Problems: Harmonic Functions and Averages
A function u is called harmonic if r2 u = uxx + uyy + uzz = 0. In this problem we will
see that the average value of a harmonic function over any sphere is exactly its value at the
center of the sphere.
For simpli

18.075 Practice Test II for Exam 2
Justify your answers. Cross out what is not meant to be part of your solution.
Total number of points: 60.
I. Consider the integral
x sin x
dx
(1).
I=
(x2 2 )(x2 + 1)
1. (2pts) By replacing x by the complex variable z,

18.075 Practice Test IV for Quiz 3
December 5, 2004
Justify your answers. Cross out what is not meant to be part of your nal
answer.
Total number of points: 80. Time: 80 min.
I. Find the domain of convergence of the following series:
1. (1 pts)
3n (x 2)n

Problems: Del Notation; Flux
1. Verify the divergence theorem if F = xi + yj + zk and S is the surface of the unit cube
with opposite vertices (0, 0, 0) and (1, 1, 1).
RR
RRR
Answer: To conrm that S Fn dS =
D divF dV we calculate each integral separately.

Math 18.305 Fall 2004/05
Assignment 6: Boundary Layers
Provided by Mustafa Sabri Kilic
1. Solve approximately the following two equations
(a) y 00 + (1 + x)2 y 0 + y = 0, 0 < x < 1, with y(0) = 0 and y(1) = 2.
(b) y 00 (1 + x2 )y 0 + y = 0, 0 < x < 1, wit

18.075 Practice Test 2 for Quiz 3
December 3, 2004
Justify your answers. Cross out what is not meant to be part of your solution.
Total number of points: 75.
I. (10 pts) Find the region of convergence of the Frobenius series for the Bessel function
Jp (x)

Triple Integrals
1. Find the moment of inertia of the tetrahedron shown about the z-axis. Assume the
tetrahedron has density 1.
z
1
R
1 y
x
1
Figure 1: The tetrahedron bounded by x + y + z = 1 and the coordinate planes.
Answer: To compute the moment of in

NAME:
18.075 Inclass Exam # 3
December 8, 2004
Justify your answers. Cross out what is not
meant to be part of your nal answer.
Total number of points: 67.5
I. Consider the ODE
xy xy y = 0
(1).
1.(2 pts) Classify the point x0 = 0.
2.(2 pts) Write the ODE

18.075 Practice Test 3 for Quiz 3
December 3, 2004
Justify your answers. Cross out what is not meant to be part of your solution.
Total number of points: 120. Time: 120 min.
I. (20 pts) Starting with the Frobenius series for the Bessel functions, show tha

18.02 Problem Set 12, Part II Solutions
1.
z
x
~
F = h x2 +z2 , y, x2 +z2 i = hP, Q, Ri.
~ ~
(a) rF =
=
i
@
@x
z
x2 +z 2
2
2 1 2 + 22x 2 2
j( x +z (x +z )
j
k
@
@y
@
@z
x
x2 +z 2
y
x
@
z
=
i(0 0) @x ( x2 +z2 )+ @z ( x2 +z2 )+k (0 0)
j( @
2
1
+ (x22z 2 )2

Problems: Curl in 3D
1. Let F = hx, y, zi. Calculate and interpret curlF.
Answer:
curlF = r F
i
j
@
@
=
@x
@y
x y
= h0, 0, 0i.
k
@
@z
z
We can interpret F as a velocity eld in which particles race away from (0, 0, 0) with speed
equal to their distance fro

Problems: Extended Stokes Theorem
Let F = h2xz + y, 2yz + 3x, x2 + y 2 + 5i. Use Stokes theorem to compute
C is the curve shown on the surface of the circular cylinder of radius 1.
H
C
F dr, where
Figure 1: Positively oriented curve around a cylinder.
Ans