AEP 3200 - Introductory Mathematical Physics

General Information

Covers review of vector analysis, tensor calculus, Dirac Delta functions, complex variable theory, Cauchy-Rieman conditions, complex Taylor and Laurent series, Cauchy integral formula and residue techniques, conformal mapping, calculus of variations, Fourier Series.

Prerequisites

Prerequisite: MATH 2930. Corequisite: MATH 2940

Topics Covered

1. Review of Vector and Matrix Algebra
   1. Matrix Representation of Vectors
   2. Subscript/Summation notation
   3. Vector and Matrix Products
      1. Kronecker Delta
      2. Levi-Civita Symbol
   4. Differential Operators
      
      1. Gradient
      2. Divergence
      3. Curl
   5. Vector Integration
      
      1. Line Integrals
      2. Surface Integrals
      3. Volume Integrals
   6. Theorems for Vector Analysis
      1. Gauss's Theorem
      2. Green's Theorem
      3. Stoke's Theorem
      4. Helmholtz's Theorem
   7. Non-Cartesian Coordinate Systems
      
      1. Cylindrical and Spherical Systems
      2. The Metric
      3. General, Orthogonal Systems
         
         
2. Tensor Calculus Part I
   
   1. Conductivity Tensor
   2. Rank
   3. Rank 1 Transformations
   4. Unit Vector Transformations
   5. Tensor Transformations
   6. Diagonalization
      1. Eigenvalues
      2. Eigenvectors
         
         
3. Dirac Delta Functions
   1. Definition of $\delta$(x)
   2. Approximations and Sequence Functions
   3. Integration Operations--Sifting
   4. The Step Function
   5. Derivatives - Doublet, Triplet
   6. Delta Function Calculus
   7. Density Functions
   8. The Infinitesimal Dipole
      
      
4. Complex Variables Part I
   
   1. Complex Numbers
   2. Functions of a Complex Variable
      1. Derivatives
      2. Cauchy-Riemann Conditions
   3. Cauchy Integral Theorem
   4. Cauchy Integral Formula
   5. Complex Series
      1. Taylor Series
      2. Laurent Series
      3. Convergence
      4. Relevance to Cauchy Integral Formula
      5. Higher Order Poles
   6. Residue Theory
   7. Principal Part of an Integral
   8. Integrals of Complex Functions
      1. Closure
      2. Analytic Continuation
   9. Conformal Mapping-Schwartz Christoffel Transformations
      
      
5. Fourier Series, Fourier and Laplace Transforms
   1. Fourier Sine and Cosine Series
      1. Periodic Functions With Period 2$\pi$
         a. Orthogonality
         b. Convergence
      2. Periodic Functions with Period 2$\pi$
      3. Expansion of an Arbitrary Function Over an Interval
      4. Symmetry
      5. Square and Triangle Waves
   2. Exponential Fourier Series
   3. Discrete Fourier Series for Computer Calculations
      1. Sampling f(t)
      2. Aliasing
      3. Orthogonality of Terms in Series
      4. Completeness
   4. Fourier Transforms
      1. The Fourier Transform Pair
      2. Specific Examples
         a. Square Pulse
         b.  -functions
         c. Periodic Function Limit
         d. Operations on f(t)
              i. Differentiation
              ii. Integration
              iii. Delay
              iv. Convolution
              v. Correlation
      3. Sampling Theorem
      4. Applications of Cauchy Integration Techniques
         a. Closure
         b. Analytic Continuation Leading to Laplace Transforms
   5. Laplace Transforms
      1. The Laplace Contour
         a. Causality
         b. Closure
         
      2. Double-side Laplace Transforms

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Workload

Weekly problem sets, one midterm and a final exam. [Spring 2023]

General Advice

1 weekly problem set (10 total for the semester); 1 midterm exam (90 minutes); 1 final exam (2 hr 30 minutes); weekly optional discussion section [Spring 2023]

The course is pretty much contained in the textbook. It is the best resource for practice problems that I am aware of, and the best place to review content; homework and exam problems have the same flavor as the textbook problems, and lecture notes are just the textbook in shorthand. If you can’t learn to love the book, at least learn to tolerate it, and if you can’t even do that, settle for knowing how to use it. Start homework early, and get far enough in that you know if all there are going to be troublesome problems on it. Practice often, and keep in mind things like efficiency and speed when solving problems. The VitalSource version of the textbook has serious flaws (missing/illegible diagrams and equations, etc) so consider buying a paper textbook or a pdf from another source. [Spring 2023]

Testimonials

For some people, particularly those who are interested in proof-based math, Mat Phys would not be a satisfying experience. The material is not covered very formally, and has an eye toward experiments (for an example of how this goes, the sampling theorem gets much more attention in the Fourier transform unit than convergence topics). For those students I mentioned, it is more sensible to take classes from the math department. With that out of the way, should more experimentally-inclined physics majors take Mat Phys? Mat Phys is not the most well-polished product. The textbook, while accurate and efficient, reads like a cookbook. It is spare on details, derivations, and applications. There is next to no continuity between chapters in the textbook - it’s better to think of it as a survey of topics than to try to find a signal in the noise. Textbook problems, except in rare cases, are divorced from physical context and occasionally are beyond tedious. The lectures are utilitarian - they follow the textbook, at times coming close to re-transcribing it. Problem sets are taken from textbook problems, and are usually somewhere in the neighborhood of six problems long. They very widely in difficulty - some take under an hour, and others will have a problem where an integral doesn’t converge, or you have to use L’Hopital four or five times to evaluate a limit, or you need to find a discrete fourier series with far too many points; these problems will plumb the depths of your very soul, particularly if you are like me and refuse to touch grass when you get stuck. In Spring ‘23, the midterm had a median of 62 or somewhere in there; it was long, and most people found it challenging. The final was a proportionally longer and harder (with a little extra), although grades on the exam were never released (are you sensing a pattern?). That being said, Mat Phys is just…so…useful. Complex integrals turn up everywhere. Fourier space is used in virtually every branch of physics. Formulas are much shorter and more digestible in Einstein notation, and if I had a penny for every curvilinear coordinate system I’ve come across, I wouldn’t need to think about them anymore. In short, Mat Phys can be a bitter pill to swallow, but it will make you better at math. So, take that for what it’s worth, and safe travels, whatever you decide to do about it. [Spring 2023]

Past Offerings

Semester	Professor	Median Grade	Course Page
Spring 2023	Ankit Disa	B+	AEP3200_SP23.pdf