DEPARTMENT OF PHYSICS AND ASTRONOMY -- (202) 806-6245 (main office), -5830 (fax)

Math. Methods in Physics/Intro. to Mathematical Physics II
PHYS-193 (16225) / PHYS-217 (16125): MW: 8:40 -10:00, in TKH #103
[Topics][Daily Schedule][Minimal Requirements][Assignments][e-Gear][Welcome]

Instructor: Tristan Hübsch (Office hours: MW 10:00 am–12:00 noon, T 2:00–4:00 pm)

TKH 213, 806-6267 thubsch@mac.com, thubsch@mac.com

Textbook (required): G. Arfken, Mathematical Methods for Physicists (6th ed. available by Fall 2005)

(optional): J. Mathews & R.L. Walker, Mathematical Methods of Physics

(optional): F.W. Byron & R.W. Fuller, Mathematics of Classical and Quantum Physics

Grading recipe (no make-ups are given, except in cases of proven medical emergency):

Component	Time	Remark	% of Grade
Homework	See in daily schedule	Late HW = 0 credit !!!	20%
Classwork/Quizzes	2-3/week	current material	20%
Exams (two midterms)	See in daily schedule	not comprehensive	(each) 20%
Now drop the one worst component (for each student individually).
Final exam	Last week of semester	comprehensive	40%

The aim of the course is to introduce, develop and discuss various methods of solving first, second and higher order ordinary and partial differential equations subject to a variety of boundary, initial and final conditions. The emphasis is always on the adaptation of the standard mathematical methods and techniques to their application in solving typical problems in physics and engineering. Applications range from classical mechanics, hydrodynamics, electromagnetism and statistical physics through the quantum counterparts of these fields, and various engineering applications of these physics models, but also some less obviously related models such as variants of the predator-prey model in social economy and marketing.

“Success = 1% inspiration + 99% perspiration”--T.A. Edison
But, learning is still 100% learning!

  §9: Differential Equations
§10: Sturm-Liouville Theory (Orthogonal Functions)
1st Midterm exam--§8+9 02/02 (open text, in-class) + take-home due 02/09
§11: Bessel Functions
§12: Legendre Functions
§13: Special Functions
§14: Fourier Series
2nd Midterm exam--§11-14 03/30 (open text, in-class) + take-home due 04/02
§15: Integral Transforms
§16: Integral Equations
§17: Calculus of Variations
Final exam--comprehensive: (almost) everything, given 04/15, due 04/22.

01/07: Partial differential equations; first order differential equations, § 9.1–2
01/12: Separation of variables and singularities, § 9.3–4
01/14: Series solutions--Frobenius’ method, § 9.5
01/19: Observed holiday: Martin Luther King’s birthday
01/21: A Second solution, non-homogeneous equation and Green’s function, § 9.6–7 [HW1 due]
01/26: Self-adjoint differential operators and equations, § 10.1–2
01/28: Orthogonalization and completeness of eigenfunctions, § 10.3–5 [HW2 due]
02/02: 1st Midterm Exam: Chapters 8, 9; 1-hour in-class, open-book + take-home due 02/09
02/04: Bessel functions of the first and second kind, §11.1–4
02/09: Modified and spherical Bessel functions and asymptotics, § 11.5–7 + more [HW3 due]
02/11: Legendre functions, § 12.1–4
02/16: Observed Holiday: President’s day
02/18: Spherical harmonics and angular momentum, §12.5–7 [HW4 due]
02/23: Products and integrals with spherical harmonics, § 12.8–11
02/25: Hermite and Laguerre functions, § 13.1–2
03/02: Orthogonal polynomials and hypergeometric functions, § 13.3–4 [HW5 due]
03/04: Confluent hypergeometric functions, § 13.5
03/09: General properties and roots of the Fourier transform, § 14.1 [HW6 due]
03/11: Advantages and application of Fourier series, § 14.2-14.3
--Spring recess: March 14th, close of classes, through March 22nd, 8:00 AM
03/23: Functional properties of the Fourier transform, § 14.4 [HW7 due]
03/25: Gibbs Phenomenon, discrete Fourier transform, aliasing and music, § 14.5–6 + more
03/30: 2nd Midterm Exam: Chapters 11-14; 1-hour in-class, open-book + take-home due 04/02
04/01: The Fourier integral transform and its functional properties, § 15.1–7 [HW8 due]
04/06: Laplace transform, § 15.8-15.12
04/08: Integral equations, transforms and generating functions, § 16.1-16.2
04/13: Neumann series and separable kernels, § 16.3 [HW9 due]
04/15: Hilbert-Schmidt theory, § 16.4
           Final Exam handed out, due 04/22
04/20: Calculus of variations--many independent variables, § 17.1–6 [HW10 due]
04/22: Constrained and Rayleigh-Ritz variation, § 17.7-17.8

To pass the course with a grade B or better, a graduate Student must by the time of the final exam be able to demonstrate the ability to:

1. recognize and solve any separable first order ordinary differential equation;
2. solve any linear ordinary differential equation with rational polynomial coefficients by means of power series and verify convergence;
3. if possible, separate a linear partial differential equation into a system of ordinary differential equations -- in Cartesian, spherical and cylindrical coordinates;
4. determine the orthogonality relation for the eigenfunctions of any Sturm-Liouville second order differential operator, and orthonormalize any given collection of functions with respect to a given (integral) scalar product;
5. write down the general expression for the Green’s function for any Sturm-Liouville second order differential operator in terms of its eigenfunctions, and solve inhomogeneous differential equations in terms of the Green’s function;
6. expand any function into a complete set of orthogonal functions and determine the coefficients in the expansion;
7. solve a linear system of differential equations by converting it into a system of algebraic equations through an integral transform.

