Mathematical Physics Vol 1

4.6 Examples

219

in terms of parabolic cylindrical coordinates, where m , h and E are constants.

Solution

1 u 2 + v 2

∂ 2 ψ ∂ v 2

∂ 2 ψ ∂ u 2

∂ 2 ψ ∂ z 2

8 π 2 m h 2

( E − W ( u , v , z )) ψ = 0 ,

+

+

+

where W ( u , v , z )= V ( x , y , z ) .

Exercise 182 Express the equation

∂ U ∂ t = κ ∇ 2 U in terms of spherical coordinates if U is independent

of a) ϕ , b) ϕ and θ , c) r and t , d) ϕ , θ and t .

Solution

= κ

∂ ∂ r

∂ U ∂ r

∂ ∂θ

∂ U ∂θ

= κ

∂ ∂ r = 0 .

∂ U ∂ r

∂ U ∂ t

∂ U ∂ t

1 r 2

1 r 2 sin θ

1 r 2

r 2

r 2

sin θ

a)

b)

+

∂ ∂θ

∂ U ∂θ

d d r

d U d r

∂ 2 U ∂φ 2

r 2

c) sin θ

sin θ

= 0

d)

+

Exercise 183 Prove that in every coordinate system div rot A = 0 and rot grad φ = 0 is true.

Exercise 184 a) If x = 3 u 1 + u 2 − u 3 , y = u 1 + 2 u 2 + 2 u 3 , z = 2 u 1 − u 2 − u 3 , find the volume of the cuboid bounded by x = 0 , x = 15, y = 0 , y = 10, z = 0 , z = 5. b) Find the relation between the volume and the Jacobian of the transformation.

Solution

a) 750,75;

b) Jacobian=10.

Exercise 185 Let ( x , y , z ) and ( u 1 , u 2 , u 3 ) be the coordinates of the same point in two systems.