The Laplacian operator can be defined, not only as a differential operator, but also through its averaging properties. Such a definition lends geometric significance to the operator: a large Laplacian at a point reflects a “nonconformist” (i.e., different from average) character for the function there. This point of view is used to motivate the wave equation for a drumhead.
References
1.
Richard P.
Feynman
, Robert B.
Leighton
, and Matthew
Sands
, The Feynman Lectures on Physics
(Addison-Wesley
, Reading, MA
, 1964
), Vol. II
, pp. 12-12
–12-13
.2.
The geometrical approach to vector operators has been particularly advocated by
H. M.
Schey
, Div, Grad, Curl: All That: An Informal Text on Vector Calculus
(W.W. Norton & Company
, New York
, 1973
). But Schey does not provide any geometrical insight into the Laplacian.3.
J. E.
McDonald
, “Maxwellian interpretation of the Laplacian
,” Am. J. Phys.
33
, 706
–711
(1965
).4.
Kalman B.
Pomeranz
, “Two theorems concerning the Laplace operator
,” Am. J. Phys.
31
, 622
–623
(1963
).5.
Harry F.
Davis
, “The Laplace operator
,” Am. J. Phys.
32
, 318
–319
(1964
).6.
C.
Leubner
, “Coordinate-free interpretation of the Laplacian
,” Eur. J. Phys.
8
, 10
–11
(1987
).7.
Edward M.
Purcell
, Electricity and Magnetism
(McGraw-Hill Book Company
, New York
, 1965
), pp. 61
–62
, 103–104, and 416–419.8.
Richard V.
Southwell
, Relaxation Methods in Theoretical Physics
(Oxford U. P.
, Oxford, UK
, 1946
).9.
David A.
Hastings
, “Computational method for electrical potential and other field problems
,” Am. J. Phys.
43
, 518
–524
(1975
).10.
For example,
William C.
Elmore
and Mark A.
Heald
, Physics of Waves
(McGraw-Hill Book Company
, New York
, 1969
), chap. 2.11.
James
Clerk Maxwell
, A Treatise on Electricity and Magnetism
, 2nd ed. (Clarendon Press
, Oxford, UK
, 1881
), Vol. I
, pp. 29
–30
.12.
This argument is modified from the one presented in Paul Lorrain and Dale Corson
, Electromagnetic Fields and Waves
, 2nd ed. (W.H. Freeman
, San Francisco
, 1970
), pp. 49
–51
.13.
Eric W.
Weisstein
, “Hypersphere,” from MathWorld—A Wolfram Web Resource; <http://mathworld.wolfram.com/Hypersphere.html>.© 2015 American Association of Physics Teachers.
2015
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