Mathematical Physics Vol 1

Chapter 4. Field theory

196

Solution a) The relation between Cartesian and spherical coordinates ( r , θ , ϕ ) are

x = r sin θ cos ϕ , y = r sin θ sin ϕ , z = r cos θ .

By differentiating we obtain

d x = − r sin θ sin ϕ d ϕ + r cos θ cos ϕ d θ + sin θ cos ϕ d r , d y = r sin θ cos ϕ d ϕ + r cos θ sin ϕ d θ + sin θ sin ϕ d r , d z = − r sin θ d θ + cos θ d r ,

and thus the square of the arc element is

( d s ) 2 = d x 2 + d y 2 + d z 2 =( d r ) 2 + r 2 ( d θ ) 2 + r 2 sin 2 θ ( d ϕ ) 2 . From here we obtain Lame’s coefficients h 1 = h r = 1, h 2 = h θ = r and h 3 = h ϕ = r sin θ . b) The relation between Cartesian and parabolic coordinates ( u , v , z ) are

1 2

( u 2 − v 2 ) ,

x =

y = uv , z = z .

By differentiating we obtain

d x = u d u − v d v , d y = u d v + v d u , d z = d z .

The square of the arc element is ( d s ) 2 = d x 2 + d y 2 + d z 2 =( u 2 + v 2 )( d u ) 2 +( u 2 + v 2 )( d v ) 2 +( d z ) 2 , and hence the Lame’s coefficients are h 1 = h u = √ u 2 + v 2 , h 2 = h v = √ u 2 + v 2 and h 3 = h z = 1.

Exercise 150 Determine the infinitesimal part of a volume d V in a) cylindrical, b) spherical coordinates.