phy201_in_class_problems
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School
Arizona State University *
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Course
201
Subject
Mathematics
Date
Feb 20, 2024
Type
Pages
38
Uploaded by KidValor13382
PHY 201 Course Supplement
Carl Covatto
Arizona State University
ii
Preface
This document contains all of the in-class problems for Spring 2024.
iii
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iv
PREFACE
Contents
Preface
iii
Agenda #1
1
Agenda #2
3
Agenda #3
4
Agenda #4
5
Agenda #5
6
Agenda #6
7
Agenda #7
8
Agenda #8
9
Agenda #9
10
Agenda #10
12
Agenda #11
13
Agenda #12
14
Agenda #13
16
Agenda #14
17
Agenda #15
18
Agenda #16
19
Agenda #17
20
v
vi
CONTENTS
Agenda #18
21
Agenda #19
22
Agenda #20
23
Agenda #21
24
Agenda #22
25
Agenda #23
26
Agenda #24
27
Agenda #25
28
Agenda #26
29
Agenda #27
30
Agenda #28
31
Agenda #29
32
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PHY 201
Agenda #1
1.
Reading
Tutorial Complex Arithmetic (CA) sections 1-5.2 due in Perusall.
2.
Tutorial Assignment
Work through Exercises 1–19 and prepare Exercises 3, 6, 10, 12, 13, 15, 17, and 18 to
submit for grading in Canvas.
3.
In-Class Exercises
The elementary physics problems in this exercise provide challenges to your abilities to
compute,
by hand
, relatively accurate quantities that are difficult to extract by ordinary
techniques. The goal is to develop skills to enable one to make such calculations quickly,
using differentials, simple Maclauren series (Taylor series about
x
= 0) expansions, or
algebraic tricks.
Instructions:
Each of the problems in this exercise is to be solved to at least three
significant figure accuracy
without the use of a calculator or computer, at least until you
identify the small parameter in the exact result and carry out an appropriate expansion.
Each study team should begin with the problem assigned it and then, when finished,
move on to another problem.
Everyone should ultimately be able to do problems a-
c.
Problems d and e are for those who have studied special relativity.
Problem f is
challening. Try it, if you have time. We will discuss the solutions during the next class
period.
(a) Estimate the percentage variation in the acceleration due to gravity between the
surface of the Earth and an altitude of 5 miles.
(b) The bathroom floor in a student’s apartment is 1
◦
off level. If the bathroom scales
are perfectly calibrated and show a weight of 130 lb, how far off is this from the
student’s true weight?
(c) A 5-
µ
F capacitor is charged to 10 V and then discharged through a 200 Ω resistor.
How long does it take for the current through the resistor to decrease to 99.9% of
its maximum value?
(d) It took the Apollo astronauts about 66 hours to get from the Earth to the Moon
and about the same to get back. By how much were their watches lagging ours at
the time they returned?
(e) A proton has total energy
E
= 10
12
MeV. How close is its velocity to that of light?
(f) In an accident report, the investigating officer wrote that the victim, a pedestrian,
was struck by a truck while in a crosswalk. The victim, whose weight was 150 lb,
was propelled in the direction of the truck’s forward motion a distance of 130 ft.
1
Witnesses said his body rose almost to the height of a flag which was hung 25 ft
above the ground. The 3-ton truck was carrying in addition a load of 2 tons. There
were no skid marks. Estimate the truck’s speed (This requires the use of more than
one conservation law).
2
PHY 201
Agenda #2
1.
Reading
Tutorial Complex Arithmetic (CA) sections 5.3–6.3 due in Perusall.
2.
Tutorial Assignment
Work through Exercises 20–35 and prepare Exercises 20, 21, 24, 25, 28, 30, 31, 32, 34,
and 35 to submit for grading in Canvas.
3.
In-Class Exercises
Perform, as teams, the following computations, reducing the result to standard
a
+
ib
form for complex numbers:
1
.
(1
−
i
) + (3 + 4
i
)
2
.
(2 + 3
i
)
−
(5
−
7
i
)
3
.
(5
−
4
i
) + (5 + 4
i
)
4
. i
×
(2
−
2
i
)
5
.
(3 + 4
i
)
×
(4
−
3
i
)
∗
6
.
(1 +
i
)
3
7
.
|
1 + 2
i
|
8
.
|
(1
−
i
)
×
(1 +
i
)
|
9
.
|
i
×
(2
−
3
i
)
|
10
.
(1
−
i
)
−
1
11
.
(
i
3
)
−
1
12
.
(2 + 3
i
)
−
1
13
.
1 +
i
1
−
i
14
.
2 +
i
i
−
1
15
.
4 + 5
i
5
−
3
i
16
.
2 +
i
3 + 2
i
+
i
+ 4
5
i
∗
17
.
i
1 +
i
−
1 +
i
i
18
.
