Answers to Homework 8, Math 308
(1) Calculate
F d where is the solid x2 + y 2 + z 2 25 and F =
2
2
2
(x + y + z )(xi + yj + zk).
If is the sphere of radius 5, then by divergence theorem, we have,
F d =
F n d
5
2
=0
=0
r5 sin dr d d
=
=
r=0
(x2 + y 2 +
Solutions to Homework 9, Math 308
(1) Calculate the Fourier coecients ak , bk for the periodic function, f (x) =
+ x, < x < 0, f (x) = x, 0 < x < . Calculate the sum of the
series,
1
1
1 + 2 + 2 +
3
5
1
Deduce the sum of the series (2) = k=1 k2 .
4
bk =
Solutions to Homework 12, Math 308
(1) Solve the Laplace equation 2 u(x, y) = 0 with the boundary conditions
u(x, 0) = 0 and u(0, y) = sin y in the form u(x, y) = f (x)g(y).
u(x, y) = ex sin y.
(2) Solve the Laplace equation 2 u(x, t) = 0 in Fourier serie
Solutions to Homework 10, Math 308
(1) Find the general solutions for the following dierential equations.
(a) x2 y + 3xy = 1.
C
1
y = 2x + x3 .
(b) y = cos(x + y).
tan 1 (x + y) = x + C.
2
(c) y 2y + y = 0.
y = Aex + bxex .
(d) y + y = 0.
y = Aex + Bex/2
Solutions to Homework 11, Math 308
(1) Write down series solutions for the following diferential equations.
(a) y y = f (x) where f (x) = n=0 an xn with initial condition y(0) = 0.
Let y =
bn xn be a series solution. The initial condition says that
b0 = 0
Solutions to Homework 5, Math 308, Spring 2010
(1) Calculate
(9 + 2y 2 )1 dxdy over the quadrilateral with vertices (1, 3), (3, 3), (2, 6), (6, 6).
log 3
.
6
(2) Find the volume in the rst octant bounded by the coordinate
planes and the plane x + 2y + z =
Homework 6, Math 308, due March 22nd
(1) Let A = ai + bj + ck be a vector with its tail at the origin. Assume that it
rotates around the x-axis with angular velocity . Calculate (as a vector)
the linear velocity of the head of A.
Since the rotation is aro
Solutions to Homework 3, Math 308, Spring 2010
2
2
u
u
(1) For For u = ex cos y verify that xy = yx and
We calculate all the required derivatives.
u
x
u
y
2u
x2
2u
xy
2u
yx
2u
y 2
2u
x2
+
2u
y 2
= 0.
=u
= ex sin y
x
=
x
=
y
=
y
=
u
x
u
y
u
x
u
y
=u
Answers to Homework 7, Math 308
(1) Compute the diveregence and curl of the vector eld V = x sin yi + cos yj +
xyk.
div V = 0 and curl V = xi yj x cos yk.
(2) Calculate the Laplacian
2
1
x2 +y 2 +z 2
.
This is zero.
(3) Calculate the line integral xdyydx
Solutions to Homework 4, Math 308, Spring 2010
(1) Given x2 + y 2 = 2st 10 and 2xy = s2 t2 , nd x , x , y and
s t s
y
at (x, y, s, t) = (4, 2, 5, 3).
t
We dierentiate the two equations with respect to s, t to get
four equations.
x
y
2x
+ 2y
= 2t
s
s
x
y
2
Answers to Homework 1, Math 308, Spring 2010
(1) Decide whether the limits exist and nd them when you can.
3n2 +4n+1
(a) limn 9n4 +n3 +n+1
This limit is 1. Dividing the numerator and denominator
by 3n2 , we get,
3n2 + 4n + 1
=
9n4 + n3 + n + 1
1+
1+
4
3n
Answers to Homework 2, Math 308, Spring 2010
(1) Write the following complex numbers in the form rei .
(a) i3
ei/2
(b) 3+i
2+i
1
r = 2 and tan = 7 .
2
(c) (i + 3)
r = 4 and tan = 3.
(2) Complex numbers have many of the properties of real numbers,
but it