A second semester graduate course primarily intended for students in mechanical and aerospace engineering, biomedical engineering, and other engineering programs. There will be three parts:
1. Complex variable theory, including the differential and integral calculus of functions of a complex variable, conformal mapping, Taylor series, Laurent series and the residue theorem. Introduction to the calculus of variations.
2. Calculus of variation, including the motivation of variational principles from physical laws, derivation of Euler-Lagrange equations, stability criterion, linearization, and brief introduction of the foundation of finite element method.
3. Perturbation methods, including applications to ode systems, examples of boundary layer, multiple-scale problems, and eigenvalue problems.
Michael D. Greenberg, Advanced Engineering Mathematics (second edition), Prentice-Hall, 1998. Further references will be provided in class.