3 + 5
i
5 + 3
i
+ 4
4.
Computer Activity (40 minutes)
You will be given access to Mathematica and a brief introduction to using it to do com-
plex arithmetic. Use Mathematica to check your answers to today’s in-class exercises.
3
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PHY 201
Agenda #3
1.
Reading
Tutorial First Order Differential Equations (FODE) sections 1–2.2.2 due in Perusall.
2.
Tutorial Assignment
Work through Exercises 1–11 and prepare Exercises 3, 6(a, c, e, g), and 11(b, c, d) to
submit for grading in Canvas.
3.
In-Class Exercises
(a) Perform the following calculations using the Euler (polar) form for the complex
numbers involved.
1
.
(1 +
i
) (1
−
i
)
2
.
1 +
i
1 +
i
√
3
3
. i
(3 + 4
i
)
4
.
1
−
3
i
i
5
.
1 +
i
√
2
·
(
1 +
i
√
3
)
6
.
(1 +
i
)
5
(b) Express the following quantities in Cartesian (
a
+
ib
) form and Euler
(
re
iθ
)
form.
Indicate where there are multiple values.
i. exp
(
1 +
i
√
3
)
ii. cos (1 +
i
)
iii. sin (1 +
i
)
iv. ln(2
i
)
v. ln (1
−
iπ
)
vi. cosh (1 +
iπ
)
(c) Compute the following roots and rational powers. Be sure to find all the solutions
and plot them on an Argand plane.
1
. i
1
/
2
2
.
(
−
i
)
1
/
2
3
.
2
1
/
3
4
.
2
2
/
3
5
.
(1 +
i
√
3)
3
/
5
6
.
(1
−
i
)
−
2
/
3
4
PHY 201
Agenda #4
1.
Reading
Tutorial First Order Differential Equations (FODE) section 2.2.3 due in Perusall.
2.
Tutorial Assignment
Work through Exercises 12–15 and prepare Exercises 13 and 15(a, c) to submit for
grading in Canvas.
3.
In-Class Exercises
(a) Find the general solutions to the following inhomogeneous first-order linear differ-
ential equations using the particular solution method:
i.
y
′
+ 3
y
=
e
2
x
ii.
y
′
+ 3
y
=
e
3
x
iii.
y
′
+ 3
y
=
e
−
3
x
iv.
y
′
+ 3
y
= 4
v.
y
′
−
y
= 2
x
2
vi.
y
′
−
y
= cos
x
vii.
y
′
+
y
= sinh
x
viii.
y
′
+
y
=
xe
2
x
(b) Use a math package to check your answers to today’s agenda problems.
(c) Complete the complex arithmetic exercises from Agenda #3.
5
PHY 201
Agenda #5
1.
Reading
Tutorial First Order Differential Equations (FODE) section 2.3 due in Perusall.
2.
Tutorial Assignment
Work through Exercises 16–24 and prepare Exercises 19, 20, 21, 22, and 24(a, b) to
submit for grading in Canvas.
3.
In-Class Exercises
(a) Find the general solutions to the following inhomogeneous first-order linear differ-
ential equations by using the integrating factor method. Compare the relative ease
with the solution to the same problems on the previous agenda by the particular
solution method.
i.
y
′
+ 3
y
=
e
2
x
ii.
y
′
+ 3
y
=
e
3
x
iii.
y
′
+ 3
y
=
e
−
3
x
iv.
y
′
+ 3
y
= 4
v.
y
′
−
y
= 2
x
2
vi.
y
′
−
y
= cos
x
vii.
y
′
+
y
= sinh
x
viii.
y
′
+
y
=
xe
2
x
(b) Use a math package to check your answers to today’s agenda problems, if you have
not already done so.
6
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PHY 201
Agenda #6
1.
Reading
Tutorial First Order Differential Equations (FODE) sections 2.4 – 4 due in Perusall.
2.
Tutorial Assignment
Work through Exercises 25–28 and prepare Exercises 25(a, b, c) and 28(a, b, c, d) to
submit for grading in Canvas.
3.
In-Class Exercises
(a) Test each of the following equations to show that they are scale invariant. Find
their general solutions (It is not necessary to do the antiderivatives. Note that not
all these equations are scale invariant in the simplest sense, i.e.,
n
= 1).
i.
xy
′
+
y
2
/x
=
xe
y/x
ii. (
x
2
+
y
2
)
dy
+ (
y
2
−
x
2
)
dx
= 0
iii. (
x
+
y
2
)
dy
+
y dx
= 0
iv. (
y
+
x
2
)
dy
−
x
3
dx
=
xy
sin (
x
2
/y
)
dx
(b) Build an Euler engine for solving first order differential equations numerically in
Excel or Maple (see FODE section 3).
There will be a minilecture on this next
class.
