
“real: chapter_12” — 2011/5/23 — 1:04 — page 35 — #35
Real-valued Functions of Two Real Variables 12-35
C
1
C
2
C
3
C
4
X
Y
R
(0,1)
(1,1)
(1,0)
(0,0)
Figure 12.2
Since the region is a square and 4x is continuous on R, the iterated
integrals and the double integral are equal. We have
R
4xdxdy=
1
0
1
0
4xdx
dy = 2
as can be easily verified.
12.8 GENERALIZATIONS
In this section, we shall deal with different kinds of differentiability
conditions for a function whose domain and ranges are in
R
2
. Since the
topology of
R
2
and the complex plane C are one and the same, we shall
use the flexibility of using points of
R
2
as points of the complex plane
C and write z = x + iy with the