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CURVILINEAR COORDINATES

Def. Reciprocal sets of vectors. The sets of vectors a, b, c and a', b', c' are called reciprocal sets or systems of vectors if

a∙a' = b∙b' = c∙c' = 1

a'∙b = a'∙c = b'∙a = b'∙c = c'∙a = c'∙b = 0

The sets a, b, c and a', b', c' are reciprocal sets if and only if

Example. The sets

a = (2, 3, -1), b = (1, -1 -2), c = (-1, 2, 2)

and

a' = (2/3, 0, 1/3), b' = (-8/3, 1, -7/3), c' = (-7/3, 1, -5/3)

are reciprocal set of vectors.

Coordinate transformations. Many problems in physics and elsewhere can be more easily and naturally formulated, analyzed and solved in some non-rectangular coordinate system such as a cylindrical or spherical coordinate system than they can in the usual rectangular Cartesian coordinate system. When using different coordinate systems we often wish to convert coordinates from one system to another. We do this using transformation equations which relate the coordinates of a general point P as referred to the two coordinate systems. For coordinate transformations in three dimensional space these transformation equations have the general form

u_{1} = u_{1}(x, y, z)

1) u_{2} = u_{2}(x, y, z)

u_{3} = u_{3}(x, y, z)

along with an inverse transformation

x = x(u_{1}, u_{2}, u_{3})

2) y = y(u_{1}, u_{2}, u_{3})

z = z(u_{1}, u_{2}, u_{3}) .

For example, cylindrical coordinates are related to rectangular coordinates by the transformation

and the inverse transformation

x = r cos θ

y = r sin θ

z = z

where r, θ, and z correspond to the u_{1}, u_{2}, and u_{3 }of systems 1) and 2).

A general characteristic of a system of equations of type 1) above that is being used in this way --- i.e. used to relate the coordinates of some general point P in space as referred two different coordinate systems — is that it represents a one-to-one mapping over that domain on which it will be used and thus possesses an inverse on that domain.

The various non-rectangular coordinate systems have given rise to a generalization of the concept of a coordinate system in the idea of a generalized (curvilinear) coordinate system employing curvilinear coordinates as follows.

Curvilinear coordinates. Consider the system of equations

3) u = u(x, y)

v = v(x, y) .

This system can be viewed as a function that assigns a number pair (u, v) to each number pair (x,
y). Let us suppose that the system effects a one-to-one mapping over some region R of the xy-plane, mapping region R into region R' of the uv-plane. This will be the case if the
transformation is continuously differentiable and the Jacobian does not vanish at any point within
region R. In this case this system of equations associates a unique pair of numbers (u, v) to each
number pair (x, y) in region R. If a point P_{0} has coordinates (x_{0}, y_{0}) in the xy-plane
transformation 3) associates with point P_{0} the number pair (u_{0}, v_{0}). Why can we not use (u_{0}, v_{0})
as the coordinates of point P_{0}? Why can we not regard system 3) as defining a transformation
between the x-y coordinate system and some other unknown coordinate system (a coordinate
system whose nature and characteristics we don’t know) and refer to point P by its coordinates in
that system? This is the idea underlying a generalized (curvilinear) coordinate system. An
important stipulation on the use of this idea, however, is that it can only be used in a region
where the mapping is one-to-one. Depending on the defining equations 3), the domain of the
mapping might be the entire xy-plane or it might be necessary to limit the domain to a relatively
small region in the plane or even just the immediate neighborhood of some selected point. The
domain must be chosen in such a way so as to avoid any points where the Jacobian vanishes. For
insight into the type of situation one may encounter consider the following question

Question. Given some selected point (u_{0}, v_{0}) in the uv-plane, what points in the xy-plane will
map into it?

Answer. Assume the loci of the equations

u(x, y) = u_{0}

v(x, y) = v_{0}

are those shown in Fig. 1.
Then the function u(x, y)
maps all points on the
locus of u(x, y) = u_{0} into
the number u_{0}. Similarly,
the function v(x, y) maps
all points on the locus of
v(x, y) = v_{0} into the
number v_{0}. Thus all points
of intersection of the
curves u(x, y) = u_{0} and
v(x, y) = v_{0} map into the
point (u_{0}, v_{0}) in the uv-plane. We see in the figure
that four points in the xy-plane map into the single point (u_{0}, v_{0}) in the uv-plane. Obviously the
domain for our mapping in this case cannot be chosen as the entire xy-plane. One could,
however, choose some point, such as the point P shown in the figure, and then use as the domain
of the mapping a relatively small region in the vicinity of P. Within this relatively small region
one could employ curvilinear coordinates.

u-curves and v-curves. Let us now suppose that the mapping 3) effects a one-to-one mapping over some region R of the xy-plane, mapping region R into region R' of the uv-plane and that the inverse transformation is given by

4) x = x(u, v)

y = y(u, v) .

