## ACMS 20550 - 20750: Applied Mathematics Method I, II, Syllabus

Textbook: "Mathematical Method in the Physical Sciences" by Mary L. Boas, Third Edition 2006, Wiley, ISBN 13-978-0-471-19826-0.

Pre-requsite: Calulus II, Math 10560 or Math 10860.

Syllabus for Intro to Applied Mathematics/Mathematical Methods

## Topics 1 to 7 in first semester 20550

1. Chapter 1. Infinite series and power series
2. Chapter 2. Complex Numbers
• basics, infinite and power series
• trig functions
• Euler formula
3. Chapter 3. Rudiments of linear algebra
• linear equations
• row reduction
• matrix operations
• eigenvalues, eigenvectors, and diagonalization
4. Chapter 4. Partial Derivatives
• total differential
• Taylor and Power Series in several variables
• Chain Rule
• Lagrange Multipliers
5. Chapter 5. Multiple Integration
• multiple integrals
• the Jacobian and change of variables
• Surface integrals
6. Chapter 6. Vector Analysis
• inner, cross, and triple products
• gradient and line integrals
• Green's theorem, the divergence theorem, and Stokes theorem
7. Chapter 7. Fourier Series and Transforms
8. ## Topics 8 to 13 in second semester 20750

9. Chapter 8. Ordinary Differential Equations I (Constant coefficient differential equations and the Laplace transform)
10. Chapter 9. Variational Calculus (Euler equation, Lagrange's Equation, and some example problems)
11. Chapter 11. Assorted Special Functions
• Gamma, Beta, and Error Functions
• Asymptotic Series and Stirling's Formula
12. Chapter 12. Ordinary Differential Equations II
• Series solution of differential equations
• Orthogonal functions, Legendre polynomials, Bessel functions
• Other classes of orthogonal functions and their ODEs
13. Chapter 13. Partial Differential Equations
• Basic types of PDEs
• model problems (heat flow, vibrating string, steady state temperature)
14. Chapter 14. Complex Function Theory
• contour integrals and Cauchy's theorem
• Laurent series and the residue calculus
• conformal maps