About The Course
This course is about the mathematics that is most widely used in the mechanical engineering core subjects: An introduction to linear algebra and ordinary differential equations (ODEs), including general numerical approaches to solving systems of equations.
Each of the first five chapters in the textbook were covered briefly in one lecture. The discussions did not cover all the topics in the chapter. Rather, they aimed to communicate the basics and make it easier for the students to read the materials in these chapters on their own. While these materials were not tested in the exams, they are fundamental and it is important for the students to learn them well. It was not a requirement for students to hand in the homeworks in the lectures, but students had to do as many as they could on their own.
The following topics were covered in detail:
- Series solutions of linear homogeneous ODE. Discussed how to make expansions around irregular as well as regular singular points.
- The WKB Method. Covered not only the lowest-order WKB approximation but also the higher-order WKB approximations.
- Asymptotic expansion of integrals. Covered the Laplace method, the method of stationary phase, and the saddle point method.
- Boundary layer theory. Demonstrated how to solve the standard boundary layer problems. Discussed some of the boundary layer problems either considered unsolvable or solved incorrectly in the literature.
- Small non-linear oscillations. This includes the two-scale method, the counter term method in renormalized perturbation and the method of renormalization group.