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
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
sequ
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
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
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 herewhi
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 i
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
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)
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 coordina
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 or
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 arbit
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.
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
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
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
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 sugges
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
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 Zamo
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 Zamorza
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 Zamor
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 Zamo
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 Zamorz
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 Zamorz
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 Assistan
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 h
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 =
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 func
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
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, prob
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 conta