A graduate student who cannot demonstrate the above listed skills by the time of the final exam automatically forfeits a grade of B or better -- regardless of the total number of points acquired in homework, quizzes and exams.

Homework assignments (93 problems)

1. Due 01/21: 9.2.5, 9.2.13, 9.3.4, 9.3.6, 9.3.10, 9.4.1, 9.4.3, 9.5.5, 9.5.6, 9.5.12
2. Due 01/28: 9.6.15, 9.6.19, 9.6.26, 9.7.3, 9.7.8, 9.7.10, 10.1.1, 10.1.11, 10.2.3, 10.2.9
3. Due 02/09: 11.1.6, 11.1.15, 11.1.25, 11.2.6, 11.2.7, 11.2.9, 11.3.5, 11.3.6, 11.4.1
4. Due 02/18: 11.5.2, 11.6.2, 11.7.10, 12.1.1, 12.1.6, 12.2.2, 12.2.5, 12.3.2, 12.3.4, 12.4.4
5. Due 03/02: 12.5.2, 12.5.11, 12.6.1, 12.6.4, 12.7.1, 12.8.2, 12.9.3, 12.10.4
6. Due 03/09: 13.1.1, 13.1.11, 13.1.13, 13.2.3, 13.2.7, 13.2.9, 13.4.1, 13.4.5
7. Due 03/23: 14.1.2, 14.1.3, 14.1.5, 14.1.6, 14.1.7, 14.2.3, 14.3.1, 14.3.2, 14.3.4, 14.3.13
8. Due 04/01: 14.4.4, 14.4.5, 14.4.9, 14.4.10, 14.4.11, 14.5.1, 14.5.2, 14.6.1
9. Due 04/13: 15.1.2, 15.3.2, 15.3.3, 15.3.5, 13.5.6, 15.8.4, 15.9.3, 15.10.1, 15.10.3, 15.12.4
10. Due 04/20: 16.1.1, 16.1.2, 16.1.4, 16.2.9, 16.2.10, 16.3.1, 16.3.2, 16.3.7, 16.3.12, 16.4.2 (bonus)

All homework assignments are due by 5:00 PM of the day indicated and should be either given to the instructor in hand, left in the instructor's mailbox in TKH#105, or slid under the instructor's office door, TKH#213. Late homework will not be accepted, except in cases of proven (medical) emergency.

Collaboration policy

Collaboration -- but not blind copying -- on the homework assignments is strongly encouraged; students should use this to learn from each other. All exams and quizzes are open text and open class-notes (including notebooks and class handouts), but no collaboration is allowed; by signing the exams and quizzes, the student implicitly agrees to abide by this policy. Violation of this policy is covered under University regulations on academic dishonesty and cheating.

Presentation and organization

While a neat presentation of home,- quiz- and exam-work is not required for full credit, it certainly makes it easier to assess the quality of the work and give the proper credit due. In all cases, include a simple sketch if it might help conveying the approach or the calculations. Where necessary, include all units and symbols such as the measure of an integral, arrow on a vector, vertical bars for the absolute value of a quantity, for the magnitude of a vector or for the determinant of a matrix, etc.

• Homework: Submitted answer sheets should be stapled (preferably in the top-left corner). The Student's full name (and ID number) should be indicated on the first (cover) page. Solutions should be labeled and ordered consecutively by the problem numbers. Include all calculation/work and give full reference to any result quoted from any source other than the text, and page/equation number for results quoted from the text. Homework assignments will be graded by a grader with whom unsatisfactory credit should be disputed prior to appealing to the instructor; the instructor may override the grader's assessment.
• Quizzes: The Student's full name (and ID number) should be indicated on the first (cover) page, unless otherwise requested at the time of the quiz. Include all calculation/work and give the page/equation number for results quoted from the text. Speed and efficiency is important, but the presentation must be legible. If submitted under a pseudonym, the quizzes will be pre-graded by a classmate; the final grade will be given by the instructor. The quizzes will be returned to the individual students.
• Mid-term exams: All exam questions are given on a question sheet and include the detailed point distribution and the place to write in the Student's full name and ID number. This question sheet is to be stapled on the top of the answer sheets, in the top-left corner. It will be used to indicate the acquired points, part by part and in total, the average score and the standard deviation. A second copy of the question sheet is to be used for the take-home part in the same manner. The formula by which the points will be summed between the to parts will be given on the question sheet. The mid-term exams will be returned to the individual students.
• Final exam: All exam questions are given on a question sheet and include the detailed point distribution and the place to write in the Student's full name and ID number. This question sheet is to be stapled on the top of the answer sheets, in the top-left corner. It will be used to indicate the acquired points, part by part and in total, the average score and the standard deviation. The final exams are stored by the instructor, but are available to the individual student for review after the exam.

ADA disclaimer

Howard University is committed to providing an educational environment that is accessible to all students.  In accordance with this policy, students in need of accommodations due to a disability should contact the Assistant Dean for Student Affairs and Records, Denise L. Spriggs (202) 806-8006), for verification and determination of reasonable accommodations as soon as possible after admission to the Law School, or at the beginning of each semester.

© Tristan Hübsch, 2009