Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 4
Date Due: 10:00 AM, Thursday, the 31st of October 2013
Mathematics is the part of physics where experiments are cheap.
V.I. Arnold, On teaching mathematics, March 7th , 19971
Exercise
Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 2
Date Due: 10:00 AM, Thursday, the 10th of October 2013
Investigation may be likened to the long months of pregnancy, and solving a problem to the day of birth. To
investigate a proble
Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 1
Date Due: 12:00 PM (noon), Sunday, the 29th of September 2013
Mathematics is learned by doing it, not by watching other people do it.
Michael Reed and Barry Simon, Methods of Modern M
Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 3
Date Due: 10:00 AM, Thursday, the 17th of October 2013
Obvious is the most dangerous word in mathematics
Eric Temple Bell
Exercise 1. Let (xn ) be a bounded real sequence.1 Let
yn = s
Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 12
This assignment contains additional exercises for your own enjoyment. All
marks on this assignment are imaginary; they will not contribute to the course
grade and the assignment will
Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 8
Date Due: 10:00 AM, Thursday, the 21st of November 2013
Logic moves in one direction, the direction of clarity, coherence and structure. Ambiguity moves in the other
direction, that o
Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 11
Date Due: 10:00 AM, Thursday, the 12th of December 2013
Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and
forthwith it is somethi
Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 9
Date Due: 10:00 AM, Thursday, the 28th of November 2013
Trigonometry, so far as this, is most valuable to every man; there is scarcely a day in which he will not resort
to it for some
Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 4
Date Due: 10:00 AM, Thursday, the 24th of October 2013
Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict
reasoning.
John Locke
Vv186 Honors Mathematics II
Sample Exercises for the Second Midterm Exam - Solutions
Exercise 1. Test the following series for convergence:
k+5
k=1
5k
,
k=1
3k k 2
5k
,
4k + 3k
k=1
k!
.
(3 Marks)
Solution.
i) Applying the ratio test,
k+1+5 1 k + 6
1
13
Vv186 Honors Mathematics 1
Sample Exercises for the First Midterm Exam - Solutions
Exercise 1.
i) Consider the set U R, where U = A B C with
A = cfw_x R : 0 < x 1,
B = cfw_x R : x = 2 1/n, n N \ cfw_0,
C = cfw_x R : x = 1/n, n N \ cfw_0.
State (without pr
Vv186 Honors Mathematics II
Sample Exercises for the First Midterm Exam
The following exercises are sample exercises of a diculty comparable to those found the rst
midterm exam. Most of these have been used in actual exams in the last few years. Solutions
Vv285 Honors Mathematics III
Functions of Several Variables
Assignment 3
Date Due: 10:00 AM, Thursday, the 11th of June 2014
Exercise 1. For an endomorphism A End(V ) of a nite-dimensional euclidean or unitary vector space V we
dene the adjoint A End(V )
Vv285 Honors Mathematics III
Functions of Several Variables
Assignment 8
Date Due: 10:00 AM, Thursday, the 16th of July 2015
Exercise 1. Consider again the problem of compression of a spring (see Exercise 3 of Assignment 7). A spring
is characterized by t
Vv285 Honors Mathematics III
Functions of Several Variables
Assignment 6
Date Due: 10:00 AM, Thrusday, the 2nd of July 2015
Exercise 1. Let C be a smooth, oriented, simple, open curve. In this exercise we use s = s(p) = (p) to denote
the length of the cur
Vv285 Honors Mathematics III
Functions of Several Variables
Assignment 7
Date Due: 10:00 AM, Thursday, the 9th of July 2015
Exercise 1. Calculate the following directional derivatives:
f : R2 \ cfw_0 R,
g : R3 R,
x 2y
3x + y
g(x, y, z) = cos(x + y + z)
f
Vv285 Honors Mathematics III
Functions of Several Variables
Assignment 11
Date Due: Not graded
This assignment will not be graded; it is for your reference only in roder to provide some practice exercises.
