Math 314 References
(This page will be continually updated)
Books
- Partial Differential Equations, by Fritz John, Springer Verlag, 1971-1982. A small book (240 pages) covering
the fundamental theory of pde.
- Partial Differential Equations, (6th ed) by Walter A. Strauss, published by John Wiley & Sons, Inc., 1992. A readable overview of the
theory, a little less technical than Fritz John's book above.
- Methods of Mathematical Physics, Vol II, by Richard Courant and David Hilbert, John Wiley and Sons, 1962.
A classic and comprehensive work on partial differential equations. Many applications.
- Partial Differential Equations of Mathematical Physics, by S.L. Sobolev, Dover Publications, 1964.
Articles
- Designing a Pleasing Sound Mathematically, (2.3MB) Erich Neuwirth, Mathematics
Magazine, Vol. 74, No. 2, April 2002. (Copyright the
Mathematical Association of America. All rights reserved.)
This article explains how Fourier series can be used to
analyze and generate sounds.
- Musical Analysis and Synthesis in Matlab (280 KB) Mark R. Petersen,
The College Mathematical Magazine, Vol. 35, No. 5, November 2004. (Copyright the
Mathematical Association of America. All rights reserved.) This article discusses how to analyze sounds with Fourier series,
why different musical instruments playing the same notes sound different, and how to synthesize (reproduce) the sound
of particular instruments electronically.
- Can One Hear the Shape of a Drum? (1.5 MB) Mark Kac, The American Mathematical
Monthly, Vol. 73, No. 4, 1966. (Copyright the
Mathematical Association of America. All rights reserved.)
In this article, Kac discusses the relation between the geometry ('shape') of a region Q and the eigenvalues lk
of the eigenvalue problem arising, for example, from the wave equation on Q; Laplacian u = -lu. (Recall that
the eigenvalues are essentially the frequencies of vibration you would hear from the drum.) He shows that
the area, at least, can be determined from the knowledge of the eigenvalues. He leaves it as an open problem whether
other characteristics of the region Q such as the length of the boundary or curvature of the boundary, can be determined
from the eigenvalues. (We know that you can determine the length of a string by listening to it. Recall that the smallest
eigenvalue there is pi/a where a is the length of the string. The lowest frequency you hear is c/2a
where c2 = T/d , T is the tension, and d the density of the string - which we
assume we know. So if you could identify the lowest frequency of vibration (by listening), you could determine the length of the
string from these formulae.)
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