18.100A - Fall 2018
18.100A Syllabus Fall 2018
(Adobe Acrobat -- pdf file)
18.100A Information and Rules Fall 2018
( Adobe Acrobat -- pdf file )
Lecturer: Arthur Mattuck
Office: 2-383 Tel: (617-25)3-4345
E-mail: mattuck@mit.edu
Office Hours: Thurs. 3-4; or after class MWF. You can also use
e-mail for questions, including weekends; it's usually checked
several times daily.
TA's: Minjae Park 2-341A; Office hour Wed. 4-5 in 2-341; minj@mit.edu
Yichi Zhang 2-239A; Office hour Thurs. 4-5 in 2-361; yichi@mit.edu
\newline\in
If you collaborate, list collaborators, and write up solutions
independently. Do not consult solutions from previous semesters or on
the internet.
Problem Set 12 (optional, not handed in or graded)
( Adobe Acrobat -- pdf file )
Problem Set 11, due Fri. Dec. 7 (complete)
( Adobe Acrobat -- pdf file )
Problem Set 10, due Fri. Nov. 30 (complete)
( Adobe Acrobat -- pdf file )
Problem Set 9, due Mon. Nov. 19 (complete)
( Adobe Acrobat -- pdf file )
Problem Set 8, due Fri. Nov. 9 (complete)
( Adobe Acrobat -- pdf file )
Problem Set 7, due Fri. Nov. 2 (complete)
( Adobe Acrobat -- pdf file )
Problem Set 6, due Fri. Oct. 19 (complete)
( Adobe Acrobat -- pdf file )
Problem Set 5, due Fri. Oct. 12
( Adobe Acrobat -- pdf file )
Problem Set 4, due Fri. Oct. 5) (complete)
( Adobe Acrobat -- pdf file )
Problem Set 3, due Fri. Sept. 28)
( Adobe Acrobat -- pdf file )
Problem Set 2, due Mon. Sept. 24 (complete))
( Adobe Acrobat -- pdf file )
Problem Set 1, due Fri. Sept. 14 (complete)
( Adobe Acrobat -- pdf file )
Problem Set 0, due Fri. Sept. 7
( Adobe Acrobat -- pdf file )
18.100A Description
This course is an introduction to devising mathematical proofs
and learning to write them up. It is primarily for students with
no prior experience with this.
The class usually contains
students from years 2,3,4,and G (grad students) -- about equal numbers
of each. Sometimes freshmen
also take it.
Graduate students (not in Course 18 (math)) should register for it using
the number 18.1001 to get graduate credit for the course. The two
numbers both represent the same subject, with one room, one set of rules
and assignments.
Students in the class are
from different
majors -- about 1/4 math majors, the rest
have been in various types of science and engineering majors, also
economics and management.
The subject matter for the first
2/3 of the syllabus is the proofs of one-variable
calculus theorems and arguments which use these theorems.
The
emphasis is on the use of inequalities to do estimation and approximation, two basic tools of
analysis.
It is assumed that students know ordinary calculus
fairly well, or once knew it and
will review it when they need to.
Calculus is used from the beginning as a
source of examples.
The last third goes beyond calculus, getting into uniform
convergence of series of functions, to justify differentiation
and integration of infinite series term-by-term.
There is similar work involving
integrals depending on a parameter, to justify differentiating
under the integral sign with respect to the parameter.
(Differentiating the Laplace transform F(s) = L(f(t)) with
respect to the s-variable is an example.)
Toward the end, there is a brief
introduction to point-set topology (18.100B and Q begin with this), which is used in upper-level courses
having an analysis prerequisite.
The textbook has a very brief
introduction to sets of measure zero and the Lebesgue integral which
may be covered at the end of the semester, depending on time available
and student interest.
In the Spring semester there are two versions of 18.100A with
different teachers: 18.100A (like the fall subject), and 18.100P,
which has a recitation slanted toward writing mathematics for various
levels, gets 15 units of credit, and can be used in satisfying
MIT's Communication Requirement.
18.100A or 18.100B?
The best way of getting a feeling for the difference
between 18.100A and 18.100B is to look at the corresponding
textbooks.
The book for 18.100B is Rudin:
"Principles of Mathematical Analysis", available for inspection at
the Kendall Square Tech Coop; or on Amazon for sample pages.
The book for 18.100A is described below; it is available
in an inexpensive printing at the Tech Coop or online at Quantum Books or Amazon.
In content
it follows the 18.100A description above closely.
Three links are given below, so you can get some idea of its
style, what's in it, and its rigorous but hopefully simplified
approach to limits, as described toward the end of the Preface
and in Chapter 3.
18.100A Textbook: Mattuck --- Introduction to Analysis
(CreateSpace, 2013; available on Amazon), (previously pub. by
Pearson/Prentice-Hall div., 1999) Website: Introduction to Analysis
Its five-page Preface can give some
idea of the book's origins and
its (hopefully) user-friendly style: Preface .
