Math 376 - Topics in Multi-Variable Calculus and Differential Equations

• Prerequisites: Math 375 or consent of instructor.
• Frequency: Fall (I), Spring (II)
• Student Body: Honors and advanced students in the sciences, engineering and Math
• Credits: 5. (N-A)
• Recent Texts: Calculus, Volume II, 2nd edition, by T. M. Apostol.
• Course Coordinator: Andreas Seeger
• Background and Goals: This course is the fourth semester of the Calculus Honors sequence developed by the Mathematics Department at the UW. The object of the course is to present subjects in multivariable calculus that were not covered in Math 375 and to give an introduction to differential equations.
• Alternatives: less advanced students can take Math 234 and Math 319 instead.
• Subsequent Courses: 5XX-level math courses

Content coverage:

• Multiple integrals
  • Partitions of rectangles and integrals of step functions
  • Upper and lower multiple integrals
  • Evaluation of multiple integrals
  • Cavalieri's principle
  • Geometric interpretations and examples
  • Changes of variables
  • Substitution rule with examples
  • Polar coordinates
• Line integrals
  • Curves and arclength
  • Paths and line integrals
  • Independence of parametrizations
  • Line integrals with respect to arclength
  • Integrability conditions
  • Potentials
  • Green's Theorem in the plane
• Surface integrals and vector analysis
  • Parametric representation of surfaces
  • Vector products and normals
  • Surface integrals
  • Divergence and curl
  • Integral theorems by Stokes and Gauss
• Linear differential equations
  • Discussion of the existence and uniqueness theorem
  • Linear differential equations of order n
  • Homogeneous and nonhomogeneous equations
  • Variations of the parameters
  • Power series as solutions of second order equations
  • Systems of linear differential equations
  • The exponential matrix
• The contraction principle
  • Successive approximations and fixed points of operators
  • Contraction of operators
  • Proof of the existence and uniqueness theorem
  • Implicit function theorem