"The physicist in preparing for his work needs three things: mathematics, mathematics and mathematics." -- Wilhelm Roentgen, The Mathematical Gazette, Volume 22, Number 252, December 1938, 1225

**Coronavirus Impact:**Masks are required in this course. This masking policy is subject to change during the semester, and any changes will be posted clearly here and in Canvas announcements. You should also be prepared to go fully online if the need should arise.**Lecturer:**Professor Randall J. Scalise**Meeting time and place:**T,Th 11:00AM-12:20PM Anette Caldwell Simmons Hall 213**Office hours:**after lecture and by appointment.**Contact:**- Call or leave a message at 768-2504, or
- send me e-mail: <scalise@smu.edu>

**Prerequisites:**- 4321: Prerequisites or corequisites: MATH 3302 (formerly MATH 2339 prior to Fall 2017), MATH 3313 (formerly MATH 2343 prior to Fall 2017).
- 7305: Working knowledge of complex variables, Fourier transforms, and partial differential equations.

**Exam Dates:**Open book, open notes, open Mathematica, closed internet.- Midterm - Thursday 3 March 2022 in class online
- Final - Saturday 7 May 2022 11:30 AM - 2:30 PM online

**Old midterm exams**4321,7305.- Practice final exam

- Syllabus

**Mathematica tutorial**PostScript 4 pages, 537511 bytes; PDF 4 pages, 32563 bytes**Grading:**- Homework - 60% (drop lowest)
- Midterm Examination - 20%
- Final Examination - 20%

PHYS 4321 and PHYS 7305 share the same lectures, but the 7305 homework assignments and examinations are more advanced than the 4321 assessments.**Course Objectives:**By the end of the course, undergraduate students (PHYS 4321) should be able to:**Describe, explain, and compare**sine, cosine, and exponential Fourier series and transforms**Describe and explain**generalized functions (distributions) and Green functions**Apply**techniques to solve linear homogenous and nonhomogeneous differential equations and**interpret**the solutions, particularly the damped, driven harmonic oscillator and Laplace's equation**Use**complex variables, complex functions, and Cauchy's theorem; matrices, determinants, and linear algebra**Practice**the calculus of variations and**construct**the Euler-Lagrange equations**Describe and explain**first and second order differential operators in Cartesian, Cylindrical, and Spherical coordinate systems- solve problems in all the areas above

**Course Objectives:**By the end of the course, graduate students (PHYS7305) should be able to:**Create and analyze**sine, cosine, and exponential Fourier series and transforms**Formulate**generalized functions (distributions) and Green functions for a specific problem**Construct**solutions to linear differential equations, homogeneous and nonhomogeneous, particularly the damped, driven harmonic oscillator and Laplace's equation using separation of variables**Analyze**complex variables, complex functions, and Cauchy's theorem; matrices, determinants, and linear algebra**Formulate**the calculus of variations and the Euler-Lagrange equations**Construct and interpret**first and second order differential operators in Cartesian, Cylindrical, and Spherical coordinate systems- solve problems in all the areas above

**Course Format:**Class time will be used for lecturing, not for problem solving.**Expectations:**Students are expected to attend all lectures and to be able to answer questions posed by the lecturer in class. Students should not use lecture time to do anything other than listen attentively and take notes. Homework assignments should be started well in advance of the due date.**Texts:**There is no course textbook, but any of the following may be useful. They are in the library and you can find them used (any edition) at abebooks.com- Mathematical Methods For Physicists by George B. Arfken and Hans J. Weber

# ISBN-10: 0120598760

# ISBN-13: 978-0120598762 - Mathematical Methods in the Physical Sciences by Mary L. Boas

# ISBN-10: 0471198269

# ISBN-13: 978-0471198260 - Advanced Engineering Mathematics by Erwin Kreyszig

# ISBN-10: 0471488852

# ISBN-13: 978-0471488859 - Advanced Engineering Mathematics by Peter V. O'Neil

# ISBN-10: 0534552080

# ISBN-13: 978-0534552084

- Mathematical Methods For Physicists by George B. Arfken and Hans J. Weber
**Lecture slides**- Fourier Series, Fourier Transform: 01, 02, 03,
Peter Olver's notes. See also
- Space
- Vector Space
- Hilbert Space
- Tone Generator
- Image Resolution
- Homer's orbit - Mathologer, complex Fourier series, epicycles

