Instructor |
Dragos Oprea
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Lectures: |
MWF 1:00-1:50PM, PETER 102 |
Course Assistants |
Michelle Bodnar
- Discussion: APM 7421,
Monday, 6:00-6:50PM
- Office: APM 6446
- Office
hour: Tuesday 3:30-4:30, Thursday 12-1.
- Email:
mbodnar at
ucsd dot edu.
|
Course Content |
Differentiation. Riemann integral. Sequences and series of
functions. Special functions. Fourier series. This corresponds to chapters
5-8 in Rudin's book.
|
Prerequisities: | Math 140A or permission of
instructor. Students will not receive credit for both Math 140 and
Math 142. Math 140
is a difficult and time consuming course, so enroll only if your course
load allows it. |
Grade Breakdown | The grade is computed as the
following
weighed average:
- Homework 20%, Midterm I 20%, Midterm II
20%,
Final Exam 40%.
|
Textbook: | W. Rudin, Principles of Mathematical
Analysis, Third Edition. |
Readings | Reading the sections of the textbook
corresponding to
the assigned homework exercises is considered part of the homework
assignment. You are responsible for material in the assigned reading
whether or not it is discussed in the lecture. It will be expected that
you read the assigned material in advance of each lecture. |
Homework |
Homework problems will be assigned on the
course
homework
page. There will be 7 problem sets, typically due on Wednesdays at
4:30PM
in
the TA's mailbox. The due date
of some of the problem sets may
change during the quarter depending on the pace at which we cover the
relevant topics. The best six problem sets will be used to compute the
final grade.
You may work together with your classmates on your
homework
and/or ask the TA (or myself) for
help on assigned homework problems. However, the work you turn in must be
your own. No late homework assignments will be accepted.
|
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Midterm Exams | There will be two midterm exams given
on January 28 and February 27. There will be no makeup
exams.
Regrading policy: graded exams will be handed back in
section.
Regrading is not possible after the exam leaves
the room.
|
Final Exam | The final examination will be held on
Friday, March 20, 11:30-2:30AM. There is no
make up final
examination. It is your responsability
to ensure that you do not have a schedule conflict during the final
examination; you should not enroll in this class if you cannot
sit for the final examination at its scheduled time. |
Announcements and
Dates |
- Monday, January 5: First lecture
- Monday, January 19: Martin Luther King Day.
No class.
- Wednesday, January 28: Midterm I
- Monday, February 16: Presidents' Day. No
class.
- Friday, February 27: Midterm II
- Friday, March 13: Last Lecture
- Friday, March 20: FINAL EXAM,
11:30-2:30.
|
Exams |
- Preparation for Midterm 1:
- Preparation for Midterm 2:
- Preparation for Final:
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Lecture Summaries | -
Lecture 1: Introduction. Definition of the derivative. Differentiable
functions are
continuous. Rules for differentiation.
- Lecture 2: Chain rule and proof. Extremal values and
derivatives.
- Lecture 3: Rolle, Cauchy and the mean value theorem.
Monotonic functions via derivatives. Example of a function with
discontinuous derivative.
- Lecture 4: Darboux functions. The derivative of a
differentiable function is
Darboux. L'Hopital's rule and proof in a particular case.
- Lecture 5: Proof of L'Hopital's rule. Taylor's theorem.
Motivation. Taylor polynomial.
Statement of the theorem.
- Lecture 6: Proof of Taylor's theorem. Integration. Lower and upper Darboux sums.
Definition of the integral. Riemann-Stieltjes integral.
- Lecture 7: Riemann's criterion for integrability. Refinements
of partitions and effects on upper and lower sums. Monotonic functions are
integrable.
- Lecture 8: Continuous functions are integrable. Functions with
finitely many discontinuities are integrable. Composition of continuous
and integrable is integrable.
- Lecture 9: Properties of the integral: sums, products,
inequalities. Change of variables formula for Riemann-Stieljes integral.
- Lecture 10: Relationship between integration and
differentiation. Fundamental theorem of calculus. Integration by parts.
Change of variables formula for Riemann integral.
- Lecture 11: Relation between Riemann-Stiltjes
integral and Riemann integral. Integration with respect to step functions.
The delta "function".
- Lecture 12: Sequences and series of functions. Pointwise
limits do not behave well with respect to
continuity, derivatives, integration. Uniform convergence and examples.
Cauchy criterion.
- Lecture 13: Uniform limit of continuous functions is
continuous. Supremum norm. The space of continuous bounded functions in
complete.
- Lecture 14: Uniform continuity and integration. Uniform
continuity and differentiation.
- Lecture 15: Series and uniform convergence. The
Weierstrass M-test. When can series be
integrated term by term and differentiated term by term.
- Lecture 16: Example of a
function which is continuous but nowhere differentiable.
When does a sequence posses an uniformly convergent
subsequence. Equicontinuity.
- Lecture 17: Connection between equicontinuity and uniform
convergence. Arzela-Ascoli theorem.
- Lecture 18: Proof that every continuous function over a
compact interval can be uniformly approximated by polynomials.
- Lecture 19: Algebras of functions. Examples: polynomials in
several variables, Fourier series, separation of variables. Algebras which separate points and vanish nowehere.
Uniform closure.
- Lecture 20: Algebras which sepate points and
vanishing nowehere: constructing functions with prescribed values
at two points. Proof of real version of Stone-Weierstrass.
- Lecture 21: Power series. Radius of convergence. Uniform
convergence within compact intervals. Power series can be differentiated
term by term. Continuity across endpoints/Abel's theorem.
- Lecture 22: Applications of power series: the exponential
function. Definition and properties.
- Lecture 23: Applications of power series: sine and
cosine. The number pi defined. Properties of the trigonometric functions.
- Lecture 24: Trigonometric polynomials. Fourier series. Fourier
coefficients of a function. Example. Statement of Parseval's theorem.
- Lecture 25: L^2 norm and L^2 convergence. The complex
exponentials form an orthogonal system for L^2. Least square
properties of Fourier partial sums.
- Lecture 26: Approximation of integrable functions by
continuous functions in L^2 norm, and eventually by trigonometric
polynomials. L^2 convergence of Fourier series. Proof of Parseval.
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