7
PHY 201
Agenda #7
1.
Reading
Tutorial Second Order Differential Equations (SOLDE) section 1 due in Perusall.
2.
Tutorial Assignment
Work through Exercises 1–11 and prepare Exercises 2, 4, 6, 7(a, c, e), 8, 10, and 11(
b) to submit for grading in Canvas.
3.
In-Class Exercises
Some of the following differential equations are exact; others are not. Solve those that
are, and attempt to solve those that are not by finding an appropriate integrating factor.
(a) (3
x
2
+
y
2
)
dx
+ 2
xy dy
= 0.
(b) (
ye
x
−
sin
x
)
dx
−
(
y
2
−
e
x
)
dy
= 0.
(c)
y
(1 +
xy
)
dx
+ (2
y
−
x
)
dy
= 0.
(d) (
e
x
+
y/x
)
dx
+ (ln
x
+ 1
/y
)
dy
= 0
, x >
0.
(e) (
x
2
−
y
2
)
dy
−
2
xy dx
= 0
.
8
PHY 201
Agenda #8
1.
Reading
Tutorial Second Order Differential Equations (SOLDE) section 2 due in Perusall.
2.
Tutorial Assignment
Work through Exercises 12–22 and prepare Exercises 15(a, d, e), 17, 19, 21, and 22(a,
c, d) to submit for grading in Canvas.
3.
In-class Exercises
(a) In each of the following, one solution to the SOLDE is given. Show that it is indeed
a solution and find a second linearly independent solution using Eqn. 9 of Tutorial
SOLDE.
i.
y
′′
−
3
x
y
′
+
4
x
2
y
= 0;
y
1
(
x
) =
x
2
for
x >
0
.
ii.
x
2
y
′′
−
xy
′
−
8
y
= 0;
y
1
(
x
) =
x
4
for
x >
0
.
(b) The following ISOLDE are given along with one solution of the associated homo-
geneous SOLDE. Find a second linearly independent solution and then use the
variation of parameters method to find the general solution of the ISOLDE.
i.
y
′′
−
4
y
′
+ 3
y
=
e
−
x
;
y
1
=
e
x
.
ii.
y
′′
+
y
= tan
x
;
y
1
= sin
x.
iii.
x
2
y
′′
−
xy
′
+
y
= 1
/x
;
y
1
=
x.
iv.
xy
′′
+
y
′
= 1 +
x
;
y
1
= 1
.
(c) Use a math package to check your answers to today’s agenda problems.
9
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PHY 201
Agenda #9
1.
Reading
Tutorial Second Order Differential Equations (SOLDE) section 3 due in Perusall.
2.
Tutorial Assignment
Work through Exercises 23–32 and prepare Exercises 28, 30, 31, and 32 to submit for
grading in Canvas.
3.
In-Class Exercises
(a) Find the general solution and apply the conditions as given.
i. ¨
y
−
6 ˙
y
+ 9
y
= 0;
y
(
−
1) = 1
,
˙
y
(
−
1) = 7
ii. ¨
y
−
2 ˙
y
−
5
y
= 0;
y
(0) = 0
,
˙
y
(0) = 3
iii.
y
′′
+ 3
y
′
−
2
y
= 0;
y
(0) = 2
, y
′
(0) =
−
3
iv. ¨
y
+ 9
y
= 0;
y
(0) = 1
, y
(1) = 0
v.
y
′′
−
9
y
= 0;
y
(0) = 0
, y
(1) = 2
vi.
y
′′
+ 2
y
′
−
3
y
= 0;
y
(0) = 0
, y
′
(0) =
−
2
vii. ¨
y
−
˙
y
−
6
y
= 0;
y
(1) = 4
,
˙
y
(1) = 7
viii.
y
′′
+ 5
y
′
= 0;
y
(0) = 1
, y
′
(0) = 4
(b) Find the general solution (homogeneous solution + particular solution) and apply
the initial or boundary conditions when given.
i.
y
′′
+ 4
y
′
+ 3
y
= 4
e
x
;
y
(0) = 0,
y
′
(0) = 2
ii.
y
′′
−
4
y
′
+ 4
y
=
e
2
x
;
y
(0) = 0,
y
′
(0) =
1
2
iii.
y
′′
+ 5
y
′
+ 6
y
=
e
−
3
x
;
y
(0) =
−
1
2
,
y
′
(0) = 0
iv.
y
′′
+ 2
y
′
+ 5
y
= 10 cos
x
;
y
(0) = 5,
y
′
(0) = 6
v.
y
′′
−
6
y
′
+ 9
y
= cosh 3
x
;
y
(0) = 1,
y
′
(0) = 0
vi.
y
′′
+ 9
y
= 5 sin 3
x
;
y
(0) = 0,
y
′
(
π/
3) = 0
vii.
y
′′
−
3
y
′
−
4
y
=
e
−
x
;
y
(0) = 1,
y
′
(0) = 0
viii.
y
′′
−
3
y
′
−
4
y
= sin
x
;
y
(0) = 0,
y
′
(0) = 0
(c) Discuss how you would solve
y
′′
−
3
y
′
−
4
y
=
e
−
x
+ sin
x
. (Hint: look at the last
two questions of part (b) above.
