Applied Analysis Preliminary Exam Syllabus
The analysis preliminary exam is designed to test students'
knowledge of basic concepts associated with the study of
functions of real variables and their ability to formulate
proofs of assertions about these basic concepts:
- Real numbers, infimum, supremum: Rudin 1.1-1.20, 1.23-1.38
- Real Line and Metric Space Topology: Rudin 2.1-2.42, 2.45-2.47
- Numerical Sequences and Series: Rudin 3.1-3.55, Buck 5.2
- Continuous Functions: Rudin 4.1-4.19, 4.25-4.33
- Differentiation: Rudin 5.1-5.19
- Riemann Integration: Rudin 6.1-6.9, 6.12-6.18, 6.20-6.27
- Sequences and Series of Functions: Rudin 7.1-7.26
- Power Series: Rudin 8.1-8.5
- Functions of Several Variables: Rudin 9.1-9.29, 9.39-9.42
Further required material is in the following recommended exercises. You may
need to consult other books to solve some of the exercises.
- Rudin Ch. 1 exercises 1-5, 8-19
- Rudin Ch. 2 exercises 1-16, 19-27, 29
- Rudin Ch. 3 exercises 1-18, 20, 21, 23-25
- Rudin Ch. 4 exercises 1-6, 8-18, 20-25
- Rudin Ch. 5 exercises 1-7, 9-14, 19, 20, 22-24
- Rudin Ch. 6 exercises 1-5
- Rudin Ch. 7 exercises 1-13,15-20,24
- Rudin Ch. 8 exercises 1-5,7-9
- Rudin Ch. 9 exercises 1-8,9 (for convex set), 11-16,20,27,30,31
The primary references are
We also recommend the following supplemental texts, which contain the
same topics covered from different vantages and some at a greater depth.
- Patrick M. Fitzpatrick, Advanced Calculus: A Course in Mathematical
Analysis, PWS, Boston, MA, 1996.
- Maxwell Rosenlicht, Introduction to Analysis, Dover, 1986.
- Georgi E. Shilov, Elementary Real and Complex Analysis,
(Ch. 1-9), Dover, revised edition, 1996.
- Herbert S. Gaskill and P. P. Narayanaswami, Elements of Real Analysis, Prentice Hall 1997.
Rudin's book is the best source, but many problems from it are
too long to be useful practice problems for the prelim
exam. Shorter problems from other references are more appropriate.
The student exam paper must be written in a way that the evaluators
can understand clearly the student's line of argument so that the
correctness of the argument can be decided.
Please do not write incorrect statements, hoping for partial credit. The evaluators will not fill
in missing arguments, make educated guesses about what the student had in
mind, or patch a solution from correct statements mixed with
incorrect ones.
The expected style of the solutions
is essentially the same as
as seen in graduate texbooks, in monographs, or in papers. In particular:
- The text and formulas, when read aloud, should form complete correct English sentences.
- Mathematical statements should be complete. There should be no fragments.
- It should be indicated how each statement is justified. For
example, if it follows from the preceding
statement words such as "so", "hence", "it follows that" should
be used. If it follows from something else, it should be explicity stated
from what it follows, such as a well-known theorem or an earlier statement.
If a statement is
assumed only temporarily,
e.g., to discuss one of several cases or to derive a contradiction, it should be indicated.
To prepare for the exam, students are
strongly advised to solve as many problems from the
recommended texts as possible, in particular the recommended exercises
above, try to find counterexamples to theorems when some assumptions are
dropped, and practice writing their solutions in the required style.
The classes
Math 5070 - Applied Analysis ,
Math 4310 - Introduction to Real Analysis I , and
Math 4320 - Introduction to Real Analysis II
can help with preparation for this exam.
However, students should note that these classes are not for PhD
students only, so the grading of students' work in these classes cannot
reflect the standards of mathematical accuracy and expression expected
on this exam. Further, these courses do not cover all of
the required material.
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