VECTOR DIFFERENTIATION. 
87 
( 1 ) = 
( 2 ) —* 
(3) =* 
*( ) J_ 
dr 1 (i‘ 
) 1 + 
1 <S( ) 
1 
*p 
1 
00 2 
r 2 00 t 
< 8 p V 
so- 
dp * 
U<d 
V 00 ) 
00 
d 2 ( 
) 1 i 
1 *( ) 
1 
1 
d<p 2 
(•py 
r 2 dcp 
(k.\ 
2 dcp 2 
*p 
\8 V ) 
\»<e) 
dcp 
(4) = ±^ > 1 
r Or 00 dp 
~se p 
(5-) = A *( ) _i_ 
' J r drd<p dp 
i n ) i 
r 2 <5(9 <5^0 
~d5 p 
i h ) i 
r 2 <5^ <5/? 
<5^ P 
(&) =*1 ^ ) 1 4-1 *( ) _L jfe. JL. 
^ ^ r <50<5r dp r dr p l 00 dp 
00 
P 80 
«) = v { ) 
( 8 ) 
' ll 8 P 
sin Op n — 
r ° 00 
1 d ( ) COS 0 1 
r 2 0<p sin 2 0 dp 
Po ~00 
<* 2 ( ^ 1 
_J_ J(_) _1_ ^ 
r d(pdr p dp r dr p 2 0<p dp 
(9) 
*i *( ) 
1 .,!«() 
1 
*p 
1 
r 2 OcpOO 
dp dp r 2 00 
( 8 P ' 
i OcpOO 
8 P' 
00 dcp 
\dO j 
1 
dcp 
When the function is scalar, that is, does not involve 
P or “ or the above operators apply without any mod¬ 
ification ; but when the function does involve the axes men¬ 
tioned, the terms marked with an * are modified in differen¬ 
tiation on account of one of the factors in the product being 
the reciprocal of an axis. 
s 
The space-coefficient of the terms involving is 
J. dp^ , _1_ dp_l_. 
p l 00 dp p 2 d(f dp 5 
00 
d(p 