10
(d) Discuss how you would solve
y
′′′
+ 3
y
′′
+ 7
y
′
+ 5
y
= 16
e
−
x
cos 2
x
;
y
′′
(0) =
−
2,
y
′
(0) =
−
4,
y
(0) = 2
(e) Use a math package to check your answers to today’s agenda problems.
11
PHY 201
Agenda #10
1.
Reading
Tutorial Trigg’s Equation (TF), Section 1 due in Perusall.
2.
Tutorial Assignment
Work Exercises 1–11 and prepare Exercises 1, 3, 4, 5, 6, 7, and 8 to submit for grading
in Canvas.
3.
In-Class Exercises
(a) Build an Euler engine for solving second-order differential equations numerically in
Excel (or Python). Use it to solve all of the following:
i.
y
′′
+ 4
y
′
+ 3
y
= 4
e
x
;
y
(0) = 0,
y
′
(0) = 2
, x
∈
[0
,
5]
ii. 2
y
′′
−
3
xy
′
+ 4
x
2
y
=
e
2
x
;
y
(0) = 0,
y
′
(0) =
1
2
, x
∈
[0
,
4]
iii.
y
′′
+ 5
y
′
+ 6
y
=
e
−
3
x
;
y
(0) =
−
1
2
,
y
′
(0) = 0
, x
∈
[0
,
2]
iv.
y
′′
+ 2
e
−
x
y
′
+ 5
e
−
2
x
y
= 10 cos
x
;
y
(0) = 5,
y
′
(0) = 2
, x
∈
[0
,
10]
v.
y
′′
+ 9
y
= cosh 3
x
;
y
(0) = 1,
y
′
(0) = 0
, x
∈
[0
,
2]
(b) Solve the above differential equations numerically using a math package. Which of
the above problems have closed form solutions?
12
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PHY 201
Agenda #11
1.
Reading
Tutorial Trigg’s Equation (TF), Sections 2–3 due in Perusall.
2.
Tutorial Assignment
Work Exercises 12–18 and prepare Exercises 12, 13, 14, and 17 to submit for grading
in Canvas.
3.
In-Class Exercises
(a) Use your Euler engine for SOLDE [use
d
(cos
x
)
/dx
=
−
sin
x, d
(sin
x
)
/dx
= cos
x
to obtain initial conditions]. Try different values of ∆
x
. Determine values of sine
and cosine functions for several values of the arguments between 0 and 2
π
(say,
π/
10,
π/
6,
π/
4,
π/
3, 2
π/
3, 4
π/
3, etc.). Assume that cos (0) = 1 and sin (0) = 0 (as
can be seen from their series expansions as defined in TF Eqs. 9 and 10). Compare
the results and estimate the accuracy of this method with values obtained from a
calculator or math package.
(b) The series as given in TF Eqs. 9 and 10 (Program the series into a Math Package;
this is Sum[] in Mathematica). How many terms do you need to include in order
to obtain the accuracy that the Math Package gives as a calculator? How much
longer does it take to obtain that accuracy? How do the answers to these questions
vary with
x
?
(c) Use a plotting program to plot products of sine and cosine in the forms sin
nθ
·
sin
mθ
, cos
nθ
·
cos
mθ
, and sin
nθ
·
cos
mθ
where
m
and
n
are integers over the
range 0
≤
θ
≤
2
π
. Note the area under the curve being positive, negative, or zero.
If you are using a math package, integrate the plotted expressions to determine the
area under the curve. What is the area under the curve when
m
=
n
and when
m
̸
=
n
? What pattern do you notice?
(d) Now for an analytic exercise. Start with the power expansions [TF Eqs. (9) and
(10)] for sine and cosine, and compute the power series of the combination sin
2
kx
+
cos
2
kx
up to and including terms of order (
kx
)
4
. Of course, you know what the
answer will turn out to be, but you will learn something by working it out in this
way.
13
PHY 201
Agenda #12
1.
Reading
Tutorial Vectors and Matrices (VAM), Sections 1–2.5 due in Perusall.
2.
Tutorial Assignment
Work Exercises 1–11 and prepare Exercises 1, 2, 3, 6, 7, and 11 to submit for grading
in Canvas.
3.
In-Class Exercises
Work as teams on the following exercises.
(a) By making an appropriate change of variable or parameter in TF Eqs. 9 and 10,
determine the even and odd series solutions to
d
2
y
dx
2
−
η
2
y
= 0
.
