Initially, we will spend some time developing various vector relationships that
will be required for writing the conservation equations in spherical geometry.
Relationships Between the Unit Vectors
In a Cartesian coordinate system the position vector r
! is given by the equation,
r
! ¼ xb x þ yb y þ zb z,
where b x, b y and b z are unit vectors in the x, y and z directions, respectively. In terms of
the spherical coordinate system as shown in Fig. 1.4, the latter equation becomes,
r
! ¼ rSinθCosϕ
ð
Þ b x þ rSinθSinϕ
ð
Þ b y þ rCosθ
ð
Þ b z,
since x ¼ rSinθCosϕ, y ¼ rSinθSinϕ and z ¼ rCosθ from Fig. 1.4. Tangent vectors in
the r, θ, ϕ directions are given by ∂ r
!
=∂r, ∂ r
!
=∂θ and ∂ r
!
=∂ϕ, respectively, and
unit vectors (b r, b θ, b
ϕ) in these directions as shown in Fig. 1.4 are given by
b r ¼
∂ r
!
=∂r
∂ r
!
=∂r
, b θ ¼
∂ r
!
=∂θ
∂ r
!
=∂θ
and b
ϕ ¼
∂ r
!
=∂ϕ
∂ r
!
=∂ϕ
:
By carrying out the differentiation of the vector r
! as prescribed by these latter
equations, we obtain the following set of relationships for the unit vectors in the
spherical coordinate system in terms of the unit vectors in the Cartesian coordinate
system,
Fig. 1.4 Spherical (r, θ, ϕ)
coordinate system is shown
with unit vectors in the b r, b θ
and b
ϕ directions
1.7 Spherical Geometry
25