Exercise 1. In electrostatics, the potential at a
Vv285 Honors Mathematics III
Functions of Several Variables
Assignment 4
Date Due: 10:00 AM, Tuesday, the 23rd of June 2014
Exercise 4.1.
i) Give an example to show that if a set A R is closed and f C(A, R), the set f (A) = ran f does not
have to be close
Vv285 Honors Mathematics III
Functions of Several Variables
Assignment 10
Date Due: 10:00 AM, Thursday, the 30th of July 2015
Exercise 1. Find the volume of
i) the ellipsoid
cfw_
2
y2
z2
3 x
E = (x, y, z) R : 2 + 2 + 2 1
a
b
c
where a, b, c > 0.
ii) the
Vv285 Honors Mathematics III
Functions of Several Variables
Assignment 1
Date Due: 12:10 PM, Friday, the 27th of May 2016
You are required to compose your solutions in neat and legible handwriting. Up to 10% of the total score may
be deducted solely due t
Vv285 Honors Mathematics III
Functions of Several Variables
Assignment 3
Date Due: 12:10 PM, Monday, the 13th of June 2016
Exercise 3.1. Let V be a nite-dimensional inner product space and B = cfw_e1 , . . . , en a basis of V . For an
endomorphism A L(V,
Vv285 Honors Mathematics III
Functions of Several Variables
Assignment 2
Date Due: 12:10 PM, Monday, the 6th of June 2016
You are required to compose your solutions in neat and legible handwriting. Up to 10% of the total score may
be deducted solely due t
Vv285 Honors Mathematics III
Functions of Several Variables
Assignment 5
Date Due: 12:10 PM, Monday, the 27th of June 2016
Exercise 5.1.
i) Give an example to show that if a set A R is closed and f C(A, R), the set f (A) = ran f does not
have to be closed
Vv285 Honors Mathematics III
Functions of Several Variables
Assignment 4
Date Due: 12:10 PM, Monday, the 20th of June 2016
Exercise 4.1. This exercise introduces the classical matrix groups. In the lecture, wehave already discussed
the general linear grou
Vv214 Lecture 3
Dr Jing Liu
UM-SJTU Joint Institute
May 20, 2016
Dr Jing Liu (UM-SJTU Joint Institute)
Lecture 3
May 20, 2016
1 / 17
Definition
- Addition of Matrices
The sum of two matrices A = [aij ] and B = [bij ] of the same size
A + B = [aij + bij ]
Vv214 Lecture 8
Dr Jing Liu
UM-SJTU Joint Institute
June 6, 2016
Dr Jing Liu (UM-SJTU Joint Institute)
Lecture 8
June 6, 2016
1 / 23
Q: Why do we have to study Vector Spaces?
- It was realized that many mathematical objects of different sorts:
vectors, ma
Vv214 Lecture 5
Dr Jing Liu
UM-SJTU Joint Institute
May 25, 2016
Dr Jing Liu (UM-SJTU Joint Institute)
Lecture 5
May 25, 2016
1 / 16
Definition
A square matrix in which all the entries below the main diagonal are 0 is called an
upper triangular matrix. In
Vv214 Lecture 16
Dr Jing Liu
UM-SJTU Joint Institute
June 29, 2016
Dr Jing Liu (UM-SJTU Joint Institute)
Lecture 16
June 29, 2016
1 / 26
- Suppose V is a finite-dimensional vector space, and
both B and B 0 are bases for V.
and let v be a vector in V, and
Vv214 Lecture 2
Dr Jing Liu
UM-SJTU Joint Institute
May 18, 2016
Dr Jing Liu (UM-SJTU Joint Institute)
Lecture 2
May 18, 2016
1 / 19
Linear Equations
Gaussian Elimination
- The most important treatise in the history of Chinese mathematics
is the Nine Chap