Course Webpage for PHYS-500A (Fall 2025) at SIU--Carbondale

Course Webpage for
PHYS 500A: Mathematical Methods (Fall 2025)
Department of Physics, Southern Illinois University--Carbondale
Last updated on .

General information:
- Instructor: K. V. Shajesh (Office: Neckers 491, Tel.: +1(618)453-5212)
- Class meetings: MF 9:35-10:50 AM, in Neckers 404.
- Office hours for instructor: MWF 11 AM-12:30 PM; MW 3-4 PM.
- Syllabus for the course. The Provost Office has put together a Syllabus Attachment to assist the student. It lists and provides web links for many elements that are important for a student.
Resources:
- References:
  - Methods of Theoretical Physics, Part 1 and 2, by Philip M. Morse and Herman Feshbach.
  - Mathematical Methods for Physicists, 6th Edition, by George B. Arfken and Hans J. Weber.
- Lectures
  - Lecture recordings: 2025F [2023F, 2021F, 2020F]
  - Informal Notes: This is not a substitute for a textbook.
- Quizzes and Homework
  - Quizzes: Due in class in the beginning of the next lecture.
  - Homework: Due in Mail Box before 4:30 PM on due date.
- Exams
  - Old Exams are available in Past course webpages: [2024F, 2023F, 2022F, 2021F, 2020F, 2019F, 2018F]
- Miscellaneous
  - Grades will be posted on SIU MyCourses (password protected).

COURSE CONTENT:

Vector calculus:
1. Vector algebra
  - Index notation,
    Scalar product, Kronecker \(\delta\)-symbol,
    Vector product, Levi-Civita symbol,
  - Vector identities involving non-commuting (quantum) operators
    Vector identities involving gradient operator \({\bf\nabla}\)
2. Vector field
  - Gradient operator, divergence, and curl
    Divergence theorem, Curl theorem
  - Dyadic notation
    Basis independent methods for gradient operations
    Identities: \( {\bf\nabla} \, {\bf r} ={\bf 1} \) and \( {\bf\nabla} \, r =\hat{\bf r} \)
3. \(\delta\)-function
  - Gauss's law, Ampere's law
Resources:
Homework No. 01
Homework No. 02
Keywords: Vector calculus, vector field, delta function

Examples:
- Gauss' law in electrostatics
Curvilinear coordinates:
1. Basis vectors, reciprocal basis vectors
  - Orthogonality, metric
  - Completeness relation
2. Transformation matrix
3. Connection, curvature
Resources:
Homework No. 03
Keywords: curvilinear coordinates, connection, curvature

Examples:
- Reciprocal basis in crystallography
Complex analysis:
1. Addition, subtraction, multiplication, subtraction, exponentiation
  - Polar representation (Argand diagram)
  - Roots of unity
2. Analytic function
  - Cauchy-Riemann conditions
  - Conjugate functions
3. Cauchy's theorem
  - Contour integrals with poles
  - Contour integrals with branch points
  - Analytic continuation of Gamma function
Resources:
Homework No. 04
Keywords: complex number, analytic function

Examples:
- Roots of a cubic equation, Cardano formula
- Analytic continuation of Beta function and Zeta function
...
Midterm Exam No. 01 (September 26, Friday)
...

More information:

Evaluation:

Homework 30%

Midterm Exams (2x20%) 40%

Classroom Quizzes 10%
Final Exam 20%

Exam schedule:

Midterm Exam 01 --- September 26 (Friday)

Midterm Exam 02 --- October 31 (Friday)

Final Exam --- Announced in 4th Week of Classes

Last updated on by K. V. Shajesh (kvshajesh@gmail.com).

Homework	30%
Midterm Exams (2x20%)	40%
Classroom Quizzes	10%
Final Exam	20%

Midterm Exam 01	---	September 26 (Friday)
Midterm Exam 02	---	October 31 (Friday)
Final Exam	---	Announced in 4th Week of Classes