Syllabus
Textbook: Boas, Mathematical Methods in the Physical Sciences, 3th edition, Wiley (2006)
Part A: Partial Differential Equations
I. Fourier Analysis
1) Fourier series, Fourier spectrum (review)
2) Fourier integral, bandwidth theorem
3) Fourier transformation
II. Partial Differential Equations
1) Two-dimensional wave equation, circular membrane
2) Bessel's function
3) Three-dimensional Laplace equation in spherical coordinates
4) Legendre polynomials
Part B: Vector Calculus
I. Introduction
1) Dot and cross products, lines and planes
2) Visualizing real-valued functions of several variables, level curves and surfaces
II. Differentiation
1) Partial differentiation
2) Taylor series in two dimensions
3) Vector-valued functions of several variables, extrema, second derivative test
4) Chain rule, total differentials, implicit differentiation
5) Directional derivative, gradient
III. Integration
1) Multiple integrals, changing the order of integration
2) Change of variables, polar coordinates
3) Triple integrals, spherical and cylindrical coordinates
IV. Vector functions and vector fields
1) Flux, divergence of a vector field, Gauss' theorem
2) Circulation, curl of a vector field, Stokes' theorem
3) Line integrals, Green's theorem
Part C: Numerical Methods
I. Introduction
1) Introduction to Matlab
2) Binary numbers, machine numbers, floating-point form of numbers
3) Computer accuracy, round-off errors, loss of significance, error propagation
II. Solution of Equations by Iteration
1) Iteration for solving x = g(x)
2) Bracketing methods for locating a root
3) The Newton-Raphson method for solving equations f(x) =0
4) Iterative methods for linear systems
III. Numerical Differentiation and Integration
1) Central-difference formulas
2) Trapezoidal rule, Simpson's rule, composite Trapezoidal and Simpson's rule
IV. Numerical Methods for Ordinary Differential Equations
1) Euler's method
2) The Runge-Kutta method
3) Systems of differential equations
Part D: Computer Simulation and Modeling
I. Selected Topics:
1) Planetary orbits
2) Chaos
3) Random walk and diffusion
4) Monte Carlo simulation