### Outline

An intermediate-level mathematics course designed for physics learners. We will cover linear algebra, Fourier analysis, special functions, and methods of numerical analysis. Prerequisites for this course include a foundational understanding of calculus, vector and matrix arithmetic, and ordinary differential equations.

After reviewing the basics of matrices, you first learn **linear algebra** with an emphasis on practical handling of matrices and **eigenvalue problems**. Motivated students are expected to reach the concept of **vector space**. You then learn basic topics of function analysis, in particular **Fourier analysis** and **special functions**.

Additionally, the course introduces fundamental concepts in numerical analysis and basic numerical methods for mathematical problems such as ordinary differential equations. Students will gain theoretical knowledge in numerical analysis as well as practical coding skills in Python, using the contemporary Python ecosystem.

This course does not cover the mathematical foundations of each topic. Instead, students are expected to be familiar with the topics and develop an appreciation of their usefulness.

#### Students’ Goals

- I am familiar with matrices and vectors. In particular, I can explain “vector space”, “basis”, “linear transformation”, “matrix rank”, and “eigenvectors”.
- I can perform Fourier analysis of functions and explain its physical interpretation.
- I know several special functions and their basic properties.
- Utilizing computers and online resources, I can numerically solve problems in linear algebra.
- I can use Python to find numerical solutions to basic differential equations.

#### Guidance document

- pm2_supplemental_2023.pdf for Academic Year 2023
- Guidelines for Using Generative AI
- Note on Sho’s Grading Convention

#### Textbook

E. Kreyszig, Advanced Engineering Mathematics, 10th ed. Taiwan custom version, Wiley (2018).

- Attendants need the textbook, preferably a physical book rather than an e-book.
- Attendants are assumed to have learned Chapters 1, 2, 9, and 10. We discuss Chapters 7, 8, 11, 5, and 21.

#### Past Exam Problems

### Schedule (2023‒2)

- 1 (Feb. 19, 22)
- Matrix and vector.
- 2 (Feb. 26, 29)
- Rank. Vector space. Determinant. / Setup of coding environment.
- 3 (Mar. 4, 7)
- Introduction to numerical analysis. / Linear-equation system.
- 4 (Mar. 11, 14)
- Vector space.
- 5 (Mar. 18, 21)
- Eigenvalue problem.
- 6 (Mar. 25, 28)
- Matrix diagonalization.
- 7 (Apr. 1)
- Summary of linear algebra.
- 8 (Apr. 8, 11)
- Midterm Exam / Exam review.
- 9 (Apr. 15, 18)
- Fourier series.
- 10 (Apr. 22, 25)
- Strum--Liouville problems.
- 11 (Apr. 29, 2)
- Fourier transformation.
- 12 (May 6, 9)
- Special functions.
- 13 (May 13, 16)
- Special functions. Frobenius Method.
- 14 (May 20, 23)
- Numerics for ODEs.
- 15 (May 27, 30)
- Numerics for ODEs.
- 16 (Jun. 3, 6)
- Term Exam / Exam review.
- 17 (Jun. 13)
- Flexible learning week (no class), compensating fo coding assignments.
- 18 (Jun. 17, 20)
- Basic group theory. Topics requested by students.