MATHEMATICS 602: Mathematical Principles in Science II

WI 2012, sections 9:30 a.m. and 2:30 p.m. MWF

**Instructor:** Dr. Rodica D. Costin,

**Office:** 436 Math Tower

**Office hours:** Monday, Friday 10:30-11:20 a.m. or
by appointment

**e-mail:** costin dot 10

Wed.
Jan 4:
Review of main concepts of Math 601 needed this quarter.

Friday Jan. 5: Sections 2.1,..., up to, and including 2.5
in the notes Chapter I.
Self-adjoint Matrices posted on Carmen.

Please review the notion of iner product and examples.

Monday
Jan. 9: Sections
2.7, 2.8, 2.9

Homework 1 (due Wednesday Jan. 18, since there are no classes on Monday Jan. 16)

All
answers must be justified. Quote a theorem and
state why it applies, or give an thorough argument (a proof).
Include all calculations, or attach a computer output to your
explanations. Even when the question askes for a yes or no answer,
justification is needed: if it is true give a proof or quote a
theorem and why it applies, or, if it is false, give a
counterexample, or show that the object does not satisfy all the
conditions of a definition etc.

An answer alone does not earn full credit!

Note. If a problem is
stated in n dimensions, you may wish start thinking about the
problem in 2 dimensions, then in three dimensions, to get a
feeling on what is going on. But a full solution to your problem
must be worked out in n dimensions.

Wed.
Jan. 11: We
finished Sec. 2.9
, then did Sec. 2.10, and from 3.1 up to Theorem 25 (including).

Fri.
Jan. 13: Sec. 3.1, 3.2, we reviewed the binomial
and multinomial formulas.

Homework 2 (due Monday Jan. 23)

Wed.
Jan. 18: Sec. 3.3, 3.4, 3.6, 3.9

Friday
Jan. 20: Sec. 3.8.1, 3.8.2 (but we did not
discuss logs and radicals yet)

Monday Jan. 23: Sections 3.8.2 continued, started 4.1 (definition of a quadratic form)

Homework 3 (due Monday Jan. 30)

Wednesday
Jan. 25: Section
4.1

My office hours will run Mon, Fri
from 10:40 a.m. until noon from now on (for those of
you who have a class at 10:30)

Friday
Jan. 27: Sections
4.1 (finished), 4.2

Homework 4 (due Monday Feb. 6)

Monday
Jan. 30: Sections
4.3, 4.4

Wednesday
Feb. 1: Sections
4.5, 4.6

Friday
Feb. 3: Sections
4.7, 4.8

Homework
5 ( not due
Monday)
New: due Friday Feb.
17 (by popular demand)

Monday
Feb. 6: Sections
5.1, 5.2

Wednesday
Feb. 8: Sections
5.3

Friday
Feb. 10: Sections
5.4, and we started SVD: the adjoint of a rectangular matrix,
that MM* and M*M are positive semidefinite

and have the same nonzero eigenvalues, the statement of Theorem
1 (p. 3) (The SVD decomposition).

The
SVD song. Thank you,
Mark!

Monday
Feb. 13: SDV

Wednesday
Feb. 15: Pseudoinverse.
Review of vector spaces, inner products. Countabe sets.

Homework 6 due Friday, Feb. 24

Monday Feb. 20: Hilbert Spaces Sections 1.5, 2.1, 2.2, 2.3, started 2.4

Wednesday Feb. 22: ended section 2 (closed sets, closure, dense sets, examples)

Friday Feb. 24: Hilbert Spaces Sections 3.2, 3.3, 3.5, 3.6

Homework 7 due Friday, March 2

Monday Feb. 27: Hilbert Spaces Sections 3.7, 4.1, and from 4.2 only Definition 36 and Theorem 37.

Wednesday Feb. 29: Theorem 38, Sec. 4.3, 4.4, 4.5.1

Friday March 2: Sec. 4.5.1 From Sturm-Liouville notes: Sec. 2.2, Lemma 1 of page 9, started Theorem 2 and Sec. 2.3

Homework 8 due Friday, March 9

Monday March 5: From Sturm-Liouville notes: Sec. 2.3, 2.4, 2.5, 3.1, 3.2, Abel's Theorem, stated that eigenvalues form an increasing sequence, with limit plus infinity.

Wednesday March 7: From Sturm-Liouville notes: Completeness of the set of egenfunctions (Theorem 4), and from 8.1 finding eigenvalues and eigenfunctios using the Raileigh's quotient, intelacing of zeroes of the eigenfunctions, Sec. 6.1, Legendre polynomials as eigenfunctions of a Sturm-Liouville problem and interlacing of their zeroes, statement Sturm's Oscillation Theorems see, periodic problem for

y''+lambda y=0 and Fourier series (Theorem 5 on p. 12)

Friday March 9: Solving PDEs by separation of variables Sec 1.1, 1.2, 1.3, 9.1, 9.2, 9.3 (we will probably not finish them all... to be continued in Math 603).

Final exam information:

Come to my office MW 436 to pick up a copy of your final exam:

Students in the 9:30 class - Monday March 12: from 10:30 a.m. to 11:40 a.m.

Students in the 2:30 class - Tuesday March 13: from 12:30 p.m. to 1:40 p.m.

If these times do not work for you please let me know.

The exam is due back according to the official university schedule.

Final examination schedule

Return you exam to my office MW 436

for the 9:30 class Wed, March 14, 9:30 AM - 11:18 AM

for the 2:30 class Thu, March 15, 1:30 PM - 3:18 PM