9Orthogonal curvilinear coordinates
IA Vector Calculus
9.1 Line, area and volume elements
In this chapter, we study funny coordinate systems. A coordinate system is,
roughly speaking, a way to specify a point in space by a set of (usually 3)
numbers. We can think of this as a function r(u, v, w).
By the chain rule, we have
dr =
∂r
∂u
du +
∂r
∂v
dv +
∂r
∂w
dw
For a good parametrization,
∂r
∂u
·
∂r
∂v
×
∂r
∂w
6= 0,
i.e.
∂r
∂u
,
∂r
∂v
and
∂r
∂w
are linearly independent. These vectors are tangent to the
curves parametrized by u, v, w respectively when the other two are being fixed.
Even better, they should be orthogonal:
Definition
(Orthogonal curvilinear coordinates)
. u, v, w
are orthogonal curvi-
linear if the tangent vectors are orthogonal.
We can then set
∂r
∂u
= h
u
e
u
,
∂r
∂v
= h
v
e
v
,
∂r
∂w
= h
w
e
w
,
with
h
u
, h
v
, h
w
>
0 and
e
u
, e
v
, e
w
form an orthonormal right-handed basis (i.e.
e
u
× e
v
= e
w
). Then
dr = h
u
e
u
du + h
v
e
v
dv + h
w
e
w
dw,
and
h
u
, h
v
, h
w
determine the changes in length along each orthogonal direction
resulting from changes in u, v, w. Note that clearly by definition, we have
h
u
=
∂r
∂u
.
Example.
(i)
In cartesian coordinates,
r
(
x, y, z
) =
x
ˆ
i
+
y
ˆ
j
+
z
ˆ
k
. Then
h
x
=
h
y
=
h
z
= 1,
and e
x
=
ˆ
i, e
y
=
ˆ
j and e
z
=
ˆ
k.
(ii)
In cylindrical polars,
r
(
ρ, ϕ, z
) =
ρ
[
cos ϕ
ˆ
i
+
sin ϕ
ˆ
j
] +
z
ˆ
k
. Then
h
ρ
=
h
z
= 1,
and
h
ϕ
=
∂r
∂ϕ
= |(−ρ sin ϕ, ρ sin ϕ, 0)| = ρ.
The basis vectors e
ρ
, e
ϕ
, e
z
are as in section 1.
(iii) In spherical polars,
r(r, θ, ϕ) = r(cos ϕ sin θ
ˆ
i + sin θ sin ϕ
ˆ
j + cos θ
ˆ
k).
Then h
r
= 1, h
θ
= r and h
ϕ
= r sin θ.
Consider a surface with
w
constant and parametrised by
u
and
v
. The vector
area element is
dS =
∂r
∂u
×
∂r
∂v
du dv = h
u
e
u
× h
v
e
v
du dv = h
u
h
v
e
w
du dv.
We interpret this as
δS
having a small rectangle with sides approximately
h
u
δu
and h
v
δv. The volume element is
dV =
∂r
∂u
·
∂r
∂v
×
∂r
∂w
du dv dw = h
u
h
v
h
w
du dv dw,
i.e. a small cuboid with sides h
u
δ
u
, h
v
δ
v
and h
w
δ
w
respectively.