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 References ========== 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.