Mathematical Physics Vol 1

4.6 Examples

215

4.6.12 Various problems

Exercise 170 Express the following surfaces in terms of spherical coordinates a) sphere x 2 + y 2 + z 2 = 9 c) paraboloid z = x 2 + y 2 b) cone z 2 = 3 ( x 2 + y 2 )

d) surface z = 0 e) surface y = x .

Solution

a) r = 3 , d) θ = π / 2 , e) the plane y = x consist of two half-planes φ = π / 4 and φ = 5 π / 4 . b) θ = π / 6 , c) r sin 2 θ = cos θ ,

Exercise 171 If ρ , φ , z are cylindrical coordinates, draw each of the following examples and write their equations in Cartesian coordinates a) ρ = 4 , z = 0 b) ρ = 4 c) φ = π / 2 d) φ = π / 3 , z = 1 .

Solution

a) A circle in the z plane, x 2 + y 2 = 16 , z = 0 , b) Cylinder x 2 + y 2 = 16 , z = z . c) The yz plane where y ≥ 0 , d) Straight line y = √ 3 x , z = 1where y ≥ 0 , x ≥ 0 .

Exercise 172 For the spherical coordinate system, find a) unit vectors e r , e θ , e φ in terms of i , j , k , b) unit vectors of the Cartesian coordinate system i , j , k in terms of e r , e θ , e φ .