vector

410 Vectors
F
= 0
2 2
2 2
2 2
= i r z r y j r z r x k r y r x
y z
x z
x y
r r r r r r
= i 2rz 2ry j 2rz 2rx k 2ry 2rx
y z
x z
x y
2 2 2 2 r x r y r z
But r x y z
, , ,
x r y r z r
y z x z x y
= i
2rz 2ry j 2rz 2rx k 2ry 2rx
r r r r r r
= i(2yz 2 yz) j(2zx 2 zx) k(2xy 2 xy)
= 0i 0 j 0k
0
F is irrotational F is conservative.
Consider scalar potential such that F = .
d = dx
dy
dz
x y z
= i j k .( i dx jdy kdz)
x y z
= i j k .( i dx jdy kdz)
x y z
= F.( idx jdy kdz)
=
= 2 2 2
[Total differential coefficient]
= .( idx jdy kdz)
2
r r.( idx jdy kdz)
( = F )
( x y z )( ix jy kz).( idx jdy kdz)
= (x 2 + y 2 + z 2 ) (x dx + y dy + z dz)
= x 3 dx + y 3 dy + z 3 dz + (x dx) y 2 + (x 2 ) (y dy)
+ (x dx)z 2 + z 2 (y dy) + x 2 (z dz) + y 2 (z dz)
3 3 3 2 2
= x dx y dy z dz [( x dx) y ( y dy) x ]
=
4 4 4
x y z 1 2 2 1 2 2 1 2 2
x y xz y z c
4 4 4 2 2 2
2 2 2 2
[( xdxz ) ( zdzx ) ] [( ydyz ) ( zdz) y ]
= 1 4 (x4 + y 4 + z 4 + 2x 2 y 2 + 2x 2 z 2 + 2y 2 z 2 ) + c Ans.
r
Example 52. Show that the vector field F is irrotational as well as solenoidal. Find
3
| r |
the scalar potential.
(Nagpur University, Summer 2008, 2001, U.P. I Semester Dec. 2005, 2001)
r xi y j zk
Solution. F =
2 2 2 3/2
3
| r |
( x y z )
Curl F xi yj
zk
= F = i j k
x y z 2 2 2 3/2
( x y z )