MATH 230
VECTOR CALCULUS AND ANALYSIS
SECTION 3
3
cost is minimised when the aquarium has length and width both equal to
3
2V /5, and height equal to
V
=
xy
3
V3
=
4V 2 /25
3
25V
.
4
Section 15.8
10. (3 points)
Use Lagrange multipliers to nd the maximum a
MATH 230
VECTOR CALCULUS AND ANALYSIS
SECTION 3
5
From the second equation we see that either y = 0 or = 1. If y = 0, then
the third equation gives xz = 9, so z = 9/x. Substituting this into the
rst and third equations, we get
2x = 9/x,
18/x = x,
which be
2
HW #9 SOLUTIONS
Find the volume of the solid enclosed by the surface z = 1 + ex sin y and
the planes x = 1, y = 0, y = , and z = 0.
Solution: This is the volume lying above the rectangle R = [1, 1] [0, ]
and below the given surface, which is given by th
MATH 230
VECTOR CALCULUS AND ANALYSIS
The volume of the solid is
r 2 y 2
r
r
2
r2
r 2 y 2
r
y 2 dx dy
=
[2x
r
r
=
r
r2
SECTION 3
x= r 2 y 2
y2 ]
x= r 2 y 2
4(r2 y 2 ) dy = 8
1
= 8 r2 y y3
3
16 3
= r.
3
r
0
r
0
3
dy
r2 y 2 dy
1
= 8 r3 r3
3
48. (2 points)
E
2
HW #11 SOLUTIONS
Solution:
29: f = 2 x, y , which always points away from the origin, so the
corresponding plot is III.
30: f = 2x + y, x has negative x-component for y < 2x and
positive x-component for y > 2x, so the corresponding plot is IV.
31: f = 2
2
HW #11 SOLUTIONS
Solution:
29: f = 2 x, y , which always points away from the origin, so the
corresponding plot is III.
30: f = 2x + y, x has negative x-component for y < 2x and
positive x-component for y > 2x, so the corresponding plot is IV.
31: f = 2
MATH 230
VECTOR CALCULUS AND ANALYSIS
SECTION 3
HW #12 SOLUTIONS
Section 17.3
6. (2 points)
Determine whether or not
F(x, y ) = (3x2 2y 2 ) i + (4xy + 3) j
is conservative, and if it is, nd f such that F = grad f .
Solution: Using P (x, y ) = 3x2 2y 2 and
MATH 230
VECTOR CALCULUS AND ANALYSIS
SECTION 3
3
Comparing with the original system gives g (x) = 1, hence
f (x, y ) = yex + x + K,
where K is constant. We check that
f (x, y ) = x + yex
satises grad f = F, and then compute
C
(1yex ) dx+ex dy =
C
grad f
MATH 230
VECTOR CALCULUS AND ANALYSIS
SECTION 3
HW #13 SOLUTIONS
Section 17.5
4. (2 points)
Find (a) the curl and (b) the divergence of the vector eld
F(x, y, z ) = cos xz j sin xy k.
Solution:
(a) We have P (x, y, z ) = 0, Q(x, y, z ) = cos xz , and R(x,
MATH 230
VECTOR CALCULUS AND ANALYSIS
SECTION 3
3
Solution: We have r = P, Q, R , where
P (x, y, z ) = x(x2 + y 2 + z 2 )p/2 ,
Q(x, y, z ) = y (x2 + y 2 + z 2 )p/2 ,
R(x, y, z ) = z (x2 + y 2 + z 2 )p/2 .
We see that
P
= (x2 + y 2 + z 2 )p/2 + x(2x)(p/2)(