MATH 171: PROBLEM SET 5 SOLUTIONS
CHRIS HENDERSON (WITH ADDITIONS BY ALEXANDER KUPERS)
1. Hand-in exercises
Problem 1.1 (35.6). Let
be the set of sequences which are bounded. Let c0 be the set of
sequences which converge to 0.
(a) Prove that 1 c0
(b) Pro
Mathematics Department, Stanford University
Math. 171Autumn 2012
Preliminary Notes
Except possibly for Theorem 2.4 and the material in 6 on countable and uncountable sets,
most of the material herewhich corresponds roughly to Ch.IVI of the textwill alread
Solution Set
Math 171
Problem Set 3
Problem 1 - 19.2 Prove that cfw_an is a Cauchy sequence if and only if for every
such that n > N implies that |an aN | < .
> 0 there exists N
Assume that cfw_an is a Cauchy sequence. Then x > 0 and nd N such that n, m
MATH 171: PROBLEM SET 2 SOLUTIONS
CHRIS HENDERSON (WITH ADDITIONS BY ALEXANDER KUPERS)
1. Hand-in exercises
Problem 1.1 (9.7). Prove that the set of polynomials with rational coecients of degree n is
countable.
Proof. We proceed by induction on n. Dene Pn
MATH 171: PROBLEM SET 1 SOLUTIONS
CHRIS HENDERSON (WITH ADDITIONS BY ALEXANDER KUPERS)
1. Hand-in exercises
Problem 1.1 (3.5). Show that xy = 0 if and only if x = 0 or y = 0.
Proof. If x = 0 or y = 0 then Theorem 3.4 (of the book) gives use that xy = 0. T
PHYSICS 230 PROBLEM SET 8 1. Consider a particle of mass m conned on a thin ring of circumference L, through which there is a magnetic ux . We can model this system by the Hamiltonian 1 e2 H= p A (1) 2m c where A = in L).
L
= constant and x
x + L (i.e.,
PHYSICS 230 PROBLEM SET 7 These problems will become clearer after the lecture on Monday. 1. Shankar Problem 8.6.2 2. Consider a free particle of mass m on a circle of radius L described by a coordinate x running from 0 to 2L. We will evaluate f (t0 ) x =
PHYSICS 230 PROBLEM SET 6 1. Consider a potential of the form V (x) = x2p . Using scaling arguments calculate En using the WKB approximation, up to an unknown integral whose answer is a constant of order one. For p = 1 do the integral and compare your ans
PHYSICS 230 PROBLEM SET 2 1. Consider the perturbed harmonic oscillator Hamiltonian 1 H = p2 /2m + m 2 x2 + x4 2 Calculate the ground state energy to second order in . Calculate the energy of an arbitrary eigenstate |n > to rst order in . 2. Consider the
PHYSICS 230 PROBLEM SET 4 1. Review the bound state problem for a single delta function potential. Then nd the bound states for a potential consisting of two delta functions separated by a distance L. V (x) = ( (x + L/2) + (x L/2) Show that this reduces t
PHYSICS 230 PROBLEM SET 3 PHYSICS 230
1. Read Shankar Section 5.1. Do Problem 5.1.3. Read Shankar Chapter 6. 2. At time t = 0 the initial state of a free particle on a line is given by the sum of two identical gaussian wave packets of width separated by a
PHYSICS 230 PROBLEM SET 2 1. UNITS Theoretical physicists often work in units where h = c = 1. What does this mean? From the equation x = ct we see that if c = 1 then [length] and [time] are the same units. From the equation E = h we see that [energy ] an
PHYSICS 230 PROBLEM SET 1
1. Linear Algebra Practice Remind yourself what a projection operator is. Show that the matrix M dened below is a projection operator by explicit diagonalization. 111 1 M= 1 1 1 3 111 Show that M 2 = M as required for a projectio
MATH 171: PROBLEM SET 4 SOLUTIONS
CHRIS HENDERSON (WITH ADDITIONS BY ALEXANDER KUPERS)
1. Hand-in exercises
Problem 1.1 (16.11). Prove directly, from denition 10.2, that lim n1/n = 1.
Proof. As suggested in the hint let n = n1/n 1. Notice that if n > 1 th
MATH 171: PROBLEM SET 6 SOLUTIONS
CHRIS HENDERSON (WITH ADDITIONS BY ALEXANDER KUPERS)
1. Hand-in exercises
Problem 1.1 (40.1). Prove Theorem 40.4 (i), (iii), (iv), (v), and (vi).
