MATH 213 Advanced Calculus I
A. What is this course about? Why is it different from everything you studied until now? What makes it difficult to most of the students?
This course should be better called Introduction to Mathematical Analysis;
We have adopted the textbook J.L. Taylor, Foundations of Analysis, Amer. Math. Soc. 2012, with some material from W.P. Wade, An Introduction to Analysis, 3rd Edition, Prentice-Hall, 2004, and W. Rudin, Principles of Mathematical Analysis, McGraw-Hill 1976, and some other sources.
I provide the PDF copies of my lecture notes with many exercises.
This is the
first opportunity to lay the foundations of a rigorous approach to
mathematical analysis. In the calculus course, students have been
trained mainly in how to handle derivatives and integrals, but
analysis is much more than this. We start with three axioms, the
Commutative Field Axiom, the Order Axiom, and the Completeness Axiom,
plus the basic objects and axioms from logic and set theory. From
here we develop an axiomatic
system and prove rigorously
many theorems that make this system.
Analysis
is a study of three combined structures: algebraic,
order and topological.
The algebraic structure is provided by the Commutative Field Axiom,
the order structure is provided by the Order Axiom, while the
topological structure is obtained by a combination of the first two.
The Completeness Axiom is the most elusive one.
The analytical ideas are present almost in all other domains of mathematics and the aim of this course is to give the basic results and concepts of the mathematical analysis that any mathematician will need under most of the circumstances.
In order to
fully understand mathematics, doing exercises is not sufficient, like
in the high school. You should understand definitions,
theorems, examples and counterexamples, and
at the last stage of education, the
proofs of the results. This does not
mean simply memorizing the definitions, the statements and the
proofs.
Definitions:
Analysis (as part of mathematics) uses mathematical objects that have
to be precisely defined. Understanding the definitions of these
objects is the first essential step in this enterprise. Do not stop
after being able to repeat (memorize) a definition: try
to fully understand the ideas behind it and have sufficiently many
relevant examples. Put these
definitions and examples in perspective, in connection with other
objects and results, try to ask yourself why are they needed (if they
really are!).
A very helpful exercise concerning definitions is to try to reformulate them with different expressions but keeping the meaning unchanged.
Statements
of the results (theorems, lemmas, propositions, corollaries,
remarks): A statement (the brick of
a mathematical theory) is generally in the form of a logical
implication: If a collection of properties A holds then some other
collection of properties B is true. In order to understand a result,
it is important to clearly separate
the hypotheses (the collection of
properties A) from the conclusions (the collection of properties B).
Many times, there are some subtle relations between the hypotheses
themselves, and/or between the hypotheses and conclusions. Always ask
yourself why the hypotheses are
needed (maybe they are not needed,
or at least not all of them!). At the end of the day, you should be
able to reproduce the statement from your memory, not necessarily
exactly as in the textbook, but the meaning should be the same. A
good exercise is to try to change the notations (use different
letters).
Proofs:
The ultimate goal of mathematics is
to provide correct proofs for the statements. This is the most
difficult part and it takes time and effort to get that skill.
Mathematics is different from all the other sciences in the sense
that it is a deductive and formal system. Roughly speaking, this
means that a statement will be accepted as a result only after at
least one convincing logically correct proof is obtained. A proof is
generally a sequence of logical implications. All the implications in
a proof should be checked; if at least one is not correct, the entire
construction will collapse (a chain
is as strong as its weakest link).
You have to be convinced by the
correctness of each step in a proof, with no exception.
You should have a clear understanding of all the steps in a proof,
why are they needed (maybe some of them are not, and the proof can be
simplified) and what are the relations between these steps. I am
trying to clearly depict the steps in a proof (especially for
complicated and longer proofs) during the lectures and, later, I
expect you should be able to do it by yourself.
Calculations:
The ability of doing calculations in
analysis is taught in Calculus (first year course). Do
not lose this ability and even more,
try to improve it. But there is something more required at the
present level: calculations hold under certain hypotheses
(assumptions) and each time you want
to do a calculation ask yourself whether the necessary assumptions
are met. If not, your calculations
may be formally correct but the final result may be wrong, or at
least not sufficiently supported by proven statements, which means
that you cannot rely on that final result. In doing correct
calculations you prove that you fully understood the results: if we
compare a calculation with a handwork, then the results and the
concepts that you know are your precious tools that you have to know
how to correctly use them in order to get your job well done.
