General curvilinear coordinates

VECTOR CALCULUS
10.10 General curvilinear coordinates
As indicated earlier, the contents of this section are more formal and technically
complicated than hitherto. The section could be omitted until the reader has had
some experience of using its results.
Cylindrical and spherical polars are just two examples of what are called
general curvilinear coordinates. In the general case, the position of a point P
having Cartesian coordinates x, y, z may be expressed in terms of the three
curvilinear coordinates u1 , u2 , u3 , where
x = x(u1 , u2 , u3 ),
y = y(u1 , u2 , u3 ),
z = z(u1 , u2 , u3 ),
u1 = u1 (x, y, z),
u2 = u2 (x, y, z),
u3 = u3 (x, y, z).
and similarly
We assume that all these functions are continuous, differentiable and have a
single-valued inverse, except perhaps at or on certain isolated points or lines,
so that there is a one-to-one correspondence between the x, y, z and u1 , u2 , u3
systems. The u1 -, u2 - and u3 - coordinate curves of a general curvilinear system
are analogous to the x-, y- and z- axes of Cartesian coordinates. The surfaces
u1 = c1 , u2 = c2 and u3 = c3 , where c1 , c2 , c3 are constants, are called the
coordinate surfaces and each pair of these surfaces has its intersection in a curve
called a coordinate curve or line (see figure 10.11).
If at each point in space the three coordinate surfaces passing through the point
meet at right angles then the curvilinear coordinate system is called orthogonal.
For example, in spherical polars u1 = r, u2 = θ, u3 = φ and the three coordinate
surfaces passing through the point (R, Θ, Φ) are the sphere r = R, the circular
cone θ = Θ and the plane φ = Φ, which intersect at right angles at that
point. Therefore spherical polars form an orthogonal coordinate system (as do
cylindrical polars) .
If r(u1 , u2 , u3 ) is the position vector of the point P then e1 = ∂r/∂u1 is a vector
tangent to the u1 -curve at P (for which u2 and u3 are constants) in the direction
of increasing u1 . Similarly, e2 = ∂r/∂u2 and e3 = ∂r/∂u3 are vectors tangent to
the u2 - and u3 - curves at P in the direction of increasing u2 and u3 respectively.
Denoting the lengths of these vectors by h1 , h2 and h3 , the unit vectors in each of
these directions are given by
ê1 =
1 ∂r
,
h1 ∂u1
ê2 =
1 ∂r
,
h2 ∂u2
ê3 =
1 ∂r
,
h3 ∂u3
where h1 = |∂r/∂u1 |, h2 = |∂r/∂u2 | and h3 = |∂r/∂u3 |.
The quantities h1 , h2 , h3 are the scale factors of the curvilinear coordinate
system. The element of distance associated with an infinitesimal change dui in
one of the coordinates is hi dui . In the previous section we found that the scale
364
10.10 GENERAL CURVILINEAR COORDINATES
u3
z
ê3 ˆ 3
u2 = c2
ˆ 2
ê1
ˆ 1
u1
u1 = c1
P
u2
ê2
u3 = c3
k
O
j
y
i
x
Figure 10.11 General curvilinear coordinates.
factors for cylindrical and spherical polar coordinates were
for cylindrical polars
for spherical polars
hρ = 1,
hr = 1,
hφ = ρ,
hθ = r,
hz = 1,
hφ = r sin θ.
Although the vectors e1 , e2 , e3 form a perfectly good basis for the curvilinear
coordinate system, it is usual to work with the corresponding unit vectors ê1 , ê2 ,
ê3 . For an orthogonal curvilinear coordinate system these unit vectors form an
orthonormal basis.
An infinitesimal vector displacement in general curvilinear coordinates is given
by, from (10.19),
∂r
∂r
∂r
du1 +
du2 +
du3
∂u1
∂u2
∂u3
= du1 e1 + du2 e2 + du3 e3
dr =
= h1 du1 ê1 + h2 du2 ê2 + h3 du3 ê3 .
(10.55)
(10.56)
(10.57)
In the case of orthogonal curvilinear coordinates, where the êi are mutually
perpendicular, the element of arc length is given by
(ds)2 = dr · dr = h21 (du1 )2 + h22 (du2 )2 + h23 (du3 )2 .
