1.2
topology
Closure of a set
Closure of a set S, denoted by S is the union of the set S together with boundary points of S: S = S S.
Interior and closure of a set are in a dynamic relationship, and most of the topological analysis takes place in
this rel
University of Toronto
Faculty of Arts and Science
Quiz 2
MAT2371Y  Advanced Calculus
Duration  50 minutes
No Aids Permitted
Surname:
First Name:
Student Number:
Tutorial:
T0101
T5101
T5102
T4/T5 R4/R5 T5/R5
Chris
Anne
Ivan
SS1074 SS1070 BA1240
This exam
University of Toronto
Faculty of Arts and Science
Quiz 1
MAT2371Y  Advanced Calculus
Duration  1 hour
No Aids Permitted
Surname:
First Name:
Student Number:
Tutorial:
T0101
T5101
T5102
T4/R4 T5/R5 T5/R5
Chris
Anne
Ivan
SS1074 SS1070 BA1240
This exam con
MAT237  Tutorial 2  26 May 2015
1
Coverage
Heres what theyve covered in class between last Tuesdays tutorial and this one, which roughly
comprises the material covered in this tutorial.
Sequences and Completeness: All of Chapter 4 of their lecture notes
MAT237  Tutorial 1
1
Coverage
Heres what theyve covered in class so far, which roughly comprises the material covered in this
tutorial.
Set Theory: Set containment, indexed families of sets, intersection, union, complement,
functions, image, preimage
Top
University of Toronto
Faculty of Arts and Science
Term Test
MAT2371Y  Advanced Calculus
Duration  1 hour 50 minutes
No Aids Permitted
Surname:
First Name:
Student Number:
Tutorial:
T0101
T5101
T5102
T4/R4 T5/R5 T5/R5
Chris
Anne
Ivan
SS1074 SS1070 BA1240
University of Toronto
Faculty of Arts and Science
Quiz 2
MAT2371Y  Advanced Calculus
Duration  50 minutes
No Aids Permitted
Surname:
First Name:
Student Number:
Tutorial:
T0101
T5101
T5102
T4/T5 R4/R5 T5/R5
Chris
Anne
Ivan
SS1074 SS1070 BA1240
This exam
University of Toronto
Faculty of Arts and Science
Quiz 3
MAT2371Y  Advanced Calculus
Duration  50 minutes
No Aids Permitted
Surname:
First Name:
Student Number:
Tutorial:
T0101
T5101
T5102
T4/T5 R4/R5 T5/R5
Chris
Anne
Ivan
SS1074 SS1070 BA1240
This exam
University of Toronto
Faculty of Arts and Science
Quiz 6
MAT2371Y  Advanced Calculus
Duration  50 minutes
No Aids Permitted
Surname:
First Name:
Student Number:
Tutorial:
T0101
T5101
T5102
T4/R4 T5/R5 T5/R5
SS1074 SS1070 BA1240
This exam contains 4 page
University of Toronto
Faculty of Arts and Science
Quiz 4
MAT2371Y  Advanced Calculus
Duration  50 minutes
No Aids Permitted
Surname:
First Name:
Student Number:
Tutorial:
T0101
T5101
T5102
T4/R4 T5/R5 T5/R5
SS1074 SS1070 BA1240
This exam contains 4 page
University of Toronto
Faculty of Arts and Science
Quiz 4
MAT2371Y  Advanced Calculus
Duration  50 minutes
No Aids Permitted
Surname:
First Name:
Student Number:
Tutorial:
T0101
T5101
T5102
T4/R4 T5/R5 T5/R5
Chris
Anne
Ivan
SS1074 SS1070 BA1240
This exam
Topology of Rn
1
1.1
Basic Set Theory
1. Let A S and B S. Prove each of the following statements
(a) A B if and only if A B = B
(b) Ac B if and only if A B = S
(c) A B if and only if B c Ac
(d) A B c if and only if A B =
2. Let A, B, and C be subsets of
MAT237: Term test 2 information
Test 2 will take place Thursday March 5, 911 am. at Ex 200, with an early seatings of 79 and 810
in EX310. Please make reservation for the early seating for logistics sake. Deadline for early seating
is Tuesday March 3 m
Partial Big List Solutions
MAT 237 Advanced Calculus Summer 2015
Solutions
# 1.6
Let f : A B be a map of sets, and let cfw_Xi iI be an indexed collection of subsets of A.
