Partial Differential Equations - Jay P. Fillmore
UC San Diego - Math 132A - Winter 1981

  I. First order PDEs
     - Envelopes of surfaces
     - Method of characteristics - Cauchy
     - Surfaces of revolution
     - Birth-death process
     - Tubular surfaces
     - Solutions as envelopes
     - Lie's picture
     - The Hamilton-Jacobi equation
     - The harmonic oscillator
     - Caustics
     - The eikonal equation
     - Geometric optics

 II. Second order PDEs
     - Characteristics
     - Propagation of singularities
     - Wave fronts and rays
     - The wave equation - dimension one
     - The wave equation - dimensions  n , three, and two
     - Huyghens's Minor Principle

III. Hamilton-Jacobi Theory
     - Calculus of variations - the simplest case
     - Lagrange's equations
     - Hamilton's equations
     - The Hamilton-Jacobi equation
     - Kepler orbits in the plane

 IV. Exams and solutions
     - Midterm
     - solutions: problem 1  problem 2
     - Final
     - solutions: problem 1  problem 2


V. I. Arnold
Mathematical Methods of Classical Mechanics
Springer Verlag, New York, 1978.

A. Cayley
"Memoirs on Caustics"
Phil. Trans. 1857, vol. 147, and 1867, vol. 157.

R. Courant, D. Hilbert
Methods of mathematical physics. Vol. II.
English ed. New York, Wiley Interscience, 1962. 

K. E. Gustafson
Partial Differential Equations and Hilbert Space Methods
John Wiley, New York, 1980.
This was the text for the 1981 course, but plays only a minor rôle in these notes.

F. John
Partial Differential Equations
Springer-Verlag, Berlin, Heidelberg, New York, 1978.

F. John
Plane waves and spherical means applied to partial differential equations
New York, Interscience Publishers, Inc., 1955.

S. Lie und G. Scheffers
Geometrie der Berhürungstransformationen
Leipzig, B. G. Teubner, 1896.

I. N. Sneddon
Elements of Partial Differential Equations
McGraw-Hill, New York, Toronto, London, 1957.

H. F. Weinberger
A first course in partial differential equations
with complex variables and transform methods.
New York, Blaisdell Pub. Co., 1965; Dover, 1995.