Do you recognize the solutions as functions you have previously seen in this course?
(“No” is not an option.)
(b) Solve Trigg’s Equation for
k
= 0.
Remember that there must be two linearly
independent solutions.
(c) Find the antiderivatives (from tables, or using CA Eqs. 26, 27)
Z
cos
mθ
cos
nθ dθ
Z
sin
mθ
sin
nθ dθ
Z
sin
mθ
cos
nθ dθ
for any
m
and
n
(not necessarily integers). Use them to derive TF Eqs. TF 22–24
directly in the case where
m
and
n
are
integers.
(d) The function
f
(
x
) is periodic with wavelength
λ
= 2. In the range 0
< x <
2, it is
given by
f
(
x
) =
+1
0
< x <
1
−
1
1
< x <
2
.
Plot this function. Then find the quantities
a
0
, a
n
, and
b
n
(
n
= 1
,
2
,
3
, . . .
) that
will permit us to write
f
(
x
) =
a
0
+
∞
X
n
=1
a
n
cos (
nπx
) +
∞
X
n
=1
b
n
sin (
nπx
)
.
14
To do this, (a) multiply each side of this equation in turn by each of 1, cos
mπx
and sin
mπx
for any integer
m >
0, (b) integrate in each case over
x
from 0 to 2
(=
λ
) and (c) use the orthogonality relations, TF Eqs. 19–21.
15
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PHY 201
Agenda #13
1.
Reading
Tutorial Vectors and Matrices (VAM), Section 3 due in Perusall.
2.
Tutorial Assignment
Work Exercises 12–18 and prepare Exercises 12, 13, 14, 15, 16, 17, and 18(a, b, c) to
submit for grading in Canvas.
3.
In-Class Exercises
Divide up the following problems amongst your group so that each person does at least
half of them.
Solve your problems individually, and then explain to your teammates
how you solved each one. You should check each other’s arithmetic.
You are given the following vectors in terms of their components in the
{
ˆx
,
ˆy
,
ˆ
z
}
basis:
A
=
ˆx
+ 2
ˆy
−
2
ˆ
z
,
B
= 3
ˆx
+
ˆy
+2
ˆ
z
,
C
= 4
ˆx
−
ˆy
+
ˆ
z
.
Take
θ
AC
to be the angle between vectors
A
and
C
, etc.
Show first, as a team,
that these vectors are
linearly independent
. Then compute the following (Some of these
require using the results of Exercise 8):
(a)
A, B
and
C.
(i.e.,
A
=
|
A
|
, etc.)
(b) 2
A
+
B
,
3
C
−
B
,
A
+
B
+
C
(c)
A
·
B
,
A
·
C
and
B
·
C
(d)
A
×
B
,
A
×
C
and
B
×
C
(e) cos
θ
AB
,
cos
θ
AC
,
cos
θ
BC
(f) sin
θ
AB
,
sin
θ
AC
,
sin
θ
BC
(g)
A
·
(
B
×
C
)
,
B
·
(
A
×
C
) and
C
·
(
A
×
B
)
(h) (
A
×
B
)
×
C
+ (
B
×
C
)
×
A
+ (
C
×
A
)
×
B
16
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PHY 201
Agenda #14
1.
Reading
Tutorial Vectors and Matrices (VAM), Section 4 due in Perusall.
2.
Tutorial Assignment
Work Exercises 19–22 and prepare Exercises 19, 20, 21, and 22 to submit for grading
in Canvas.
3.
In-Class Exercises
(a) Divide up the following problems amongst your group so that each person does
at least half of them. Solve your problems individually, and then explain to your
teammates how you solved each one. You should check each other’s arithmetic.
You are given the following vectors in terms of their components in the
{
ˆx
1
,
ˆx
2
,
ˆx
3
}
basis:
A
= (1
,
2
,
−
1)
,
B
= (1
,
2
,
2)
,
C
= (1
,
0
,
1)
,
D
= (2
,
1
,
2). Components are
given by lower-case letters, as in
A
=
a
1
ˆx
1
+
a
2
ˆx
2
+
a
3
ˆx
3
= (
a
1
, a
2
, a
3
), etc. Com-
pute the following (The summation convention is used throughout these exercises):
i.
a
i
b
i
,
a
i
c
i
,
b
i
c
i
ii.
δ
ii
iii.
a
2
+
b
i
δ
3
i
iv. (
b
i
d
j
−
d
i
b
j
)
δ
ij
v.
a
i
d
k
δ
ij
δ
jk
vi.
a
k
b
i
c
j
d
k
δ
ij
vii.
a
i
b
j
c
k
d
l
(
δ
ij
δ
kl
−
δ
ik
δ
jl
)
viii.
a
i
b
j
ϵ
ij
2
ix.