(a) Prove that f (iI Xi ) = iI f (Xi )
(b) Prove that f (iI Xi ) iI f (Xi )
(c) When
1. Some preliminaries.
1.1. Quantiers. There are two symbols that will commonly appear in the textbook and throughout the course: and . You can translate these symbols into English as follows:
= for every
= there exists
Example: Consider the following t
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MAT 237, Hints for PS2
Please note that problem sets are designed to encourage you to study the textbook. If you ignore the
textbook and try to use your own wisdom (of MAT137) then you have defeated the purpose of the problem
sets. So please try to follow
MAT 237, PS3

Due, Friday Nov. 15, 2:00 pm.
in SS Math Aid Center
FAMILY NAME:
FIRST NAME:
STUDENT ID:
Please note:
1. Your problem set must be submitted on this form. Please provide your nal, polished solutions in
the spaces provided. Remember, it is
Term Test 2  July 24, 6:10pm  7:40pm
Instructions: Each question is worth 10 marks. All answers must be submitted in the provided booklets.
Ensure that your name is on each booklet, and number the booklets according to the order in which they
should be
1. Exercises from Sections 2.72.9
Problem 1. (Folland 2.7.3) Show that  sin(x) x + x3 /6 < 0.08 for all x < /2
(1) Notice that x3 x3 /6 is the fourth order Taylor approximation of sin(x) about zero
(2) Since sin(x) is smooth (therefore C k for all k
1. Exercises from Sections 2.9
Problem 1. (Folland 2.9.1) Find the extreme values of f (x, y) = 3x2 2y 2 + 2y on the set
(x, y)  x2 + y 2 1
Extreme values can occur either on the boundary, or at critical points on the interior
On the interior of the di
1. Exercises from Sections 2.22.3
Problem 1. (Folland 2.1.8) Suppose f : S R, S Rn . If all partial derivatives j f exist and
are bounded on S, then f is continuous on S.
Remark : Notice the converse is not true  i.e. f (x, y) = x2 + y 2 on R2 has unbou
1. Exercises from Sections 1.82.1
Problem 1. We call f : S Rn Holder continuous with exponent i there exists constants
C, > 0 such that f (x) f (y) < Cx y for every x, y S. Show that f is uniformly continuous.
Proof.
(1) Fix
> 0.
(2) NTS for any x, y
1. Exercises from Sections 2.42.5
Problem 1. (Folland 2.4.5) Suppose S is a convex, connected open set and f : S R is differentiable with 1 f = 0 for all x S, then f (a) = f (b) for all a, b S such that aj = bj for all
j = 1.
Proof. We prove by contrapos
1. Exercises from Sections 1.41.7
Problem 1. (Folland 1.5.8) If S Rn is an innite bounded set then S has an accumulation point
Let cfw_xk S be any sequence (it exists, since S is innite)
Proof.
By theorem 1.19 since S is bounded there is a convergent
1. Topology of Rn
Throughout, let S Rn . Lets recall some denitions from class.
Definition. These denitions are related to subsets of Rn
(1) S is called open if and only if x S, there exists
> 0 such that B( , x) S
c
(2) S is called closed if and only if
2.3
2.3.1
Bounds of sets of real numbers
Upper bounds of a set; the least upper bound (supremum)
Consider S a set of real numbers.
S is called bounded above if there is a number M so that any x S is
less than, or equal to, M : x M . The number M is called
Math 1920
Parameterization
Tricks
V2
Definitions
Surface
Pictures
Math 1920 Parameterization Tricks
Dr. Back
Nov. 3, 2009
Please Dont Rely on this File!
Math 1920
Parameterization
Tricks
V2
Definitions
In Math 1920 youre expected to work on these topics m
Review for Exam 3.
I
Sections 15.115.4, 15.6.
I
50 minutes.
I
5 problems, similar to homework problems.
I
No calculators, no notes, no books, no phones.
I
No green book needed.
Triple integral in spherical coordinates (Sect. 15.6).
Example
Use spherical
Triple integrals in Cartesian coordinates (Sect. 15.4)
I
Review: Triple integrals in arbitrary domains.
I
Examples: Changing the order of integration.
I
The average value of a function in a region in space.
I
Triple integrals in arbitrary domains.
Review: