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 and 8 to 13 in second semester
- Infinite series and power series
- Complex Numbers
- basics, infinite and power series
- trig functions
- Euler formula
- Rudiments of linear algebra
- linear equations
- row reduction
- matrix operations
- eigenvalues, eigenvectors, and diagonalization
- Partial Derivatives
- total differential
- Taylor and Power Series in several variables
- Chain Rule
- Lagrange Multipliers
- Multiple Integration and Vector Analysis
- multiple integrals
- the Jacobian and change of variables
- inner, cross, and triple products
- gradient and line integrals
- Green’s theorem, the divergence theorem, and Stokes theorem
- Fourier Series and Transforms
- Ordinary Differential Equations I (Constant coefficient differential equations and the Laplace transform)
- Variational Calculus (Euler equation, Lagrange’s Equation, and some example problems)
- Assorted Special Functions
- Gamma, Beta, and Error Functions
- Asymptotic Series and Stirling’s Formula
- Ordinary Differential Equations II
- Series solution of differential equations
- Orthogonal functions, Legendre polynomials, Bessel functions
- Other classes of orthogonal functions and their ODEs
- Partial Differential Equations
- Basic types of PDEs
- model problems (heat flow, vibrating string, steady state temperature)
- Complex Function Theory
- contour integrals and Cauchy’s theorem
- Laurent series and the residue calculus
- conformal maps
- Probability and Statistics
- sample spaces and random variables
- discrete and continuous distributions
- the binomial, the Gaussian, and the Poisson distribution
- sampling and confidence intervalsReview of Linear Algebra: Linear Spaces, Inner Products;
- Review of Multivariable Calculus: Stokes's formula, Green's theorem;