290 Chapter 9Functions of several variables
20.For the van der Waals equation
Find (i) , (ii) , (iii) , (iv).
Section 9.4
Find the stationary points of the following functions:
- 31 − 1 x
2
1 − 1 xy 1 − 1 y
2
1 + 12 y 22.x
3
1 + 1 y
2
1 − 13 x 1 − 14 y 1 + 12 23. 4 x
3
1 − 13 x
2
y 1 + 1 y
3
1 − 19 y
Determine the nature of of the stationary points of the functions in Exercises 21−23:
- 31 − 1 x
2
1 − 1 xy 1 − 1 y
2
1 + 12 y 25.x
3
1 + 1 y
2
1 − 13 x 1 − 14 y 1 + 12 26. 4 x
3
1 − 13 x
2
y 1 + 1 y
3
1 − 19 y
27.Find the stationary value of the functionf 1 = 12 x
2
1 + 13 y
2
1 + 16 z
2
subject to the constraint
x 1 + 1 y 1 + 1 z 1 = 11 , (i) by using the constraint to eliminate zfrom the function, (ii)by the
method of Lagrange multipliers.
28.Find the maximum value of the functionf 1 = 1 x
2
y
2
z
2
subject to the constraint
x
2
1 + 1 y
2
1 + 1 z
2
1 = 1 c
2
, (i)by using the constraint to eliminate zfrom the function, (ii)by the
method of Lagrange multipliers.
- (i)Find the stationary points of the functionf 1 = 1 (x 1 − 1 1)
2
1 + 1 (y 1 − 1 2)
2
1 + 1 (z 1 − 1 2)
2
subject to
the constraintx
2
1 + 1 y
2
1 + 1 z
2
1 = 11. (ii)Show that these lie at the shortest and longest
distances of the point (1, 2, 2) from the surface of the spherex
2
1 + 1 y
2
1 + 1 z
2
1 = 11.
- (i)Show that the problem of finding the stationary values of the function
E(x, y, z) 1 = 1 a(x
2
1 + 1 y
2
1 + 1 z
2
) 1 + 12 b(xy 1 + 1 yz)
subject to the constraintx
2
1 + 1 y
2
1 + 1 z
2
1 = 11 (a and bare constants) is equivalent to solving
the secular equations
(a 1 − 1 λ)x 1 + 1 by = 0
bx + 1 (a 1 − 1 λ)y 1 + 1 bz = 10
by + 1 (a 1 − 1 λ)z= 0
These equations have solutions for three possible values of the Lagrangian multiplier:
(ii)Find the stationary point corresponding to each value of λ(assume xis positive).
(iii)Show that the three stationary values of Eare identical to the corresponding
values of λ. (This is the Hückel problem for the allyl radical, CH
2
CHCH
2
; see also
Example 17.9).
Section 9.5
Find the total differential df:
31 .f(x,y) 1 = 1 x
2
1 + 1 y
2
32.f(x, y) 1 = 13 x
2
1 + 1 sin(x 1 − 1 y) 33.f(x,y) 1 = 1 x
3
y
2
1 + 1 ln 1 y
- 35.f(r, θ, φ) 1 = 1 r 1 sin 1 θ 1 sin 1 φ
36.Write down the total differential of the volume of a two-component system in terms of
changes in temperature T, pressure p, and amountsn
A
andn
B
of the components A and
B. Use the full notation with subscripts for constant variables.
fxyz
xyz
(),, =
++
1
222
λλ λ
12 3
=,aabab= + 22 , = −
Tnp
V
,∂
∂
Vnp
T
,∂
∂
TnV
p
,∂
∂
pnV
T
,∂
∂
p
na
V
- Vnb nRT
−− =
2
2
() 0