
“real: chapter_12” — 2011/5/23 — 1:04 — page 50 — #50
12-50 Real Analysis
Theorem 12.8.22 (Stoke’s Theorem) Let be a well-behaved sur-
face whose boundary is a curve ∂.IfA(x, y, z), B(x, y, z), C(x, y, z) ∈
C
(), then
(C
y
− B
z
)dy dz + (A
z
− C
x
)dz dx + (B
x
− A
y
) dxdy
=
∂
Adx+ Bdy+ Cdz.
Theorem 12.8.23 (Gauss Divergence Theorem) Let S be a well-
behaved region in
R
3
whose boundary ∂S is a closed surface. Let
A(x, y, z), B(x, y, z), C(x, y, z) ∈ C
(S). Then
S
(A
x
+B
y
+C
z
)dx dy dz =
∂S
A dydz +B dzdx+C dxdy.
SOLVED EXERCISES
1. Examine whether the following limit exists.
lim
(x,y)→(0,0)
|y |
|x|
x
2
+ y
2
x
2
+ y
2
+
y
x
Solution: We shall show that the limit does ...