Math Methods of Applied Sciences II

Department: 
MATH
Course Number: 
6702
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every spring semester

Review of vector calculus and and its application to partial differential equations.

Prerequisites: 

MATH 6701, or MATH 2403 and one of MATH 2406, MATH 4305

Course Text: 

No text

Topic Outline: 
  • Multidimensional Calculus
    • Curves and surfaces, gradients, divergence and curl
    • Taylor expansions in IR3
    • Divergence and Stokes theorem
    • Classification of partial differential equations
    • The concept of well-posed problems
  • Potential Problems
    • Derivation of Laplace's equation; Dirichlet and Neumann problems
    • The maximum principle and uniqueness of solutions
    • Green's identities and Green's functions for selected domains
    • Connections to variational problems and complex variables
  • Parabolic Problems
    • Derivation of the heat equation in IR3; discussion of boundary and initial conditions; the maximum principle for the heat equation and uniqueness of solutions; fundamental solution for pure initial value problems; Duhamel's principle for inhomogeneous equations
  • Hyperbolic Problems
    • The concept of characteristics for a single first order equation
    • Solution of initial value problems; the concept of a shock
    • D'Alembert solution of the wave equation; Huyghen's principle and the solution of the wave equation in IR3