Page 106 - 6099

where the unit vectors
b e ρ = (cos φ, sin φ), b e φ = (− sin φ, cos φ), (10.6)

are an orthonormal set. We say that circular coordinates are orthogonal.
2
We can now use this to evaluate ∇ ,
2
∂ 2 sin φ cos φ ∂ sin φ ∂ 2 sin φ cos φ ∂
2
2
∇ = cos ϕ + + + +
∂ρ 2 ρ 2 ∂φ ρ 2 ∂φ 2 ρ 2 ∂φ
2
∂ 2 sin φ cos φ ∂ cos φ ∂ 2 sin φ cos φ ∂
2
+ sin φ − + − =
∂ρ 2 ρ 2 ∂φ ρ 2 ∂φ 2 ρ 2 ∂φ
∂ 2 1 ∂ 1 ∂ 2 1 ∂ ∂ 1 ∂ 2
= + + = (ρ ) + . (10.7)
2
2
∂ρ 2 ρ ∂ρ ρ ∂φ 2 ρ ∂ρ ∂ρ ρ ∂φ 2
A final useful relation is the integration over these coordinates.
As indicated schematically in Fig. 10.2, the surface related to a change ρ → ρ + δρ,
φ → φ + δφ is ρδρδφ. This leads us to the conclusion that an integral over x, y can be
rewritten as Z Z
f(x, y)dxdy = f(ρ cos φ, ρ sin φ)ρdρdφ. (10.8)
V V

10.3 Spherical coordinates

Spherical coordinates are defined as

x = r cos φ sin θ, y = r sin φ sin θ, z = r cos θ, (10.9)
p
2
p x + y 2
2
2
2
r = x + y + z , φ = arctan(y/x), θ = arctan( ), (10.10)
z
as indicated schematically in Fig. 10.3.
Using the chain rule we find
∂ ∂r ∂ ∂φ ∂ ∂θ ∂ x ∂ y ∂ xz ∂
= + + = − + p =
∂x ∂x ∂r ∂x ∂φ ∂x ∂θ r ∂r x + y ∂φ r 2 x + y ∂θ
2
2
2
2
∂ sin φ ∂ cos φ cos θ ∂
= sin θ cos φ − + , (10.11)
∂r r sin θ ∂φ r ∂θ
∂ ∂r ∂ ∂φ ∂ ∂θ ∂
= + + =
∂y ∂y ∂y ∂y ∂φ ∂y ∂θ
y ∂ x ∂ yz ∂
= + + p =
2
2
r ∂r x + y ∂φ r 2 x + y ∂θ
2
2
∂ cos θ ∂ sin φ cos θ ∂
= sin θ sin φ + + , , (10.12)
∂r r sin θ ∂φ r ∂θ
p
∂ ∂r ∂ ∂φ ∂ ∂θ ∂ z ∂ x + y ∂
2
2
= + + = − = (10.13)
∂z ∂z ∂r ∂z ∂φ ∂z ∂θ r ∂r r 2 ∂θ
∂ sin θ ∂
= sin θ sin φ − . (10.14)
∂r r ∂θ
99