292 Chapter 9Functions of several variables
Show that the following functions of position in a plane satisfy Laplace’s equation:
50.x
5
1 − 110 x
3
y
2
1 + 15 xy
4
- 52.r
n 1cos 1 nθ, n 1 = 1 1, 2, 3, =
Section 9.7
Test for exactness:
53.(4x 1 + 13 y)dx 1 + 1 (3x 1 + 18 y)dy 54.(6x 1 + 15 y 1 + 1 7)dx 1 + 1 (4x 1 + 110 y 1 + 1 8)dy
55.y 1 cos 1 x 1 dx 1 + 1 sin 1 x 1 dy
56.Given the total differentialdG 1 = 1 −S 1 dT 1 + 1 Vdp, show that.
Section 9.8
57.Evaluate the line integral on the liney 1 = 12 xfromx 1 = 10 tox 1 = 12.
58.When the path of integration is given in parametric formx 1 = 1 x(t),y 1 = 1 y(t)fromt 1 = 1 t
A
to
t 1 = 1 t
B
, the line integral can be evaluated as
Evaluate on the curve with parametric equationsx 1 = 1 t,
y 1 = 1 t
2
from A(0, 0) to B(1, 1).
59.Evaluate on the curvey 1 = 1 x
2
fromx 1 = 10 tox 1 = 12 (see Exercise 57).
60.Evaluate the line integral on the curve with parametric
equationsx 1 = 1 t
2
,y 1 = 1 tfrom A(0, 0) to B(1, 1) (see Exercise 58).
61.The total differential of entropy as a function of Tand pis (Example 9.27)
Given that, and for 1 mole of ideal gas,
show that the (reversible) heat absorbed by the ideal gas round the closed path shown in
Figure 9.10 is equal to the work done by the gas; that is, (see Example 9.25).
- (i)Show thatFdx 1 + 1 GdyforF 1 = 19 x
2
1 + 14 y
2
1 + 14 xyandG 1 = 18 xy 1 + 12 x
2
1 + 13 y
2
is an exact
differential. (ii)By choosing an appropriate path, evaluate from
(x, y) 1 = 1 (0, 0)to(1, 2). (iii)Show that the result in (ii) is consistent with the differential
as the total differential of
z(x,y) 1 = 13 x
3
1 + 14 xy
2
1 + 12 x
2
y 1 + 1 y
3
.
Z
C
[]Fdx Gdy+
IITdS pdV=
Tp
S
p
V
T
R
p
∂
∂
=−
∂
∂
=−
p
S
T
C
T
R
T
p∂
∂
==
5
2
dS
S
T
dT
S
p
dp
pT
=
∂
∂
∂
∂
Z
C
()()x y dx y x dy
22
+++ 2
Z
C
xydx ydy+
2
Z
C
()()x y dx y x dy
22
+++ 2
ZZ
C
A
B
F dx G dy F
dx
dt
G
dy
dt
dt
tt
=+
.
Z
C
xydx ydy+
2
Tp
S
p
V
T
∂
∂
=−
∂
∂
Ar
B
r
sinθ