ϵ
ijk
ϵ
ijk
x.
a
i
b
j
c
k
ϵ
ijk
xi.
a
i
b
j
c
k
ϵ
ijl
ϵ
3
kl
xii.
a
i
c
k
ϵ
ij
1
δ
jk
xiii.
ϵ
ijl
ϵ
kml
ϵ
ijm
a
k
17
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PHY 201
Agenda #15
1.
Reading
Tutorial Orthogonal Functions and Fourier Series (OFFS), Section 1–1.2 due in Pe-
rusall
2.
Tutorial Assignment
Work Exercises 1–11 and prepare Exercises 2, 4, 5, 8, 9, 10, and 11 to submit for
grading in Canvas.
3.
In-Class Exercises
Use the following vectors:
A
= (1
,
2
,
−
1)
,
B
= (1
,
2
,
2)
,
C
= (1
,
0
,
1).
(a) Test to see that these three vectors are linearly independent.
(b) Construct,
using the Schmidt orthogonalization method
, three different sets
of orthonormal three-dimensional basis vectors. In the first one, take one basis vec-
tor in the direction of
A
, the second in the
A
-
B
plane and the third perpendicular
to that plane in a right-handed sense. In the second one, start with
B
, take the
second basis vector to be in the
B
-
C
plane, etc.
(c) Write the vector
A
in each of the three bases that you constructed [Hint: in the
first basis,
A
=
√
6(1
,
0
,
0)]. Do the same for
B
and
C
.
18
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PHY 201
Agenda #16
1.
Reading
Tutorial Orthogonal Functions and Fourier Series (OFFS), Section 1.3 due in Perusall
2.
Tutorial Assignment
Work Exercises 12–20 and prepare Exercises 14, 16(a, b, c, d) and 17(a, b) to submit
for grading in Canvas.
3.
In-Class Exercises
(a) Problem 1, Tutorial OFFS, p. 5
(b) Problem 2, Tutorial OFFS, p. 5
19
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PHY 201
Agenda #17
1.
Reading
Tutorial One Dimensional Wave Equation (ODWE), Sections 1–2 due in Perusall
2.
Tutorial Assignment
Work Exercises 1–10 and prepare Exercises 4, 5, 6, 7, 8, and 9 to submit for grading
in Canvas.
3.
In-Class Exercises
Find the Fourier series for the following periodic functions, defined over one period as
given. Use a math package to plot the function and to sum and plot the Fourier series
for 1
,
5
,
10, and 50 terms over the same range.
(a)
f
(
t
) = 3 sin (
πt/
3)
,
−
1
< t <
1
(b)
f
(
t
) =
t
(1
−
t
)
,
0
< t <
1
(c)
f
(
t
) =
t
(1
−
t
)
,
−
1
/
2
< t <
1
/
2
20
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PHY 201
Agenda #18
1.
Midterm
21
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PHY 201
Agenda #19
1.
Reading
Tutorial One Dimensional Wave Equation (ODWE), Section 3 due in Perusall
2.
Tutorial Assignment
Work Exercises 11–23 and prepare Exercises 12, 13, 15, 17, 20, and 21 to submit for
grading in Canvas.
3.
In-Class Exercises
Find the
complex
Fourier series for the following periodic functions, defined over one
period as shown. Use a math package to plot the function itself (It might be a good
idea to do this first before you derive any coefficients) and to sum and plot the Fourier
series over two periods for, say, 1
,
5
,
11
,
and 51 terms (
i.e.
, sum
n
from
−
N
to
N
for
N
= 0
,
2
,
5, and 25).
(a)
f
(
t
) = cosh (2
t
)
,
0
< t <
1
(b)
f
(
x
) =
(1
−
2
x
)
2
0
< x <
1
−
(3
−
2
x
)
2
1
< x <
2
22
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PHY 201
Agenda #20
1.
Reading
Tutorial Vectors and Matrices (VAM), Sections 5–5.3.2 due in Perusall.
2.
Tutorial Assignment
Work Exercises 23–31 and prepare Exercises 23, 25, 26, and 27 to submit for grading
in Canvas.
3.
In-Class Exercises
(a) As you have seen in Section 3.2 of
ODWE
, the general solution for the transverse
displacement of a string stretched between two pegs a distance
L
apart is
y
(
x, t
) =
∞
X
n
=1
sin (
k
n
x
) [
B
n
cos (
ω
n
t
) +
C
n
sin (
ω
n
t
)]
,
where
k
n
=
nπ/L
and
ω
n
=
vk
n
, v
being the velocity of propagation of waves on
the string.
B
n
and the
C
n
are expansion coefficients that can be determined by the
initial conditions (
i.e.
, boundary conditions in time).
Calculate (by hand) these
coefficients for the following boundary conditions:
y
(
x,
0)
=
2
x
(
L
−
x
)
/L,
˙
y
(
x,
0)
=
0
.