If we regard u as fixed in system 4) i.e. u = c_{1}, a constant, then system 4) becomes a system in a
single variable v describing a curve where v is the varying parameter. For each different value of
u there is a separate curve. These curves are called v-curves. Similarly one can let v = c_{2}, a
constant,
and
obtain
a u-curve
where
u is
the
varying
parameter.
One
thus obtains families of u-curves and v-curves. The u-curves and v-curves are called coordinate
curves. These coordinate

curves form a curvilinear net on the surface similar to the coordinate net on a plane.

Fig. 2 shows a mesh of quadrilaterals formed by the u-curves and v-curves in the xy-plane by a
parameterization of u and v in the immediate vicinity of a point of interest P. The u-curves and
v-curves map into mutually perpendicular lines in the uv-plane. Points on the v-curve u(x, y) =
c_{1} map into the line u = c_{1} and points on the u-curve v(x, y) = c_{2} map into the line v = c_{2.}

Curvilinear coordinates of a point on a surface in 3-space. Let S be a surface element defined by the one-to-one mapping

x = x(u, v)

5) y = y(u, v)

z = z(u, v)

of a region R of the uv-plane into xyz-space. The parametric equations 5) assign a point (x, y, z) to each number pair (u, v). The number pair (u, v) can be considered a set of coordinates for a point P(x, y, z). The numbers (u, v) are called the curvilinear coordinates of point P. If we regard u as fixed in system 5) i.e. u = c, a constant, then system 5) becomes a system in a single variable v describing a space curve where v is the varying parameter. For each different value of u there is a separate space curve. These curves are called v-curves. Similarly one can let v = k, a constant, and obtain a u-curve where u is the varying parameter. One thus obtains families of u-curves and v-curves. The u-curves and v-curves are called coordinate curves. These coordinate curves form a curvilinear net on the surface similar to the coordinate net on a plane. See Fig. 3.

Curvilinear coordinates for a 3-space to 3-space mapping. Let us now consider the one-to-one mapping

x = x(u_{1}, u_{2}, u_{3})

6) y = y(u_{1}, u_{2}, u_{3})

z = z(u_{1}, u_{2}, u_{3})

defined over some region R of u_{1}-u_{2}-u_{3} space, mapping R into region R' in xyz-space. Let the
inverse transformation be

u_{1} = u_{1}(x, y, z)

7) u_{2} = u_{2}(x, y, z)

u_{3} = u_{3}(x, y, z) .

Region R might be all of u_{1}-u_{2}-u_{3} space or only some small, localized region. Within region R
the mapping is continuously differentiable and the Jacobian is nonvanishing.

The parametric equations 6) assign a point
(x, y, z) to each number triple (u_{1}, u_{2}, u_{3}).
The number triple (u_{1}, u_{2}, u_{3}) can be
considered a set of coordinates for a point
P(x, y, z). The numbers (u_{1}, u_{2}, u_{3}) are
called the curvilinear coordinates of
point P.

The three surfaces

u_{1}(x, y, z) = c_{1}

u_{2}(x, y, z) = c_{2}

u_{3}(x, y, z) = c_{3}

where c_{1}, c_{2}, c_{3} are constants, are called
coordinate surfaces and each pair of these surfaces intersect in curves called coordinate
curves. See Fig. 4. Each of the three surfaces represents one of a family of surfaces generated
by different values of the parameter c_{i} (constant term). The family corresponding to u_{1}(x, y, z) =
c_{1} are called the u_{1} surfaces, the family corresponding to u_{2}(x, y, z) = c_{2} are called the u_{2}
surfaces, etc. The curve corresponding to the intersection of a u_{1} surface with a u_{2} surface is
called a u_{3}-curve (it is also sometimes called a u_{1}u_{2}-curve, depending on the author). See Fig. 3.
Similarly for the other coordinate curves.

If we regard u_{2} and u_{3} as fixed in system 6) i.e. u_{2} = c_{2}, u_{3} = c_{3}, then 6) becomes a system in a
single variable u_{1} describing a space curve where u_{1} is the varying parameter, namely the u_{1}-curve. If we regard u_{1} and u_{3} as fixed in system 6) i.e. u_{1} = c_{1}, u_{3} = c_{3}, then 6) becomes a system
in a single variable u_{2} describing a space curve where u_{2} is the varying parameter, namely the u_{2}-curve. Similarly for the u_{3}-curves.

The u_{1}, u_{2}, and u_{3} coordinate curves of a curvilinear system correspond to the x, y and z axes of a
rectangular system. Since the three coordinate curves are generally not straight lines, as in the
rectangular coordinate system, such coordinate systems are called curvilinear coordinate
systems.