(10.58)
The volume element for the coordinate system is the volume of the infinitesimal
parallelepiped defined by the vectors (∂r/∂ui ) dui = dui ei = hi dui êi , for i = 1, 2, 3.
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VECTOR CALCULUS
For orthogonal coordinates this is given by
dV = |du1 e1 · (du2 e2 × du3 e3 )|
= |h1 ê1 · (h2 ê2 × h3 ê3 )| du1 du2 du3
= h1 h2 h3 du1 du2 du3 .
Now, in addition to the set {êi }, i = 1, 2, 3, there exists another useful set of
three unit basis vectors at P . Since ∇u1 is a vector normal to the surface u1 = c1 ,
a unit vector in this direction is ˆ 1 = ∇u1 /|∇u1 |. Similarly, ˆ 2 = ∇u2 /|∇u2 | and
ˆ 3 = ∇u3 /|∇u3 | are unit vectors normal to the surfaces u2 = c2 and u3 = c3
respectively.
Therefore at each point P in a curvilinear coordinate system, there exist, in
general, two sets of unit vectors: {êi }, tangent to the coordinate curves, and {ˆi },
normal to the coordinate surfaces. A vector a can be written in terms of either
set of unit vectors:
a = a1 ê1 + a2 ê2 + a3 ê3 = A1 ˆ 1 + A2 ˆ 2 + A3 ˆ 3 ,
where a1 , a2 , a3 and A1 , A2 , A3 are the components of a in the two systems. It
may be shown that the two bases become identical if the coordinate system is
orthogonal.
Instead of the unit vectors discussed above, we could instead work directly with
the two sets of vectors {ei = ∂r/∂ui } and {i = ∇ui }, which are not, in general, of
unit length. We can then write a vector a as
a = α1 e1 + α2 e2 + α3 e3 = β1 1 + β2 2 + β3 3 ,
or more explicitly as
a = α1
∂r
∂r
∂r
+ α2
+ α3
= β1 ∇u1 + β2 ∇u2 + β3 ∇u3 ,
∂u1
∂u2
∂u3
where α1 , α2 , α3 and β1 , β2 , β3 are called the contravariant and covariant components of a respectively. A more detailed discussion of these components, in
the context of tensor analysis, is given in chapter 26. The (in general) non-unit
bases {ei } and {i } are often the most natural bases in which to express vector
quantities.
Show that {ei } and {i } are reciprocal systems of vectors.
Let us consider the scalar product ei · j ; using the Cartesian expressions for r and ∇, we
obtain
∂r
ei · j =
· ∇uj
∂u
i
∂uj
∂uj
∂uj
∂y
∂z
∂x
=
i+
j+
k ·
i+
j+
k
∂ui
∂ui
∂ui
∂x
∂y
∂z
∂y ∂uj
∂z ∂uj
∂uj
∂x ∂uj
.
+
+
=
=
∂ui ∂x
∂ui ∂y
∂ui ∂z
∂ui
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10.10 GENERAL CURVILINEAR COORDINATES
In the last step we have used the chain rule for partial differentiation. Therefore ei · j = 1
if i = j, and ei · j = 0 otherwise. Hence {ei } and {j } are reciprocal systems of vectors. We now derive expressions for the standard vector operators in orthogonal
curvilinear coordinates. Despite the useful properties of the non-unit bases discussed above, the remainder of our discussion in this section will be in terms of
the unit basis vectors {êi }. The expressions for the vector operators in cylindrical
and spherical polar coordinates given in tables 10.2 and 10.3 respectively can be
found from those derived below by inserting the appropriate scale factors.
Gradient
The change dΦ in a scalar field Φ resulting from changes du1 , du2 , du3 in the
coordinates u1 , u2 , u3 is given by, from (5.5),
∂Φ
∂Φ
∂Φ
du1 +
du2 +
du3 .
