Mathematical Physics Vol 1

4.6 Examples

197

Solution a) From Figure 4.22 we can see that the sides of the shaded body are ρ d ϕ , d ρ and d z . Given that the system of cylindrical coordinates is orthogonal, it follows that the elementary volume is d V = d s 1 d s 2 d s 3 (see p. 113, relation (4.147)). As the lengths of the sides are

1 = 1 · ( d ρ )= d ρ , 2 = ρ · ( d ϕ )= ρ d ϕ , 3 = 1 · ( d z )= d z ,

d s 1 = h 1 d u d s 2 = h 2 d u d s 3 = h 3 d u

the volume is

1 d u 2 d u 3 ⇒

d V = h 1 h 2 h 3 d u

or

d V = 1 · ρ · 1d ρ d ϕ d z = ρ d ρ d ϕ d z . b) From Figure 4.23 we can see that the sides of the shaded body are d r , rd θ and r sin θ d ϕ . Given that the system of spherical coordinates is orthogonal, it follows that the elementary volume is d V = d s 1 d s 2 d s 3 . As the lengths of the sides are d s 1 = h 1 d u 1 = 1 · ( d r )= d r , d s 2 = h 2 d u 2 = r · ( d θ )= r d θ , d s 3 = h 3 d u 3 = r sin θ · ( d ϕ )= r sin θ d ϕ , the volume is d V = h 1 h 2 h 3 d u 1 d u 2 d u 3 , or d V = 1 · r · r sin θ d r d ϕ d ϕ = r 2 sin θ d r d ϕ d ϕ .

Exercise 151 Find Lame’s coefficients and the volume element d V in spheroidal coordinates.

Solution The relation between the Cartesian and the spheroidal coordinate systems is

x = a ch ξ cos η cos ϕ , y = a ch ξ cos η sin ϕ ,

(4.221)

z = a sh ξ sin η . The procedure for determining Lame’s coefficients and calculating the volume element is the same as for the previously observed coordinate systems: calculating d x , d y , d z , and then determining d s and Lame’s coefficients, and finally determining the volume element.