(i). First, we claim that if a, b R then |a| |b| |a b|. To see this notice
MATH 171: PROBLEM SET 7 SOLUTIONS
CHRIS HENDERSON (WITH ADDITIONS BY ALEXANDER KUPERS)
1. Hand-in exercises
Problem 1.1 (43.1). Prove that the set S = cfw_x M : d(x, 0) = 1 is closed and bounded in M ,
but not compact if M is 2 , c0 or .
Proof. Suppose th
Math 171 Homework 3
Due Friday April 22, 2016 by 4 pm
Please remember to write down your name and Stanford ID number, and to staple your solutions. Solutions are due to the Course Assistant, Alex Zamorzaev, in his office, 380-380M
(either hand your soluti
Math 171 Homework 5
Due Friday May 6, 2016 by 4 pm
Please remember to write down your name and Stanford ID number, and to staple your solutions. Solutions are due to the Course Assistant, Alex Zamorzaev, in his office, 380-380M
(either hand your solutions
Math 171 Homework 1
Due Friday April 8, 2016 by 4 pm
Please remember to write down your name and Stanford ID number, and to staple your solutions. Solutions are due to the Course Assistant, Alex Zamorzaev, in his office, 380-380M
(either hand your solutio
Math 171 Homework 2
Due Friday April 15, 2016 by 4 pm
Please remember to write down your name and Stanford ID number, and to staple your solutions. Solutions are due to the Course Assistant, Alex Zamorzaev, in his office, 380-380M
(either hand your soluti
Math 171 Homework 8
Due Friday May 27, 2016 by 4 pm
Please remember to write down your name and Stanford ID number, and to staple your solutions. Solutions are due to the Course Assistant, Alex Zamorzaev, in his office, 380-380M
(either hand your solution
Math 171 Homework 6
Due Friday May 13, 2016 by 4 pm
Please remember to write down your name and Stanford ID number, and to staple your solutions. Solutions are due to the Course Assistant, Alex Zamorzaev, in his office, 380-380M
(either hand your solution
Math 171 Homework 7 (half weight)
Due Friday May 20, 2016 by 4 pm
Please remember to write down your name and Stanford ID number, and to staple your solutions. Solutions are due to the Course Assistant, Alex Zamorzaev, in his office, 380-380M
(either hand
Math 171 Homework 2 Solutions
8.2
Let f : X Y be a bijection. Dene the functions
F : P(X) P(Y ) : A cfw_f (a) : a A,
AX
and
G : P(Y ) P(X) : B cfw_f 1 (b) : b B,
BY
We show F and G are inverses, and hence F is a bijection. For any A X,
GF (A) = Gcfw_f (a)
Math 171 Homework 1 Solutions
3.3
By denition x + (x) = 0. So
x = x + (x + (x) = (x x) + (x) = (x).
3.5
If x = 0 or y = 0 then xy = 0 by theorem 3.4. Conversely, if xy = 0 but
x = 0, then y = x1 xy = x1 0 = 0.
4.1
(iii) x > y means x y P . Hence (x + z) (
MATH 171: PROBLEM SET 9 SOLUTIONS
CHRIS HENDERSON (WITH ADDITIONS BY ALEXANDER KUPERS)
1. Hand-in exercises
Problem 1.1 (48.4). (a) Prove that f is dierentiable at a if and only if there exists a function
E (h) dened on an open interval (, ) and a constan
MATH 171: PROBLEM SET 8 SOLUTIONS
CHRIS HENDERSON (WITH ADDITIONS BY ALEXANDER KUPERS)
1. Hand-in exercises
Problem 1.1 (45.3). Let f be a continuous function from a compact, connected metric space M
into R. Prove that f (M ) is a closed interval.
Proof.
Physics 170: Problem Set 1 (Due Thursday, October 7 in class) This assignment is very simple. Do all 6 problems at the end of Kittel and Kroemer, chapter 2. Other news: 1) Section times and locations are: Tuesday, 5:15-6:15 pm, Hewlett 101 Thursday, 6:00-
Physics 170: Problem Set 7 (Due Thursday, December 2 in TAs mailbox) 1. Kittel Ch. 9, problem 1 2. Kittel Ch. 9, problem 2 3. Kittel Ch. 9, problem 3 4. Kittel Ch. 10, problem 1 5. Kittel Ch. 10, problem 4 6. Kittel Ch. 10, problem 8 In the last few lectu
DFS/BFS, MST, Dynamic Programming
November 16, 2010
Homework 5
Due Date: Tuesday, November 30, 2010, 2:05pm 10pts Problem 5-1. Explain how a vertex u of a directed graph can end up in a DFS tree containing only u, even though u has both incoming and outgo