Course Materials of Professor Jeffrey RAUCH


Math 558. Advanced Ordinary Differential Equations and Dynamical Systems, Fall 2014

2014 Course information

2014 Daily Syllabus

2014 Homework Assignments

Old Exams

Dynamics in Dimension 1

Dynamics in Dimension 1.5

Causality and Ordinary Differential Equations

The Steps of Perturbation Theory

Science Text Linearization (Linearization I)

Linearization at an Equilibrium. Peturbation Theory Approach (Linearization II)

Fundamental Theorems of Linear Systems

Zoom Zoom Zoom (Linearization III)

Zeno's Paradox and Uniqueness

Distinct eigenvalues implies eigenbasis

Phase plane for linear 2 by 2 systems

Ellipse axes, eccentricity, and direction of rotation

Spectral Theory with Repeated Eigenvectors

Multiple Roots Algorithm

Kronecker's Theorem

Parametric Resonance

Asymptotic Stability and Linearization

Asymptotic Stability by Linearization : alternate approach

The Turing Instability

Stable Curve Lemma

Undamped Pendulum Energy Landscape

Exercises in 1-d Mechanical Systems

Linearization Unstable and Nonlinear Equilibrium Stable



Lyapunov and LaSalle Theorems

Theorems on Gradient Systems

Derivatives of the Poincare Map

Conjugation of Logistic and Tent Maps

Boundary Value Problems for Partial Differential Equations

University of Pisa, March-April 2014

Course proposal

Course Outline, so far

Complements and Exercises

Positive Subspaces Handout

K.O. Friedrichs, Symmetric Hyperbolic Linear Differential Equations, CPAM 1954

K.O. Friedrichs, Symmetric Positive Linear Differential Equations, CPAM 1958

P.D. Lax and R.Phillips, Local Boundary Conditions for Dissipative Symmetric Differential Operators, CPAM 1960

K.O. Friedrichs and P.D. Lax, Systems of Conservation Equations with a Convex Extension, PNAS 1971.

J. Rauch, Symmetric Positive Systems with Boundary Characteristic of Constant Multiplicity, TAMS 1985


Pseudodifferential Operators and Propagation of Singularities

Universite de Paris VII,

28 January - 4 February, 2014

Proof of Continuity of Pseudodifferential Operators

Sections II.7 and II.8 of Taylor PDE vol. II

Taylor reflection article with diagonalization argument

Mike Taylor's Commutator Lemma for diagonalization

Nicolas Laillet's course notes, 28 January


Math 555. Applied Complex Analysis, Fall 2013

Fall 2013 Course information.

Fall 2013 Daily Schedule.

Fall 2013 Homework Assignments

Fall 2011 Daily Schedule.

Fall 2011 Homework Assignments

Old Exams

Image of Circles by 2x2 Matrices

Conformal Matrices

Orientation of the Plane

Image of Spheres by Linear Transformations

Open Mapping Theorem

Outlines of the derivation of everything from Cauchy's Theorem

Laurent Expansion Yields Partial Fractions

Laurent Expansion Yields Fourier Series

Partial Fractions and the Inverse Laplace Transform

More Fourier Analysis from Complex Analysis

The Dog on a Leash Principal

Conformal Mapping Primer

The Dirichlet Problem in a Half Space and Corners

The Neumann Problem, Insulator Boundary Conditions

Some Fluid Flows

The Dirichlet Problem in the Disk

Electrostatic Screening

Encounters with Partial Differential Equations

Universite de Paris Nord

janvier 13, 20, 27, fevrier 4, 2011


Quick Haar measure

The infintesimal Laplacian for distributions

Convergence of series for means on balls

Exercises on the Laplacian

The Partial Difference Equations of Mathematical Physics, by Courant, Friedrichs, and Lewy

Fourier Analysis from Complex Analysis

Introduction of geometric optics from Hyperbolic PDE book

Spring energy from Noether's Theorem

The Turing Instability

Math 556 Lecture Notes


The Wave Equation and Geometric Optics

Universite de Paris 13, January, 2010


Course notes

A second example of the approximation of geometric optics

Additional problems


Math 256. Honors Applied Ordinary Differential Equations

Course information

Daily syllabus


m- files and computer related

Wronskian Theorem statement

Integrating factor review

Matlab tutorials

The Steps of Perturbation Theory

Multiple Roots Algorithm

The Turing Instability

K. Miller's Linear Algebra Lecture Notes

Nonlinear Resonance

Universita di Pisa, March-April 2007


Chapter 9. The first classes.

Chapter 10. April 12.

Chapter 6. Sections 6.4 and 6.6 are cited in Chapter 9.

Chapter 11. April 19,26.


Dispersive Properties of Hyperbolic Partial Differential Equations

Universita di Pisa, March-April 2006




Math 556. Applied Linear Analysis

Mathematics 556 Notes

Fall 1994 Homework Assignments

Math 571. Numerical Linear Algebra


Help with the proof of Theorem 6.2-3.

Homework Assignment 1.

Homework Assignment 2.

Homework Assignment 3.

Homework Assignment 4.

Homework Assignment 5.

Homework Assignment 6.

Homework Assignment 7.

Homework Assignment 8.

Homework Assignment 9.

Homework Assignment 10.

Homework Assignment 11.

Homework Assignment 12.

Study problems.

Math 471. Introduction to Numerical Methods


Textbook's m.files

IEEE floating point standard.

Improved Theorem for 6.4.1.

FFT Handout.

Matrix Norms Handout.