Page 511 - Modelling in Transport Phenomena A Conceptual Approach

Appendix A

Mat hemat ical Preliminaries

A.l THE CYLINDRICAL AND SPHERICAL
COORDINATE SYSTEMS

For cylindrical coordinates, the variables (r, 8, z) are related to the rectangular
coordinates (x, y, z) as follows:
x=rcos~ r=Jw (A.1-1)
y = r sin 8 0 = arctan(y/x) (A.l-2)
z=x z=z (A. 1-3)
The ranges of the variables (r, 0, z) are

Olrlcu 0<0<2.rr -oo<z<oo

For spherical coordinates the variables (r, B,4) are related to the rectangular
coordinates (x, y, z) as follows:

x=rsinBcos4 r= ,/x2+y2+z2 (A.1-4)
y = r sin e sin 4 e = arctan (,/~/z) (A.l-5)
z = rcose 4 = arctan(y/x) (A.l-6)

The ranges of the variables (r, 8,d) are

O<r<cu 05e<.rr 054<2~
The cylindrical and the spherical coordinate systems axe shown in Figure A.l.
The differential volunaes in these coordinate systems are given by
dV= { r drdedz cylindrical (A.l-7)

r2 sin 8 drddd4 spherical

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