# Syllabus

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## Prerequisites

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

## 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

18.085 has regular homeworks, three one-hour quizzes, and no final exam.

## Texts

This course as taught during the Fall 2005 term on the MIT campus used the following text: