M 672 Applied Mathematics

Andrej V. Cherkaev, professor
Department of Mathematics
University of Utah
Salt Lake City, UT 84112
USA
Email: cherk@math.utah.edu
Tel : +1 801 - 581 6822
Fax : +1 801 - 581 4148

Attention! The final class schedule is: 11:00 MWF, in WBB 707.

TEXT: Principles of Applied Mathematics

by James P. Keener.

Syllabus: additional references:

Material will be covered this quarter that roughly corresponds to chapters 4, 5, and 7 of text book.

-- Calculus of variations

Chapter 5 of text book plus instructor's notes; also I would recommend any of Dover books:

• Calculus of Variations, by Robert Weinstock,
• The variational principles of mechanics, by Cornelius Lanczos
• Calculus of Variations with Applications, by George M. Ewing

(actually, there are tons of good and affortable books for this classical stuff).

--Distributions and Greens function

Chapter 4 of text book, also I would recommend an excellent text book:
• Greens Functions and Boundary Value Problems by Ivar Stakgold; John Wiley, 1979. chapters 0-3, 5, 7, 8. (one copy will be placed in math. library for references and for work in the library only),

-- Spectrum theory of differential operators and PDE

(we will start PDE material this quarter if time permits)

Chapters 7, 8 of text book, the mentioned book by Ivar Stakgold.

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Syllabus: the plan.

We start with calculus of variations, because (i) it is a self consistent branch; (ii) it provides motivations for the distribution theory, for spectrum theory, for PDE, etc; (iii) it is my favorite subject.

Then, we will go through theory of distributions and spectrum theory towards partial differential equations.

I expect that everybody will attend a math seminar during the quarter. In the end of the quarter everybody submits a report about one-two talks.

Grading policy:

The grade will be based entirely on homework assignments and on the report after a lecture in a mathematical seminar that you will attend during this quarter.

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Homework assignment for chapter 5:

Section 5.1 ## 1a, 1c, 2, 5, 8, 10.

Section 5.2 ## 5, 6

Section 5.3 ## 1, 3, 6.

Section 5.4 ## 1 a, 5, 7-8.

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Homework assignment for chapter 4:

Section 4.1 ## 2, 6, 7, 12.

Section 4.2 ## 2, 4, 7

Section 4.3 ## 2, 11.

Section 4.4 ## 1, 3, 7.

Section 4.5 ## 1, 3, 9, 13.

Consider the a delta-sequence and its Fourier transform. Find the limit of this transform.

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Homework assignment for chapter 6:

Section 6.1 ## 3, 4.

Section 6.2 ## 8, 11

Section 6.4 ## 2, 3, 6, 15.

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Homework assignment for chapter 7:

Section 7.2, ## 1, 2, 7

Section 7.3, ## 1 d, 3, 5, 13

Section 7.4, # 1

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Notes and comments to lectures:

Calculus of variation part: topics (chapter 5 of the text and some extras)

Variational problems with non-convex integrants. Relaxation.