Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
with similar expressions for r…A 2^u 2 † and r…A 3^u 3 †. Adding these together,
we get rA in orthogonal curvilinear coordinates:
rA ˆ ^u
1 @
… A
h 2 h 3 @u 3 h 3 † @
… A
2 @u 2 h 2 † ‡ ^u
2 @
… A
3 h 3 h 1 @u 1 h 1 † @
… A
3 @u 3 h 3 †
1
‡ ^u
3 @
… A
h 1 h 2 @u 2 h 2 † @
…A
1 @u 1 h 1 † : …1:64†
2
This can be written in determinant form:
h 1^u 1 h 2^u 2 h 3^u 3
rA ˆ 1
@ @ @
h 1 h 2 h 3
@u 1 @u 2 @u 3
: …1:65†
A 1 h 1 A 2 h 2 A 3 h 3
We now express the Laplacian in orthogonal curvilinear coordinates. From
Eqs. (1.58) and (1.62) we have
r ˆ grad ˆ 1 @
^u
h 1 @u 1 ‡ 1 @
^u ‡ 1 @
^u
1 h 2 @u 2 h 3 @u 3 ;
3
rA ˆ div A ˆ 1
h 1 h 2 h 3
If A ˆr, then A i ˆ…1=h i †@=@u i , i ˆ 1, 2, 3; and
@
…h
@u 2 h 3 A 1 †‡ @ …h
1 @u 3 h 1 A 2 †‡ @ …h
2 @u 1 h 2 A 3 †
3
rA ˆrr ˆr 2
ˆ 1
@ h 2 h 3 @
‡ @
h 3 h 1 @
‡ @
h 1 h 2 @
: …1:66†
h 1 h 2 h 3 @u 1 h 1 @u 1 @u 2 h 2 @u 2 @u 3 h 3 @u 3
:
Special orthogonal coordinate systems
There are at least nine special orthogonal coordinates systems, the most common
and useful ones are the cylindrical and spherical coordinates; we introduce these
two coordinates in this section.
Cylindrical coordinates …; ; z†
u 1 ˆ ; u 2 ˆ ; u 3 ˆ z; and ^u 1 ˆ e ; ^u 2 ˆ e ^u 3 ˆ e z :
From Fig. 1.17 we see that
x 1 ˆ cos ; x 2 ˆ sin ; x 3 ˆ z
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