∂b r
∂ϕ
¼ À SinθSinϕ
ð
Þ b x þ SinθCosϕ
ð
Þ b y ¼ b
ϕSinθ
∂ b θ
∂ϕ
¼ À CosθSinϕ
ð
Þ b x þ CosθCosϕ
ð
Þ b y ¼ b
ϕCosθ
∂ b
ϕ
∂ϕ
¼ À Cosϕ
ð
Þb x À Sinϕ
ð
Þb y ¼ À Sinθ
ð
Þb r À Cosθ
ð
Þ b θ
Incremental Vector Path in Spherical Coordinates
An expression for a small increment in the vector path length d r
! in terms of the
spherical coordinates is given by
d r
! ¼ d rb r
ð Þ ¼ b rdr þ rdb r ¼ b rdr þ r
∂b r
∂r
dr þ
∂b r
∂θ
dθ þ
∂b r
∂ϕ
dϕ
and by using the previous relationships for the derivatives of these unit vectors we
can write the previous equation in the form,
d r
! ¼ b rdr þ b θrdθ þ b
ϕrSinθdϕ:
The Vector Differential Operator del, Written as —
→
, in Spherical Coordinates
The differential operator ∇
!
in a Cartesian coordinate system is
∇
! ¼ b x
∂
∂x
þ b y
∂
∂y
þ b z
∂
∂z
,
and we need to express this operator in terms of the unit vectors b r, b θ, b
ϕ in the
spherical coordinate system. In order to so this, let us consider a function Ψ that
depends on r, θ, ϕ, that is, Ψ ¼ Ψ(r, θ, ϕ), then the derivative dΨ is
dΨ ¼
∂Ψ
∂r
dr þ
∂Ψ
∂θ
dθ þ
∂Ψ
∂ϕ
dϕ
and this can also be written as
dΨ ¼ ∇
! Ψ Á d r
!
1.7 Spherical Geometry
27