Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR INTEGRATION AND INTEGRAL THEOREMS
so
ds 2 ˆ dr dr ˆ…dr† 2 ‡ r 2 …d† 2 ‡ r 2 sin 2 …d† 2 :
Equating the two ds 2 , we ®nd the scale factors: h 1 ˆ h r ˆ 1, h 2 ˆ h ˆ r,
h 3 ˆ h ˆ r sin . We then ®nd, from Eqs. (1.58), (1.62), (1.64), and (1.66), the
gradient, divergence, curl, and the Laplacian in spherical coordinates:
@
r ˆ ^e r
@r ‡ ^e 1 @
r @ ‡ ^e
1 @
r sin @ ;
…1:72†
rA ˆ 1
r 2 sin sin @ @r …r2 A r †‡r @ @ …sin A †‡r @A
; …1:73†
@
^e r r^e r sin ^e
rA ˆ 1
@ @ @
r 2 sin
@r @ @
; …1:74†
A r rA r r sin A
"
r 2 ˆ 1
r 2 sin sin @
r 2 @
‡ @ #
@
sin ‡ 1 @ 2
@r @r @ @ sin @ 2 : …1:75†
Vector integration and integral theorems
Having discussed vector di€erentiation, we now turn to a discussion of vector
integration. After de®ning the concepts of line, surface, and volume integrals of
vector ®elds, we then proceed to the important integral theorems of Gauss,
Stokes, and Green.
The integration of a vector, which is a function of a single scalar u, can proceed
as ordinary scalar integration. Given a vector
A…u† ˆA 1 …u†^e 1 ‡ A 2 …u†^e 2 ‡ A 3 …u†^e 3 ;
then
Z
A…u†du ˆ ^e 1
Z
A 1 …u†du ‡ ^e 2
Z
A 2 …u†du ‡ ^e 3
Z
A 3 …u†du ‡ B;
where B is a constant of integration, a constant vector. Now consider the integral
of the scalar product of a vector A…x 1 ; x 2 ; x 3 ) and dr between the limit
P 1 …x 1 ; x 2 ; x 3 ) and P 2 …x 1 ; x 2 ; x 3 †:
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