Mathematical Physics Vol 1

4.5 Special coordinate systems

115

In this case, the relation between coordinates, according to Fig. 4.23, is

Figure 4.23: Spherical coordinate system.

x = r sin θ cos ϕ , y = r sin θ sin ϕ , z = r cos θ , where r > 0 , 0 ≤ ϕ < 2 π , 0 < θ < π .

In a similar way as in the previous case we obtain

h r = 1 h ϕ = r sin θ h θ = r .

3. PARABOLICAL-CYLINDRICAL ( u , v , z )

x = u 2 − v 2 , y = uv , z = z where − ∞ < u < ∞ , v ≥ 0 , − ∞ < z < ∞ h u = h v = p u 2 + v 2 , h z = 1 , 1 2

while the relation with cylindrical coordinates is given by

ϕ 2

ϕ 2

u = p 2 ρ cos

, v = p 2 ρ sin

, z = z .