Since [0
, L
] covers only
half
a wavelength (Why?), you need to use OFFS Eqs. (17)-
(18).
(b) The steady-state temperature distribution,
T
(
x, y
), in a flat metal sheet obeys the
partial differential equation
∂
2
T
∂x
2
+
∂
2
T
∂y
2
= 0
.
Separate the variables in this equation just as we did in the one-dimensional wave
equation and find
T
everywhere on a square flat plate of sides
S
with the boundary
conditions
T
(0
, y
) =
T
(
S, y
) =
T
(
x,
0) = 0
, T
(
x, S
) =
T
0
.
23
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PHY 201
Agenda #21
1.
Reading
Tutorial Vectors and Matrices (VAM), Sections 5.3.3 – 5.3.4 due in Perusall.
2.
Tutorial Assignment
Work Exercises 32–43 and prepare Exercises 32, 33, 35, 39(a, c, e), and 42 to submit
for grading in Canvas.
3.
In-Class Exercises
Given the following matrices and their transposes, find, by hand,
all
possible sums and
products.
A
=
0
1
−
1
0
;
B
=
2
1
3
−
1
2
−
3
;
C
=
−
1
2
3
−
1
−
2
3
;
D
=
0
1
−
1
−
1
0
1
1
−
1
0
.
Repeat all of the operations using a math package to check your results.
24
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PHY 201
Agenda #22
1.
Reading
Tutorial Vectors and Matrices (VAM), Section 6 due in Perusall.
2.
Tutorial Assignment
Work Exercises 44–48 and prepare Exercises 44, 46, and 48 to submit for grading in
Canvas.
3.
In-Class Exercises
Determine which of the following matrices have inverses by computing their determi-
nants using Gauss-Jordan reduction. For those which have non-vanishing determinants,
compute the inverses using the algorithm described in Section 5.3.4. Use a math package
to check your results.
(a)
3
1
5
−
1
−
3
−
1
2
2
3
; (b)
6
−
2
3
1
1
1
2
−
3
1
; (c)
4
2
2
−
1
3
−
1
3
4
5
;
(d)
2
3
−
1
1
3
−
4
3
−
1
2
−
1
1
−
3
3
1
−
2
4
; (e)
1
2
3
4
2
3
4
5
3
4
5
6
4
5
6
7
.
25
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PHY 201
Agenda #23
1.
Reading
Tutorial Vectors and Matrices (VAM), Sections 7.1–7.2 due in Perusall.
2.
Tutorial Assignment
Work Exercises 49–59 and prepare Exercises 50, 52, 53, 54, 55, and 59 to submit for
grading in Canvas.
3.
In-Class Exercises
As teams, do each of the following problems. We will try to reserve time for solutions
to be reported.
(a) Find an equation of the plane through the points (0
,
0
,
3)
,
(1
,
1
,
1) and (
−
1
,
1
,
2).
(b) Find the equation of the plane that contains the point (2
,
1
,
−
4) and is perpendic-
ular to the vector 3
ˆx
−
2
ˆy
+
ˆ
z
.
(c) Determine
b
such that the line through (5
,
0
,
3) and (
−
1
,
−
10
, b
) will be perpen-
dicular to (
ˆx
+
ˆy
)
×
(
ˆx
+
ˆ
z
).
(d) Use vectors to find the equation of the straight line containing the points (1
,
−
2
,
4)
and (6
,
1
,
1)
.
Write this equation in
parametric form
,
i.e.
, in the form
r
=
r
0
+
t
V
,
where
r
0
is the vector describing a given point on the line,
V
is a vector oriented
along the line, and
t
is a variable that has range (
−∞
,
∞
).
26
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PHY 201
Agenda #24
1.
Reading
Tutorial Vectors and Matrices (VAM), Section 7.3 due in Perusall.
2.
Tutorial Assignment
Work Exercises 60–64 and prepare Exercises 61, 62, and 63 to submit for grading in
Canvas.
3.
In-Class Exercises
Note that, as in the tutorial, all
sans serif
characters represent matrices. Consider the
following vectors:
A
=
2
3
,
B
=
1
5
,
C
=
−
2
1
.
(a) Rotate (
i.e.
, find the necessary rotation matrix)
A
into the direction of
B
.
(b) Construct a basis in which the
ˆx
′
1
basis vector is in the direction of
C
.
Express the
new basis vectors as individual column matrices in the original (“home”) basis:
ˆ
x
′
i
=
Rˆ
x
i
, i
= 1
,
2
,
and write out the matrix
R
.
The new basis should be right-handed.
(c) Express the new basis vectors in terms of the home basis in the manner:
ˆx
′
1
ˆx
′
2
=
M
ˆx
1
ˆx
2
,
where
M
is a matrix. Compare
M
to
R
.
