Math 210A (Mathematical Methods in Physics and Engineering I)

Instructor: Prof. Michael Holst (5739 AP&M, mholst@math.ucsd.edu)
Term: Fall 2015
Lecture: 1:00p-1:50p MWF, 5402 AP&M
Office Hours: 2:00p-2:50p M, 5739 AP&M
Class webpage: http://ccom.ucsd.edu/~mholst/teaching/ucsd/210a_f15/

Textbooks/References:
  • Required [DeMi90] L. Debnath and P. Mikusinski. Introduction to Hilbert Spaces with Applications. Academic Press, San Diego, CA, 1990.
  • Recommended [StHo10] I. Stakgold and M. Holst. Green's Functions and Boundary Value Problems. John Wiley & Sons, Inc., NY, NY, third edition, 2010.
  • Recommended [Hass06] S. Hassani. Mathematical Physics: A Modern Introduction to Its Foundations. Springer, NY, NY, 2006.
Printable Syllabus: Can be found [ here ].

Note Regarding the Three Editions of the Debnath-Mikusinski Book:
As you have probably discovered, there are three editions of the main book we are using. Since many of you have found a very inexpensive copy of the first edition, but a few of you did buy the third edition (its seems very few have the second edition), let me assure you that you can use any of the three editions. Here is a quick summary of the differences in the three editions:
  • First Edition: Core book consisting of these eight chapters:
    • Part I: Theory
      1. Normed Vector Spaces
      2. The Lebesgue Integral
      3. Hilbert Spaces and Orthonormal Systems
      4. Linear Operators on Hilbert Spaces
    • Part II: Applications
      1. Applications to Integral and Differential Equations
      2. Generalized Functions and Partial Differential Equations
      3. Mathematical Foundations of Quantum Mechanics
      4. Optimization Problems and Other Miscellaneous Applications
  • Second Edition: The primary change was the insertion of a new Chapter 8, pushing the existing Chapter 8 to Chapter 9:
    • Part II: Applications
      1. Wavelets and Wavelet Transforms
      2. Optimization Problems and Other Miscellaneous Applications
  • Third Edition: The primary changes are slight expansions of some of the sections; the most signficant change is probably the inclusion of a section in Chapter 6 on Sobolev spaces.

In order to allow you to use any of the three editions, we need to take care of two things:
  1. Homework numbering differences between the three editions. My homework problems will be of two types:
    • Problems that are self-contained in the HW PDF file.
    • Problems that are assigned from the first edition exercises. If you have one of the other two editions, in order to make sure we are all looking at the same problems, I will post a scanned copy of the set of exercises from the first edition with each homework. Since nearly all of these exercises from the first edition appear in all three editions (just with different numberings), this is just a matter of dealing with the number differences in each edition.
  2. We will talk about both Sobolev spaces (mainly 3rd edition) and wavelets (both 2nd and 3rd editions). You do not need the book for this material; my lecture notes will sufficient. I will make sure to assign problems on that material without refering to the third (or second) edition.



COURSE DESCRIPTION: The following is a brief description of the course. Note that the UCSD catalog currently has a very dated description of this course that was written a couple of decades ago. Our goal for the course remain the same as for the original course: provide graduate researchers in applied mathematics, physics, science, and engineering with a more sophisticated and powerful set of mathematical tools for performing their research. However, we will update some of the topics to reflect changes in the mathematical tools needed for modern science and engineering research; obvious examples being the growing importance of simulation algorithms over the last two decades, and the more recent emergence of data science techniques in all areas of science and engineering research. The prerequisites for the course remain the same as for the older description of the course.

210A. Mathematical Methods in Physics and Engineering (4)
Prerequisites: Math 20DEF, 140A/142A or consent of instructor.

PLANNED TOPICS FOR 210A (FALL 2015):
  1. Basic ideas of real analysis (Mathematical Background):
    Sets, fields, ordered fields, the real and complex fields. Equivalence relations, maps and functions, domain, range, injection, surjection, bijection, inverse. Convergent and Cauchy sequences of real numbers, pointwise convergence of sequences of real-valued functions of a single real variable. Derivatives of real-valued and vector-valued functions of several variables, gradient, Hessian, Jacobian.
  2. Normed spaces (Chap. 1, bit of Chap. 2):
    Vector spaces, norms, normed spaces, Banach spaces, Linear maps, linear functionals, the dual space, Banach Fixed-Point Theorem. Some examples involving the Lp spaces (Pass 1).
    • HW1, covering Topics 1-2, is posted [ here ]
    • For Reference, the Chapter 1 Exercises from First Edition are posted [ here ]
  3. Inner-Product spaces (Chap. 3, bit of Chap. 2):
    Inner-products, Inner-product spaces, Hilbert spaces, Schauder bases, orthonormal systems, Fourier series, reflexivity, separability. Orthogonal complement, Projection, Riesz Representation Theorem. Some examples involving the space L2 (Pass 1).
  4. Linear Operators (Chap. 4, bit of Chap. 6):
    Linear and Bilinear functionals, quadratic forms, linear operators, adjoints, invertibility, normality, isometric, unitary, positive, projection, Lax-Milgram Theorem. Some examples involving a general linear elliptic boundary value problem.
    • HW2, covering Topics 3-4, is posted [ here ]
    • For Reference, the Chapter 3 Exercises from First Edition are posted [ here ]
    • For Reference, the Chapter 4 Exercises from First Edition are posted [ here ]
  5. Some Tools from Nonlinear Analysis (Chap. 8 in 1st Ed. or Chap. 9 in 3rd Ed., and bit of Chap. 6):
    Local extrema of real-valued functionals on normed spaces. Gateaux and Frechet variations, differentials, and derivatives of general maps on normed spaces. Stationary points of functionals, 1st-order condition for local extrema, Euler-Lagrange equations. Minimization of quadratic functionals. The calculus of variations. The theory of Best Approximation in Banach spaces.
    • HW3, covering Topic 5, is posted [ here ]
    • For Reference, the Chapter 8 Exercises from First Edition are posted [ here ]
      NOTE: Hints for some of the homework problems above can be found in the back of each of the three editions of the book. The third edition contains more complete hints for some of the problems that appear in both books (which are the ones I tried to focus on), so you might want to borrow the third edition briefly if you have the first (or second) edition just to look at the hints.