Critical Thinking: Mathematics is by excellence a human activity where critical thinking is present all times. The most important question in mathematics is why. You have to ask yourself at each step why the operation/reasoning that you do is correct, and the answer should be given in terms of known concepts and results that you have checked by yourself. In a way, mathematicians are probably the most paranoic people, not trusting anybody and anything until checked, sometimes more than once.
How to study: Four hours of lecture should be accompanied by at least four hours of study at home, every week. This means reviewing the material taught in class, trying to understand all the details, doing exercises and homework assignments.
I use the following scale for
calculating the letter grades with respect to the overall calculated
grade (1st Term Exam 25%, 2nd Term Exam 25%, Final Exam 30%,
Homework 10%, Quiz 10%):
Points |
Letter-grade |
85-100 |
A |
80-84.9 |
A- |
75-79.9 |
B+ |
70-74.9 |
B |
65-69.9 |
B- |
60-64.9 |
C+ |
55-59.9 |
C |
50-54.9 |
C- |
45-49.9 |
D+ |
40-44.9 |
D |
0-39.9 |
F |
However, in the last years the passing grade has been lowered to 35 points.
What is covered in this course:
Basically mathematical analysis on the real line: Real Numbers, Supremum and Infimum, Transcendental Functions, Sequences and Subsequences, Limit Supremum and Limit Infimum, Cauchy Sequences, Numerical Series, Topology on R, Continuous Functions and Limits of Functions, Uniform Continuity and Uniform Convergence, Sequences and Series of Functions, Differentiable Functions, Mean Value Theorems and Lâ€™Hospitalâ€™s Rule, The Riemann Integral and the Darboux Integral, The Fundamental Theorem of Calculus, Applications of the Riemann Integral.
B. A Short Description of the Teaching Style
1. Teaching: The teaching is mainly by lectures, 4 hours every week. The material is presented and discussed by writing on the board but recently I started to mix this with video presentations of PDF files in AMS-LaTeX. Many times I initiate dialogues about the definitions, examples, statements and the proofs in an inquiry style.
Sometimes I digress to historical comments and to relations with questions coming from other domains of mathematics or even other disciplines. If students ask questions while teaching, the lecture can deviate from the original plan to a certain extent. However, the amount of material to be taught is rather large and I have to keep the pace to a reasonably high speed in order to cover the syllabus during the semester.
Some smaller results, or those that use similar ideas or techniques from the known results and proofs, are left as exercises. More difficult proofs are split in steps and the main ideas are pointed out. The statements that we study are analysed firstly in terms of the relations between the assumptions and the conclusions, and then at a deeper level.
2. Description of Assessments: Each exam has a duration of 2 hours, with extra 30 minutes if needed, is made of five questions, two of them routine/easier and the other of increasing difficulty, in which the emphasize is on reasoning, making the connection between the concepts and the results, some of them requiring imagination. Two questions from the exam are either in the exercises list of the lecture notes or closely related to some of these.
Sometimes, questions are split into subquestions that are meant to pave the way to a solution when the question requires imagination or may be too difficult. Questions may be related and be parts of a chain of statements that, taken individually, may be too difficult. Anyhow, I consider each exam as an opportunity for students to learn new things and open new perspectives.
Each homework assignment is made of five questions, similar to the structure of the exam and meant to be a preparation for the exams. Only two questions are graded, selected randomly. I do not provide answers to exercises leaving the students to try solving them by themselves. However, I encourage students to come to office hours and show to me their ideas/solutions and check the correctness, and I provide hints by asking appropriate questions.
A quiz has one or at most two related questions. If there are two questions, then the first one is a hint, or a small step, for the second one. The questions in a quiz are usually simpler than those in the homework assignments but there is a time constraint of 20 up to 30 minutes (if done online). Solutions of quizzes are provided or explained in class.
Aurelian Gheondea 2022 September 7th