vector

Vectors 407
=
i j k
x y z
2 2
y 2xy z
= i(0) j(0) k(2y 2 y)
= 0
Hence, A is irrotational. To find the scalar potential function f.
A
= f
df = f dx
f dy
f
dz
x y z
= i j k f.
dr
x y z
= Adr .
f f f
= i j k .( i dx jdy kdz)
x y z
= f.
dr
2 2
= ( y i 2 xy j z k).( i dx jdy kdz)
= y 2 dx + 2xy dy – z 2 dz = d (xy 2 ) – z 2 dz
=
f = d 2 2
( xy ) z dz
xy
2
3
(A = f)
z
C
Ans.
3
Example 47. A vector field is given by
A = (x 2 + xy 2 ) i + (y 2 + x 2 y) j . Show that the field
is irrotational and find the scalar potential.(Nagpur Univeristy, Summer 2003, Winter 2002)
Solution. A is irrotational if curl A = 0
i j k
Curl A =
Hence, A
A
A
x y z
2 2 2 2
x xy y x y
is irrotational. If is the scalar potential, then
= grad
d = dx
dy
dz
x y z
= i j k .( i dx jdy kdz)
= grad . dr
x y z
0
2 2 2 2
= i(0 0) j(0 0) k(2xy 2 xy) 0
[Total differential coefficient]
= Adr . = [( x xy ) i ( y x y) j].( idx jdy kdz)
= (x 2 + xy 2 ) dx + (y 2 + x 2 y) dy = x 2 dx + y 2 dy + (x dx)y 2 + (x 2 ) (y dy)
=
2 2 2 2
xdx y dy [( xdx ) y ( x )( y dy )] =
2 2
3 3 2 2
x y x y
c Ans.
3 3 2
Example 48. Show that V( x, y, z) 2 x yzi ( x z 2 y)
j x yk is irrotational and find a
scalar function u(x, y, z) such that V
= grad (u).
Solution. V (x, y, z) =
2 2
2 xyzi ( x z 2 y)
j x yk