Syllabus

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Prerequisites

Calculus of Several Variables (18.02) and Differential Equations (18.03) or Honors Differential Equations (18.034)

Calendar

The full schedule, organized by session number. (PDF)

Course Outline

This course has three major topics:

Applied Linear Algebra

• Second difference matrices K, T, B, C
• Positive definiteness: pivots, eigenvalues, energy
• ATCA framework for equilibrium problems
• Springs and masses
• Least squares and covariance matrix
• Graphs, networks, Kirchhoff's laws
• Deformation of trusses (and mesh generation)
• Minimum principles and constraints
• Finite elements in one dimension

Boundary Value Problems

• Ordinary differential equations
• Boundary conditions and delta functions
• Dynamics: Mu'' + Ku = F(t)
• Beam equations and cubic splines
• Partial differential equations
• Laplace and poisson equations
• Special solutions from (x + iy)n and f(x + iy)
• Potential, stream function, Cauchy-Riemann equations
• Finite differences and boundary conditions
• Finite element method and weak form

Fourier Methods and the FFT

• Fourier series (and orthogonal polynomials)
• Orthogonality and Parseval's formula
• Laplace equation on a circle
• Discrete Fourier series
• Fourier matrix and the fast Fourier transform
• Convolution and filtering in signal processing
• Fourier integral
• Shannon sampling theorem
• Differential equations
• Integral equations (convolution kernel)

Assignments and Exams

This course has nine problem sets, three one-hour exams, and no final exam. You may use your textbook and notes on the exams.

Text

The textbook for this course is: