Advanced Calculus Page

General Presentation

A few words about Analysis:


    This is the first opportunity to lay the foundation of a rigorous approach to 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 four postulates, plus the basic objects and axioms from the set theory. From here we develop an axiomatic system and prove rigorously all the theorems that make this system.

    Analysis is about combined structures: algebraic, order and topological. 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 mathematical objects of analysis that any mathematician will need under most of the circumstances.

Homework: 


It is  extremely important to perform weekly the homework (normally, five questions  from the textbook). Additionally, we recommend to all students (at least try) to solve all the other questions in the textbook. There is no other way to understand mathematics except  doing it.

Understanding mathematics:

In order to fully understand mathematics, doing exercises is not sufficient, like in the high school. You shuold 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 and have sufficiently many relevant examples. Put these defintions and examples in perspective, in connection with other objects and results; try to ask yourself why are are they needed (if they really are!).

Statements of the results (theorems, lemmas, propositions, remarks): A statement (the bricks of a mathematical theory) is generally in the form of a logical implication: If a collection of properties A holds then some other colllection 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 ideas 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 colapse (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) durng the lectures and you should learn how to do it by yourself later.

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 argumented 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 that you know are your precious tools that you have to know how to correctly use in order to get your job well done.

Grading

   We use the following scale for calculating the letter grades with respect to the overall calculated grade (1st Midterm 20%, 2nd Midterm 20%, Final 30%, Homework 20%, Quiz/Attendance 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

Midterms:

There will be two midterms and a final written examination, acknowledged in advance.


What is covered in this course:

I. ONE-DIMENSIONAL THEORY (Fall Semester)
  1. The real number system: the four postulates, functions, countability, and the algebra of sets;
  2. Sequences in R: limits of sequences, limit theorem, the Bolzano-Weierstrass Theorem, fundamental sequences, limits supremum and limits infimum;
  3. Limits of functions: two-sided limits, one-sided limits, continuity, and uniform continuity;
  4. Differentiability: derivatives, differentiability theorems, mean value theorems, monotone functions and the derivative of an inverse function;
  5. Integrability: the Riemann integral, Riemann sums, the Fundamental Theorem of Calculus, improper integrals, functions of bounded variation, and convex functions;
  6. Series of numbers: convergence, series with nonnegative terms, absolute convergence, alternating series, estimation of series, tests for convergence;
  7. Series of functions: uniform convergence, power series, Taylor expansions, analytic functions;
  8. Fourier series:  summability of Fourier series, growth of Fourier coefficients, convergence of Fourier series.

II. MULTI-DIMENSIONAL THEORY (Spring Semester) Syllabus
  1. Euclidean spaces: algebraic structure, subspaces and linear transformations, topology ( interior, closure, boundary, connectedness, and path connectedness).
  2. Convergence in Rn : limits of sequences, limits of functions, continuous functions, compact sets, Dini's Theorem, Lebesgue characterization of Riemann integrability, Closed Graph Theorem.
  3. Differentiability in Rn : partial derivatives and partial integrals, differentiability, derivatives and tangent planes, the Chain Rule, the Mean Value Theorem, the Taylor's Formula, the Inverse Function Theorem.
  4. Integration in Rn : Jordan regions, Riemann integration on Jordan regions, iterated integrals, change of variables, partitions of unity, the Gamma function.

  5. Vector Calculus: Curves, oriented curves, surfaces, oriented surfaces, Stokes type theorems.