PLANNED TOPICS FOR 210B (WINTER 2016):
  1. Some Additional Tools Needed for ODE and PDE (Chap. 2, bit of Chap. 4):
    Lebesgue measure and integral, Lp spaces (Pass 2). Closed and compact linear operators, Bounded and Unbounded Linear Operators, Spectral Theorem.
  2. Integral Equations and ODEs (Chap. 5):
    Fredholm alternative, Riesz-Schauder theory, Fredholm and Volterra integral equations, initial- (IVP) and boundary-value (BVP) problems in ordinary differential equations (ODE), Green's functions, Fourier Transform.
  3. Generalized Functions and PDEs (Chap. 6):
    Test functions, convergence of sequences of test functions, distributions, convergence in space of distributions, distributional derivative, multiplication by functions, weak solutions of PDE, Fourier transform, Sobolev spaces.
  4. Mathematical Foundations of Quantum Mechanics (Chap. 7)
    Classical mechanics, Poisson Brackets, postulates of quantum mechanics, Heisenberg Uncertainty Principle, Schroediner Equation and Picture, Heisenberg Equation and Picture, Interaction Picture, linear harmonic oscillator.
  5. Wavelets, Wavelet Transforms, and Related Topics (Chap. 8 in 3rd Ed.)
    Continuous wavelet transform, discrete wavelet transform, multiresolution analysis, orthonormal wavelet bases, wavlet examples.
PLANNED TOPICS FOR 210C (SPRING 2016):
  1. Topology
  2. Differentiable Manifolds
  3. Vector Fields, Tensor Fields, Vector Bundles
  4. Exterior Derivative, Lie Derivative
  5. Riemannian Manifolds, Covariant Derivative, Curvature
Applications appearing throughout the year: Linear and Nonlinear Ordinary Differential Equations (ODE), Partial Differential Equations (PDE), Integral Equations, Wavelets and related tools.


GRADES, HOMEWORKS, EXAMS, AND IMPORTANT DATES: Course information, such as the planned lecture topics for the week, any homework assignments, and so forth, will be maintained on this class webpage. Note that I sometimes make minor changes to both the lectures and homework assignments as the quarter progresses, based on how much I am able to cover in the lectures, and which directions we go based partially on the interests of the students. Therefore, CHECK THE WEBPAGE FREQUENTLY.

The course will be graded on attending most of the lectures, doing some homework assignments, and a final "take home" examination, according to the following guidelines:

Participation in Class (I.e., coming to most of the lectures): 25% of grade
Homeworks (3 written homeworks, each covering about 1/3 of the course): 25% of grade
Final Exam (take home exam) WHICH IS POSTED [ HERE ] 50% of grade

The "participation in class" part of the course is that I simply want you to try to come to most of the lectures; that will give you full marks on that part of the grade. (I will not actually take attendence; you are all adults.)

The "homeworks" will be based on the lectures, and will just give you the opportunity to use some of the tools we go over in the lecture. They will not be particularly time-consuming; they are for your benefit. Similar to the participation metric, if you make a good attempt on the homework then I will give you full marks. (I do not plan to formally grade the homeworks, but will try to give you feedback if you feel you need it.)

The final will be a "take home" exam, which will simply be some problems from the list of homework problems (that you will probably have already attempted during the quarter), plus just a few other problems you will not have seen. I know some people made travel plans right after the final exam period, so I will post this list of "take home final exam" problems to the class website by the middle of the 10th week, so that you have plenty of time to finish the take home exam by the time of the final. In any case, please turn in the take home exam to me before the end of finals week. During the actual scheduled time of the final exam for 210A (Monday December 7, 11:30a-2:30p), I will be in my office and can answer any last-minute questions about problems on the take home final. (I will also have some coffee available for anyone who comes by...)

Here are some important dates:

First lecture: FRI 9/25
Holiday: WED 11/11 (Veterans Day)
Last lecture: FRI 12/4
Finals week: MON-FRI, 12/7-12/11
Take Home Final Exam Due Date: FRI, 12/11 (by 3pm in my office)