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r 2 u D @ 2 u
@x 2
C @ 2 u
@y 2
C @ 2 u
@ z 2
:
(7.3)
Equation (7.2) can then be more compactly written as
@ u
@t D k r 2 u C f:
(7.4)
The r 2 u term is a meaningful notation also in one-dimensional and two-dimensional
problems; if u is not a function of z , i.e. a two-dimensional problem, the z deriva-
tive vanishes, and if u D u .x; t /, as in one-dimensional problems, both the terms
involving the y and z derivatives are omitted in the expression for r 2 u .
The diffusion equation without a time derivative is known as the Poisson 4
equation :
r 2 u D f:
(7.5)
If the source term is also neglected, the resulting equation is referred to as the
Laplace 5 equation :
r 2 u D 0:
(7.6)
This is perhaps the most famous of all PDEs.
Cylindrical Symmetry
Many geometrical objects have the shape of a cylinder (a can of beer being one
example). Computations in such geometries can benefit from working with cylin-
drical coordinates . Instead of the Cartesian coordinates x, y,and z , we introduce
.r; ; O z /.Ther coordinate measures the distance from the z axis, and the coordinate
measures the rotation of a point around the z axis. The relation between cylindrical
and Cartesian coordinates reads as 6
x D r cos ;
y D r sin ;
z D z :
The z axis is supposed to coincide with the central axis in the cylinder.
4 Simeon-Denis Poisson, 1781-1840, French mathematician and physicist, contributed to many
topics in mathematics and physics, including celestial mechanics, Fourier series and integrals,
probability, and the theory of electricity and magnetism. He also independently derived the Navier-
Stokes equations ( 7.29 )-( 7.30 ) governing the flow of fluids.
5 Pierre-Simon, marquis de Laplace, 1749-1827, was a French mathematician and astronomer.
Besides formulating the famous Laplace equation, he worked on a wide range of topics in astron-
omy and mathematics (including statistics). He is considered one of the greatest scientists of all
time.
6 See your favorite topic on vector calculus for more details about cylindrical (and spherical)
coordinates.
 
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