Mathematical Physics Vol 1

4.6 Examples

217

Solution

a) ρ , c) u 2 + v 2 , d) a 2 ( sh 2 u + sin 2 v ) , e) a 3 ( sh 2 ξ + sin 2 η ) sh ξ sin η b) r 2 sin θ ,

Exercise 177 Calculate R V p x and z = 8 − ( x 2 + y 2 ) .

2 + y 2 d x d y d z ,where V is a region bounded by the surfaces z = x 2 + y 2

Solution

256 π 15

Exercise 178

a) Describe the coordinate surfaces and coordinate lines for the system x 2 + y 2 = 2 u 1 cos u 2 , xy = u 1 sin u 2 , z = u 3 . b) Show that the system is orthogonal. c) Calculate the Jacobian of the transformation. d) Show that u 1 and u 2 are related to cylindrical coordinates ρ and ϕ and find these relations.

Solution

1 2

1 2 ρ

2 , u

2 = 2 φ

c)

, d) u 1 =

Exercise 179

∂ r ∂ u 1

∂ r ∂ u 2

∂ r ∂ u 3

, ∇ u 1 , ∇ u 2 , and ∇ u 3 in

Find

,

,

a) cylindrical, b) spherical and c) parabolic cylindrical coordinates. Showthat e 1 = E 1 , e 2 = E 2 and e 3 = E 3 for these systems.