Mathematical Physics Vol 1

4.6 Examples

127

Solution

∂ ∂ y

∂ ∂ z

∂ ∂ x

( x 2 z )+ ( xy 2 z )= = 2 xz + 6 y 2 z 2 − xy 2 = − 2 + 6 − 1 = 3 . ( 2 y 2 z 3 ) −

div A =

Problem46 For Φ = 3 x 2 z − y 2 z 3 + 4 x 3 y + 2 x − 3 y − 5 find ∇ 2 Φ .

Solution

∇ 2 Φ = 6 z + 24 xy − 2 z 3 − 6 y 2 z .

Problem47 For A =( 3 x 2 y − z ) i +( xz 3 + y 4 ) j − 2 x 3 z 2 k find ∇ ( ∇ · A ) at point (2,-1,0).

Solution

∇ ( ∇ · A )= − 6 i + 24 j − 32 k .

Problem48

For A = 3 xyz 2 i + 2 xy 3 j − x 2 yz k and Φ = 3 x 2 − yz find a) ∇ · A , b) A · ∇Φ , c) ∇ · ( Φ A ) , d) ∇ · ( ∇Φ ) , at point (1,-1,1).

Solution a) 4, b) -15, c) 1, d) 6.

Problem49

Show that the excess volume of fluid that flows out, in a unit of time, from a unit volume of fluid space, represents the divergence of the velocity v of the fluid.