Math/Physics 216: Mathematics for the Sciences II

Course Description
Download the Syllabus (.doc)

Syllabus

Part A:  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 B:  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 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