(d) Find
A
and
B
in this new basis. Demonstrate that their scalar product is the same
in each basis.
(e) Express the home basis vectors in terms of the new basis.
(f) Find the rotation matrix from Exercise (a) in the new basis. Check it (
i.e.
, does it
rotate
A
into
B
when both are expressed in the new basis?).
27
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PHY 201
Agenda #25
1.
Reading
Tutorial Matrix Theory (MT), Sections 1–1.3 due in Perusall.
2.
Tutorial Assignment
Work Exercises 1–7 and prepare Exercises 1, 2, 3, 4, 5, and 6 to submit for grading in
Canvas.
3.
In-Class Exercises
(a) Find the rotation matrix that rotates the vector
A
into the vector
B
, where, in the
home representation,
A
=
1
√
13
1
3
−
4
,
B
=
1
√
15
2
−
1
5
(b) Find the inverse of the rotation matrix.
(c) What happens to the scalar product of
A
and
B
under this rotation?
28
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PHY 201
Agenda #26
1.
Reading
Tutorial Matrix Theory (MT), Section 2 due in Perusall.
2.
Tutorial Assignment
Work Exercises 8–17 and prepare Exercises 9, 10, 12, 13, 14, 15, 16, and 17 to submit
for grading in Canvas.
3.
In-Class Exercises
(a) For each of the following matrices, (i) compute the trace, (ii) write down the Her-
mitian adjoint, and (iii) determine whether it is Hermitian, unitary or neither.
1
.
1
0
−
i
0
−
2
4
−
i
i
4 +
i
3
2
.
1
√
2
1
√
6
1
√
3
0
−
2
√
6
1
√
3
1
√
2
−
1
√
6
−
1
√
3
3
.
2
i
0
i
1
−
i
0
−
i
2
4
.
i
√
2
0
1
√
2
0
1
0
1
√
2
0
i
√
2
5
.
i
√
2
0
1
√
2
0
1
0
1
√
2
0
−
i
√
2
6
.
1
√
2
0
i
√
2
0
1
0
−
i
√
2
0
1
√
2
(b) Compute the determinants of the matrices.
(c) Use a math package to check your answers to today’s agenda problems.
29
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PHY 201
Agenda #27
1.
Reading
Tutorial Matrix Theory (MT), Section 3–3.1.3 due in Perusall.
2.
Tutorial Assignment
Work Exercises 18–25 and prepare Exercises 19, 21, 24, and 25 to submit for grading
in Canvas.
3.
In-Class Exercises
(a) Write the following matrices as linear combinations of Pauli matrices and the 2
×
2 identity matrix. Find the coefficients by taking traces.
i.
i
−
2
3
4
ii.
2
3
i
−
3
i
−
2
iii.
1
√
2
i
√
2
i
√
2
1
√
2
!
(b) Find the following matrices:
i. exp(
iπσ
2
/
4)
ii. exp(
−
iπ
σ
·
ˆx
)
iii. exp(
−
iπ
σ
·
ˆx
/
2)
(c) Define
s
i
=
1
2
σ
i
and
s
2
=
s
2
1
+
s
2
2
+
s
2
3
. Find the commutation relations of the
s
i
.
Show that
s
2
commutes with all of the
s
i
.
30
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PHY 201
Agenda #28
1.
Reading
Tutorial Matrix Theory (MT), Section 3.2 due in Perusall.
2.
Tutorial Assignment
Work Exercises 26–31, but none are assigned for submission.
The solutions will be
made available soon.
3.
In-Class Exercises
(a) Work as teams to find the eigenvectors and eigenvalues of the following matrices.
Examine the properties of the sets of eigenvectors.
i.
0
1
1
1
0
1
1
1
0
ii.
1
2
0
1
0
1
0
2
1
iii.
0
1
i
1
0
1
i
1
0
iv.
0
1
i
1
0
1
−
i
1
0
(b) Use a math package to check your answers to today’s agenda problems.
31
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PHY 201
Agenda #29
1.
Reading
None
2.
Tutorial Assignment
None
3.
In-Class Exercises
(a) During the last class period, we found the eigenvectors and eigenvalues of the
following Hermitian matrices. Construct the unitary transformations that will di-
agonalize them,
i.e.
, the matrices
U
such that
U
†
HU
=
Λ
, where
Λ
is a diagonal
matrix whose elements are the eigenvalues of
H
.
i.
0
1
1
1
0
1
1
1
0
ii.
0
1
i
1
0
1
−
i
1
0
(b) Here are some more matrices to practice on. Find their eigenvalues and orthonormal
sets of eigenvectors.
i.
2
0
0
0
1
1
0
1
1
ii.
1
1
1
1
1
1
1
1
1
iii.
5
0
√
3
0
3
0
√
3
0
3
(c) Use a math package to check your answers to today’s agenda problems.
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