Curvilinear trihedrals. In a rectangular
Cartesian coordinate system we have the base
vectors i, j, k which are unit vectors in the
directions of the x, y and z axes. We can
introduce a similar set of unit base vectors for
curvilinear coordinate systems. In fact, we
introduce two sets. The first set is the set of unit
vectors e_{1}, e_{2}, e_{3} tangent at P to the coordinate
curves through P. The second set is the set of
unit vectors E_{1}, E_{2}, E_{3} normal at P to the
coordinate surfaces through P. See Fig. 5. Both
sets are analogous to the i, j, k unit vectors in
rectangular coordinates but are unlike them in that they may change direction from point to point.
It can be shown that the two sets, e_{1}, e_{2}, e_{3} and E_{1}, E_{2}, E_{3}, represent reciprocal systems of vectors.

Expressions for the unit base vectors e_{1}, e_{2}, e_{3} and E_{1}, E_{2}, E_{3}.. Let r = x i + y j + z
k be a position vector of a point P. Then, from 5),

r = x(u_{1}, u_{2}, u_{3}) i + y(u_{1}, u_{2}, u_{3}) j + z(u_{1}, u_{2}, u_{3}) k .

● The three unit vectors e_{1}, e_{2}, e_{3} tangent to the u_{1}, u_{2}, and u_{3} coordinate curves are given by

where

The quantities h_{1}, h_{2} and h_{3} are called scale factors. The unit vectors e_{1}, e_{2}, e_{3} are in the
directions of increasing u_{1}, u_{2}, u_{3}, respectively.

● The three unit vectors E_{1}, E_{2}, E_{3} normal to the u_{1}, u_{2}, and u_{3} coordinate surfaces are given by

where

Vector representation in terms of base vectors. A vector A can be represented
in terms of the unit base vectors e_{1}, e_{2}, e_{3} or E_{1}, E_{2}, E_{3} in the form

A = A_{1}e_{1} + A_{2}e_{3} + A_{3}e_{3} = a_{1}E_{1} + a_{2}E_{3} + a_{3}E_{3}

where A_{1}, A_{2}, A_{3} and a_{1}, a_{2}, a_{3} are the respective components of A in each system.

We can also represent A in terms of the base vectors

which are called unitary base vectors but are not unit vectors, in general. In this case

and

where

C_{1}, C_{2}, C_{3} are called the contravariant components of A and c_{1}, c_{2}, c_{3} are called the covariant
components of A.

Orthogonal curvilinear coordinates. A curvilinear coordinate system is called
orthogonal if the coordinate curves are everywhere orthogonal. In this case, the three vectors e_{1},
e_{2}, e_{3} are mutually orthogonal at every point i.e.

e_{1}•e_{2} = e_{2}•e_{3} = e_{3}•e_{1} = 0 .

We shall also assume that e_{1}, e_{2}, e_{3} form a right handed system.

● In an orthogonal curvilinear coordinate system the base vectors e_{1}, e_{2}, e_{3} and E_{1}, E_{2}, E_{3}
become identical i.e.

e_{1} = E_{1}, e_{2} = E_{2}, e_{3} = E_{3} .

● In an orthogonal curvilinear coordinate system the following hold:

Arc length and volume elements. From

r = x(u_{1}, u_{2}, u_{3}) i + y(u_{1}, u_{2}, u_{3}) j + z(u_{1}, u_{2}, u_{3}) k

we obtain

The differential of arc length ds is determined from ds^{2} = dr•dr. For orthogonal systems, e_{1}•e_{2}
= e_{2}•e_{3} = e_{3}•e_{1} = 0 and

ds^{2} = h_{1}^{2} du_{1}^{2} + h_{2}^{2} du_{2}^{2} + h_{3}^{2} du_{3}^{2}

Along a u_{1} curve, u_{2} and u_{3} are constants so dr =
h_{1} du_{1} e_{1}. Thus the differential of arc length ds_{1}
along u_{1} at P is h_{1} du_{1} and the differential arc
lengths along u_{2} and u_{3} are ds_{2} = h_{2} du_{2} and ds_{3} =
h_{3} du_{3}.

Thus, referring to Fig. 6, the volume element for an orthogonal curvilinear coordinate system is given by

Expressions for the gradient, divergence and curl in terms of
curvilinear coordinates. If Φ is a scalar function and A = A_{1}e_{1} + A_{2}e_{3} + A_{3}e_{3} is a
vector function of orthogonal curvilinear coordinates u_{1}, u_{2}, u_{3} then the following hold.

Special orthogonal curvilinear coordinate systems. The following are examples of some orthogonal curvilinear coordinate systems.

1. Cylindrical coordinates

2. Spherical coordinates

3. Parabolic cylindrical coordinates

4. Parabolic coordinates

5. Elliptic cylindrical coordinates

6. Prolate spheroidal coordinates

7. Oblate spheroidal coordinates

8. Ellipsoidal coordinates

9. Bipolar coordinates

See Murray R. Spiegel. Vector Analysis. (Schaum) for more information on these coordinate systems.

References.

Spiegel. Vector Analysis. Chap. 7.

Hsu. Vector Analysis.

Taylor. Advanced Calculus.

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