vector

Vectors 409
= i j k .( idx jdykdz)
= i j k . d r .
d r = vdr .
x y z
x y z
= [( y z) i ( z x) j ( x y) k].( i dx jdy kdz)
= (y + z) dx + (z + x) dy + (x + y) dz
= y dx + z dx + z dy + x dy + x dz + y dz
= xy + yz + zx + c
= ( ydx x dy) ( zdy y dz) ( zdx x dz)
Velocity potential = xy + yz + zx + c
Example 50. A fluid motion is given by
v = (y sin z – sin x) i + (x sin z + 2yz) j + (xy cos z + y 2 ) k
is the motion irrotational? If so, find the velocity potential.
Solution. Curl v = v
=
=
i j k ( ysin zsin x) i +( xsin z+2 yz) j+( xy cos z + y ) k
x y z
i j k
2
x y z
ysin z sin x xsin z 2yz xy cos z y
= (x cos z + 2y – x cos z – 2y) i – [y cos z – y cos z] j + (sin z – sin z) k = 0
Hence, the motion is irrotational.
So,
2
Ans.
v = where is called velocity potential.
d = dx
dy
dz
[Total differential coefficient]
x y z
= i j k .( i dx jdy kdz)
= .d r = v.
dr
x y z
= [(y sin z – sin x) i + (x sin z + 2yz) j + (xy cos z + y 2 ) k
].[ idx jdy kdz]
= (y sin z – sin x) dx + (x sin z + 2 y z) dy + (x y cos z + y 2 ) dz
= (y sin z dx + x dy sin z + x y cos z dz) – sin x dx + (2 y z dy + y 2 dz)
= d (x y sin z) + d (cos x) + d (y 2 z)
2
= d ( xy sin z) d (cos x) d( y z)
= xy sin z + cos x + y 2 z + c
Hence, Velocity potential = xy sin z + cos x + y 2 z + c. Ans.
Example 51. Prove that
Solution. Given F =
Consider
F =
2
F r r is conservative and find the scalar potential such that
F = . (Nagpur University, Summer 2004)
i j k
x y z
2 2 2
r x r y r z
2
r r =
2 2 2
r 2 ( xi y j zk)
= r xi r yj
r zk