |detDh(x)|dx−∫

XU (x) f− (h(x)) |detDh(x)|dx

=∫

XU (x) f (h(x)) |detDh(x)|dx. ■