∂u1
∂u2
∂u3
dΦ =
For orthogonal curvilinear coordinates u1 , u2 , u3 we find from (10.57), and comparison with (10.27), that we can write this as
dΦ = ∇Φ · dr,
(10.59)
where ∇Φ is given by
∇Φ =
1 ∂Φ
1 ∂Φ
1 ∂Φ
ê1 +
ê2 +
ê3 .
h1 ∂u1
h2 ∂u2
h3 ∂u3
(10.60)
This implies that the del operator can be written
∇=
ê2 ∂
ê3 ∂
ê1 ∂
+
+
.
h1 ∂u1
h2 ∂u2
h3 ∂u3
Show that for orthogonal curvilinear coordinates ∇ui = êi /hi . Hence show that the two
sets of vectors {êi } and {ˆi } are identical in this case.
Letting Φ = ui in (10.60) we find immediately that ∇ui = êi /hi . Therefore |∇ui | = 1/hi , and
so ˆ i = ∇ui /|∇ui | = hi ∇ui = êi . Divergence
In order to derive the expression for the divergence of a vector field in orthogonal
curvilinear coordinates, we must first write the vector field in terms of the basis
vectors of the coordinate system:
a = a1 ê1 + a2 ê2 + a3 ê3 .
The divergence is then given by
∂
1
∂
∂
∇·a=
(h2 h3 a1 ) +
(h3 h1 a2 ) +
(h1 h2 a3 ) .
h1 h2 h3 ∂u1
∂u2
∂u3
367
(10.61)
VECTOR CALCULUS
Prove the expression for ∇ · a in orthogonal curvilinear coordinates.
Let us consider the sub-expression ∇ · (a1 ê1 ). Now ê1 = ê2 × ê3 = h2 ∇u2 × h3 ∇u3 . Therefore
∇ · (a1 ê1 ) = ∇ · (a1 h2 h3 ∇u2 × ∇u3 ),
= ∇(a1 h2 h3 ) · (∇u2 × ∇u3 ) + a1 h2 h3 ∇ · (∇u2 × ∇u3 ).
However, ∇ · (∇u2 × ∇u3 ) = 0, from (10.43), so we obtain
ê3
ê1
ê2
= ∇(a1 h2 h3 ) ·
∇ · (a1 ê1 ) = ∇(a1 h2 h3 ) ·
×
;
h2
h3
h2 h3
letting Φ = a1 h2 h3 in (10.60) and substituting into the above equation, we find
∇ · (a1 ê1 ) =
1
∂
(a1 h2 h3 ).
h1 h2 h3 ∂u1
Repeating the analysis for ∇ · (a2 ê2 ) and ∇ · (a3 ê3 ), and adding the results we obtain (10.61),
as required. Laplacian
In the expression for the divergence (10.61), let
a = ∇Φ =
1 ∂Φ
1 ∂Φ
1 ∂Φ
ê1 +
ê2 +
ê3 ,
h1 ∂u1
h2 ∂u2
h3 ∂u3
where we have used (10.60). We then obtain
∂
h2 h3 ∂Φ
h3 h1 ∂Φ
h1 h2 ∂Φ
1
∂
∂
∇2 Φ =
+
+
,
h1 h2 h3 ∂u1
h1 ∂u1
∂u2
h2 ∂u2
∂u3
h3 ∂u3
which is the expression for the Laplacian in orthogonal curvilinear coordinates.
Curl
The curl of a vector field a = a1 ê1
coordinates is given by
1 ∇×a=
h1 h2 h3 + a2 ê2 + a3 ê3 in orthogonal curvilinear
h1 ê1
h2 ê2
∂
∂u1
h1 a1
∂
∂u2
h2 a2
h3 ê3 ∂ .
∂u3 h3 a3 Prove the expression for ∇ × a in orthogonal curvilinear coordinates.
Let us consider the sub-expression ∇ × (a1 ê1 ). Since ê1 = h1 ∇u1 we have
∇ × (a1 ê1 ) = ∇ × (a1 h1 ∇u1 ),
= ∇(a1 h1 ) × ∇u1 + a1 h1 ∇ × ∇u1 .
But ∇ × ∇u1 = 0, so we obtain
∇ × (a1 ê1 ) = ∇(a1 h1 ) ×
368
ê1
.
h1
(10.62)