SOLUTION OF QUIZ 5
MINGFENG ZHAO
April 07, 2013
1. [3 Points] Let f (x, y) = (x + y, xy). Use the Inverse Function Theorem to compute Df −1 (f (x, y))
for (x, y) ∈ {(x, y) : 0 < y < x}.
Proof. Let S = {(x, y) : 0 < y < x}. Since f (x, y) = (x + y, xy), then


 1

=

y
Df (x, y)
1 

.

x
Then for all (x, y) ∈ S, we have
det Df (x, y) = x − y > 0.
So Df (x, y) is invertible and

−1
[Df (x, y)]


=


x
x−y
1

− x−y

.

y
− x−y
1
x−y
Since Df (x, y) is invertible, by the Inverse Function Theorem, we know that f −1 exists near (x, y),
and f −1 ◦ f (x, y) = (x, y), which implies that

 1

Df −1 (f (x, y)) · Df (x, y) = 

0

0 

.

1
So we know that

Df
−1
−1
(f (x, y)) = [Df (x, y)]


=


x
x−y
y
− x−y
1

− x−y

.

1
x−y
1
2
MINGFENG ZHAO
2. [7 Points] (1) Let E be a Jordan region in Rn . Prove that
Vol(E 0 ) = Vol(E) = Vol(E).
(2) Suppose that E1 and E2 are Jordan regions in Rn . Prove that if E1 , E2 are non-overlapping, then
Vol(E1
[
E2 ) = Vol(E1 ) + Vol(E2 ).
Proof. Claim I: Vol(E 0 ) = Vol(E) = Vol(E).
It is clear that E 0 ⊂ E ⊂ E, then we know that
Vol(E 0 ) ≤ Vol(E) ≤ Vol(E).
On the other hand, we know that
E = E0
[
∂E.
Since E is a Jordan region, then
Vol(∂E) = 0.
So we know that
Vol(E)
[
=
Vol(E 0
≤
Vol(E 0 ) + Vol(∂E)
=
Vol(E 0 ) + 0
=
Vol(E 0 ).
∂E)
In summary, we have
Vol(E 0 ) = Vol(E) = Vol(E).
Claim II: Let E1 and E2 be non-overlapping, then Vol(E1
S
E2 ) = Vol(E1 ) + Vol(E2 ).
SOLUTION OF QUIZ 5
3
First, we know that
[
Vol(E1
E2 ) ≤ Vol(E1 ) + Vol(E2 ).
Since E1 and E2 are non-overlapping, that is, Vol(E1
grid G = {Ri : 1 ≤ i ≤ n} such that E1
[
n
[
E2 ⊂
T
E2 ) = 0. So for any > 0, we can find a
Ri and
i=1
Vol(E1
[
E2 ) ≥
n
X
|Ri | − ,
X
and
i=1
Rj
|Rj | < .
T
T
(E1 E2 )6=∅
So we have
X
Vol(E1 ) + Vol(E2 ) ≤
Rj
≤
T
n
X
j=1
<
n
X
X
|Rj | +
E1 6=∅
Rj
T
X
|Rj | +
Rj
T
(E1
T
|Rj |
E1 6=∅
|Rj |
E2 )
|Rj | + .
j=1
So we know that
Vol(E1
[
E2 ) ≥ Vol(E1 ) + Vol(E2 ) − 2.
By taking → 0, we get
Vol(E1
[
E2 ) ≥ Vol(E1 ) + Vol(E2 ).
Vol(E1
[
E2 ) = Vol(E1 ) + Vol(E2 ).
Therefore, we have
Department of Mathematics, University of Connecticut, 196 Auditorium Road, Unit 3009, Storrs, CT
06269-3009
E-mail address: mingfeng.zhao@uconn.edu