- Generalized functions,Distributions: 04, 05
- Differential equations: 06, 07
- Numerical Approximations to Solutions of Differential Equations
- video/audio for Numerical Approximations to Solutions of Differential Equations
- When I was your age, we programmed in BASIC. And we liked it. Not really.
- Mathematica notebook for the quantum harmonic oscillator, PDF version
- First-order (forward) Euler method for solving differential equations
- Runge-Kutta method

- Green functions (PDF format), Green functions (1/2), Green functions (2/2), Green function for heat equation, Green function example problem
- Nonlinear Simple Pendulum
- Curvilinear Coordinates: 08 [video audio only], 09 [video audio only]. See also Curvilinear coordinates, Orthogonal coordinates, Scale factors (h's).
- First-order Differential Operators: Gradient:10, [video audio only]; Divergence, Curl:11, [video audio only];
- Second-order Differential Operators: Laplacian, etc.: 12, [video audio only].

- Fourier Series, Fourier Transform: 01, 02, 03,
Peter Olver's notes. See also
**Lecture notes**- 26Jan16. Mathematica notebook, PDF file.
- 28Jan16. See also Aleph number, Georg Cantor's diagonalization.
- 04Feb16. See also Gaussian integrals.
- 09Feb16.
- 18Feb16. See also Green function example Mathematica notebook, PDF.
**Guest lecture**: Tuesday 26 March 2020, Professor Stephen Sekula on Monte Carlo methods- 25Feb16. Second root to dD/dw=0; Green function homework solution.
- 01Mar16. Practice midterm solutions.
- 22Mar16. See also Visualizing Divergence and Curl.
- 24Mar16. See also Mixed partial derivatives.
- 29Mar16. See also Separation of variables [video audio only].
- 31Mar16. See also Particle in a two-dimensional box.
- 05Apr16. See also Separation of variables 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 8/8.
- Bessel function (first kind) J
_{n}(complete, like sin and cos for cylindrical coord's), - Weber function, Neumann function, Bessel (second kind) Y
_{n}(infinite at s=0), - modified Bessel (first kind) I
_{n}(exponential growth for cylindrical coord's), - modified Bessel (second kind) K
_{n}(exponential decay for cylindrical coord's), - Legendre Polynomials - P
_{l}(cos θ) (complete in polar angle for spherical coord's) - Spherical harmonics - Y
_{lm}(θ, φ) (complete in polar and azimuthal angles for spherical coord's) - Spherical Bessel (first kind) j
_{n}(complete in radius r for spherical coord's) - Calculus of variations,
- The Chain Rule for Functions of Two Variables.

- Bessel function (first kind) J
- 07Apr16. See also Variational Calculus, 1/2, 2/2
- Evolutionary computation (Genetic Algorithms)
- "Evolving Inventions" by John R. Koza, Martin A. Keane and Matthew J. Streeter, Scientific American February 2003 p53
- War of the Weasels An Evolutionary Algorithm Beats Intelligent Design by Dave Thomas in Skeptical Inquirer, Vol 34, No 3.
- 15 Real-World Uses of Genetic Algorithms from Brainz.org

- 12Apr16. Complex functions of a complex variable
1/2. See also
- Quantum Mechanics requires complex numbers
- Venn (Euler) diagram for number sets
- Relationships between various basic mathematical structures by Max Tegmark, https://arxiv.org/pdf/gr-qc/9704009.pdf
- A Gentle Introduction to Abstract Algebra by B.A. Sethuraman
- Complex number,
- Analytic complex (holomorphic) function,
- Conformal (angle-preserving) mapping.

