It is not that polar co-ordinates are complicated,
it is simply that cartesian co-ordinates are simpler than they have a right to
be.
Kleppner & Kolenhow, An Introduction to
Mechanics
Spherical Polar co-ordinates
Particularly when studying situations symmetric under rotations about at
least some axis through some point (axial symmetry) and especially when any axis
through that point would suffice (spherical symmetry), it is helpful to describe
three-dimensional space in terms of a distance from the given point and a
description of direction in terms of position on a standard sphere about the
point. One standard system of co-ordinates of this form is known as
spherical polar co-ordinates. The central point is taken as origin and
an axis through it, about which all rotations are symmetries of the system, is
selected. Distance from the central point is known as radius and written r; but
the word radius is also used for the line (or displacement vector) from
the origin to the point; r is the length of the radius, in this sense of the
word. Spherical polar co-ordinates describe direction in terms of one
co-ordinate up and down the axis – called latitude and defined by an angle
relative to the axis – and annother co-ordinate about the axis
– called longitude and defined relative to some half-plane whose boundary
is the axis.
The latitude is commonly called θ but (since typing θ is
rather fiddly) I generally call it m if I'm going to refer to it a lot. It
takes the value zero on a plane through our central point and perpendicular to
the axis; its magnitude is the angle between the radius and this plane; it is
positive on one side of the plane and negative on the other side. It is thus
minus a quarter turn on one half of the axis and plus a quarter turn on the
other. The longitude is commonly called φ but (since typing φ is
rather fiddly) I generally call it n if I'm going to refer to it a lot. It
takes the value zero on a (usually arbitrary) half-plane whose boundary is our
axis; its magnitude is the angle through which one must rotate about the axis to
bring the radius into this half-plane; it is positive for one sense of rotation
and negative for the other. If an observer on the half-axis with m positive is
so oriented as to deem the direction of increasing m to be up and the
direction, along the ray on which m and n are both zero, away from the origin to
be forwards, then small negative values of n must fall on the right side,
and small positive values of n on the left side, of the half plane on which n is
zero.
Since angles are intrinsically periodic, with period one turn, our longitude
is in principle multi-valued; its values at any given point differ from one
another by integer multiples of a turn and are equivalent (as angles). For the
sake of clarity, it is usual to select a particular interval of length one turn
and render the longitude single-valued by always using its value in that
interval. When this is appropriate, I use the interval from minus a half turn
to plus a half turn, so that the half-plane on which longitude is zero falls in
mid-interval; other authors commonly use the interval from zero to one turn (and
measure angles in radians, so describe this as the interval from 0 to 2.π; or
describe the interval I use as running from −π to +π).
Note that the restriction to the plane of zero latitude yields plane polar
co-ordinates. Simple projection parallel to the axis onto this plane yields a
related co-ordinate system for three dimensions, which is particularly apt to
the case where there is rotational symmetry only about this one axis, or where
the plane of zero longitude has special significance: cylindrical polar
co-ordinates, in which r and m (a.k.a. θ) are replaced by an axial radius,
the distance from the axis, and a co-ordinate parallel to the axis. The latter
is generally called z (it's the same co-ordinate as shall have that name below)
and I'll call the former R (though it's often called r) to distinguish it from r
here. These are related to r and m by: R = r.cos(m), z = r.sin(m); R.R +z.z =
r.r, tan(m) = z/R.
Relation to Cartesian co-ordinates
We can also construct rectilinear co-ordinates (also known as Cartesian
co-ordinates, after René Descartes, who popularised their use). Take the
ray on which both our angular co-ordinates are zero as x axis, the ray of zero
latitude with longitude a quarter turn as y axis and the ray on which latitude
is a positive quarter turn as z axis. Decompose any displacement into its
components parallel to these three rays and use the lengths of these components,
negated if in the opposite direction to the relevant ray, as co-ordinates
associated with x, y and z; these co-ordinates are orthonormal and
right-handed. They are related to the spherical polar co-ordinates by:
x = r.Cos(m).Cos(n)
y = r.Cos(m).Sin(n)
z = r.Sin(m)
(Here Cos and Sin are functions ({reals}:|{angles}) related to the
exponential function by exp(i.a/radian) = Cos(a) +i.Sin(a) where i is a square
root of −1 and 2.π.radian = turn.) These imply r.r = x.x +y.y +z.z
and
as the three co-ordinates of the spatial differential operator in three
dimensions, ∇ = [∂/∂x, ∂/∂y, ∂/∂z].
The metric and the measure
The gradient fields dr, dm and dn form a basis of gradients at each point
off the axis, just as dx, dy and dz do everywhere. The metric is a tensor g for
which v·g·v is the square of the length of v, for any displacement
v; consequently, g is a linear map from displacements to gradients; as such, it
can be expressed as a sum of terms, each of which is a tensor product of two
gradients, optionally times a scalar. Indeed, since [x, y, z] are orthonormal
co-ordinates, g = dx×dx +dy×dy +dz×dz. We may thus express it
in terms of dr, dm and dn:
This has determinant (for some displacements s, w and u satisfying 1 =
dr·s = dm·w = dn·u and 0 = dr·w = dr·u =
dm·u = dm·s = dn·s = dn·w, so that [s,w,u] is the
basis of displacements dual to the basis [dr,dm,dn] of gradients)
whose positive square root (note that Cos(m) is positive everywhere off
the axis) is r.r.Cos(m).dr^dm^dn/radian/radian: this, then, is the measure,
which mediates integration by contracting with three displacements to yield the
volume of a parallelepiped whose sides are the three displacements. We might
equally have obtained this as
On the unit sphere, g's restriction is simply (dm×dm
+Cos(m).Cos(m).dn×dn)/radian/radian with determinant
power(2,Cos(m)/radian/radian).(dn^dm)×(dn^dm) yielding measure (or area
element) Cos(m).dn^dm/radian/radian, which we might equally have obtained
from the full measure by observing that dr is a true length differential, so
what it's ^-ed with for the measure must be the area element on surfaces of
constant r.
Derivatives
Our co-ordinates are scalar functions of position (albeit they are not
dimensionless; r has units of length while m and n have units of angle), so
their gradients are co-vector fields. On the axis, dm and dn are degenerate,
with r also degenerate at the origin; but everywhere else they form a basis b =
[dr, dm, dn] of co-vectors. Introduce the dual basis of vectors, p = [q, w, u],
and note that neither basis is independent of position – the basis members
change direction and (in some cases) magnitude from place to place. We use a
differential operator, D, which considers the metric constant, i.e. D(g) = 0.
We need to know the action of D on our bases in order to determine the
derivative of anything other than scalars.
We can also express the tangent to a curve specified in polar co-ordinates
in terms of q, w and u; this is actually relatively simple, though. On such a
curve r, m and n are functions of a scalar parameter along the curve, which I'll
call t (but it needn't be time). The tangent vector is then simply q.dr/dt
+w.dm/dt +u.dn/dt, as you might expect.
Written by Eddy.