The Table of Contents tells you what's in it, and what's not, and
the order of topics: Table of Contents .
Leafing through some of the early chapters of this book and of
Rudin will give you a further idea of the differences in style
and content between the A and B versions of Introductory Analysis.
Here are a few sections from Chapters 1-3 -- about 15 pages in
all --
to give you a sample of the writing style:
Sample sections.
Chapter 1: Real Numbers and Monotone Sequences
Chapter 2: Estimations and Approximations
Chapter 3: The Limit of a Sequence
Corrections to the first seven printings by Pearson are on the book's website:
List of Corrections.
The eighth printing by CreateSpace is the same book as the earlier
printings, with the following changes:
--- All significant corrections have been incorporated;
--- A three-page Appendix F has been added;
--- The book's current price on Amazon is $15.
Helpful features of the book: These are described
briefly on the Amazon website for the book (Google: Mattuck
Introduction to Analysis and select the topmost link) or in more
detail in the "Preface" link given earlier.
On the Amazon website mentioned above, there are
several reviews that you can read.
It also gives a description for students,
instructors, and general readers of what's
in the book and who it's for.
You can also find this description at
Introduction to Analysis .
Posted on the book's website (see the link just given) are the first
three chapters and Appendix A from the 8th printing,
for the convenience of students who will not have a copy of the book
before the first few class meetings.
What's below is
more MIT-specific about how the subject will be run
this semester.
Further details are given in the "Information
and Rules" handout (see link to it at the beginning of this page).
Fall 2018 Information
Exams: There will be one midterm exam and a 3-hour final, both
open book, but no electronics or other material can be used.
Assignments:
This semester there will be once weekly problem sets, posted here after class on Fridays, due
the next Friday (or the following Monday if Friday is a holiday).
They will be graded and returned the following Wed. There are several
exceptions -- the Syllabus gives the detailed schedule.
The problem set is based on the classes for Fri., Mon., and Wed., and
gives the reading assignment for each class, and the relevant problems
based on it.
Since the homework is
really where
the learning takes place, and timely feedback is essential
to improving, handing in 3/4 of the assignments when they are
due is a requirement for passing; any exceptions have to be
for cause, and arranged in advance. Students who are accepted
into the class late have to make up the missed assigments.
The textbook is by and large an adequate substitute for class
attendance; students in the past have found it sufficiently
clear. A few just read the book, get the assignments online
here, hand them in and retrieve their graded assignments in
class, just before it starts or after it ends. Others
slip the homework under
my door before class, retrieving the returned homework
from a box outside my door. See the link near the top
"Information and Rules" for more details.
Subject Prerequisites: The Subject Catalog says Calculus I and II;
or 18.014 (Calculus I with Theory) and Calculus II (Corequisite:
taken simultaneously with 18.100A). This latter alternative is
incorrect for 18.100A: since 18.014 and 18.100A have similar material,
students who have had 18.014 should not also take 18.100A; they should
take 18.100B or Q instead. Further explanation is given below.
Upperclassmen (i.e., students in years 2,3,4,G) will almost
always have credit for Calculus I and II, and will
have had at least one or more additional math subjects, like Differential
Equations, Linear Algebra, Complex Variable, Probability, Statistics,
or an introductory Discrete Math subject like 18.062J. Any of these
additional subjects provides "mathematical maturity", which helps with
18.100A, but they are not prerequisites.
Even the requirement of Calculus II
is primarily for mathematical
maturity; almost nothing is used in 18.100A except what's in
Calculus I and high-school mathematics; only in the last two weeks
is reference made to Calculus II (vector and double integral topics) and
Differential Equations (the Laplace transform topic, in ODE courses that include it).
Freshmen taking Calculus II with a standard AP BC Calculus I background and
a desire to see and make up proofs can therefore also take 18.100A instead of 18.014 (Calculus I with Theory).
18.100A and 18.014 are similar;
the difference is that 18.100A assumes Calculus I, but 18.014 does not; the corresponding textbooks are written to reflect this difference.
From the above, it follows that students (at all levels) with a good
background in proofs (from summer "math camps" for instance, or
a proof-oriented "honors" Calculus course in high school, or 18.014,
or more advanced proof-oriented math subjects) should study Analysis in the 18.100B or Q versions.
Book Prerequisites: The textbook assumes knowledge of
standard K-12 mathematics, plus one-variable
calculus -- differentations and easy integrations (including the natural
log, exp, and trig functions), with standard applications to
finding rates, maximum points, areas, and simple volumes, plus an
intuitive idea of what a limit is. These things are used from
the beginning to provide examples, but the aim of the book is to
describe the theory behind all of this -- to teach you how to read
mathematical proofs and how to find and write them down yourself,
using real analysis as the
mathematical material.
Return to
Arthur Mattuck's Homepage
Return to
MIT Math Department Homepage