- The Octonions by John C. Baez
- 14Apr16. Complex functions of a complex variable 2/2. See also Cauchy's integral theorem.
- 19Apr16. See also Residue.
- Linear Algebra, Matrix Operations: 1/3, 2/3, 3/3. See also LU Decomposition.
- Group Theory:
1/2,2/2
See also
- Klein four-group a.k.a. Z
_{2}xZ_{2}a.k.a Vierergruppe - Group
- Group Theory
- Finite Group
- Lie Group
- Lie Groups and Lie Algebras - Gang Xu 4 lectures
- The Standard Model - Paul Langacker 15 lectures

- Klein four-group a.k.a. Z

- Dirac Belt Trick
- Knot Theory Rope Trick
- Paperclip trick
- Pages 1 - 10 (PDF format)
- Pages 11 - 20 (PDF format)
- Pages 21 - 29 (PDF format)
- Green functions (PDF format), Green functions (1/2), Green functions (2/2)
- Nonlinear Simple Pendulum
- Pages 30 - 40 (PDF format)
- Pages 41 - 50 (PDF format)
- Pages 51 - 55 (PDF format)
- Pages 56 - 66 (PDF format) Separation of variables 1/6, 2/6, 3/6, 4/6, 5/6, 6/6.
- Vibrating Rectangular Membrane - solutions to the wave equation in 2 dimensions
- 2D Vibrations of a Membrane
- Pages 67 - 77 (PDF format)
- Pages 78 - 88 (PDF format)
- Variational Calculus, 1/2, 2/2, Feynman Lectures: Principle of Least Action
- Complex Numbers; Complex Variables and Functions; Conformal Mapping; Contour Integration and Residues; Laurent Expansion; Laurent Series Examples; Residue Examples; Lecture slides: 1/2, 2/2
- Group Theory: 1/2,2/2
- 26Apr16. See also
- Introduction to the general theory of relativity,
- General relativity,
- Is Middle Earth flat?,
- Curved Space - Feynman lecture II-42 audio,
- Relativistic rocket I,
- Relativistic rocket II,
- Tests of general relativity,
- What Do You Mean, The Universe Is Flat? (Part I),
- What Do You Mean, the Universe Is Flat? Part II: In Which We Actually Answer the Question,
- metric tensor and Christoffel symbols for a sphere,
- Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe,
- The Universe Never Expands Faster Than the Speed of Light by Sean Carroll,
- Covariance and contravariance of vectors.

- Breit-Wigner links
**Homework**Due dates are strictly enforced. 50% if late; 0% once the solutions are posted. You may work together, but the work that you turn in should be unique. Identical work will receive a grade that is divided among all parties. It is possible to find answers to some homework problems on the internet; do not do this. The point, after all, is not to fool me into thinking that you have learned physics, but rather actually to learn some physics.- homework #1 (PDF format) - due Friday 28 January 2022 at 11:59:59pm
- homework #2 (PDF format) - due Friday 4 February 2022 at 11:59:59pm

(Mathematica example notebook PDF format, Fourier.nb) - homework #3 (PDF format) - due Friday 11 February 2022 at 11:59:59pm
- homework #4 (PDF format) - due Friday 18 February 2022 at 11:59:59pm
- homework #5 (PDF format) - due Friday 25 February 2022 at 11:59:59pm
- No homework due Friday 4 March 2022 because of the midterm exam on 3 March
- homework #6 (PDF format) - due Friday 11 March 2022 at 11:59:59pm
- No homework due Friday 18 March 2022 because of Spring Break
- homework #7 (PDF format) - due Friday 25 March 2022 at 11:59:59pm
- homework #8 (PDF format) - due Friday 1 April 2022 at 11:59:59pm
- homework #9 (PDF format) - due Friday 8 April 2022 at 11:59:59pm pseudocode for Monte Carlo assignment #11
- No homework due Friday 15 April 2022 because of the holiday
- homework #10 (PDF format) - due Friday 22 April 2022 at 11:59:59pm
- homework #11 (PDF format) - due Friday 29 April 2022 at 11:59:59pm (counts as two homeworks)

You may find the discussion of Olbers' Paradox in chapter 2 of Barbara Ryden's Introduction to Cosmology useful. Be careful to use "flat" variables.

**Homework Solutions****Disability Accommodations, Religious and Excused Absences****Official University Calendar**- Links:
- Mathematics for Physics by Michael Stone and Paul Goldbart
- Mandelbrot Set Explorer
- Robust Op Amp Realization of Chua's Circuit by Michael Peter Kennedy
- The Chaos Game (Triangle) Mathematica notebook
- The Chaos Game (Square) Mathematica notebook
- Hennon Map Mathematica notebook

- SMU Required Syllabus Statements

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