]> Spherical Polar co-ordinates

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 &theta; 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 &phi; 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

x.dy −y.dx
dz.Cos(m) −Sin(m).(dx.Cos(n) +dy.Sin(n))
= (z.Cos(m) −x.Sin(m).Cos(n) −y.Sin(m).Sin(n)).dr/r
= (r.Sin(m).Cos(m) −r.Cos(m).Sin(m).(Cos(n).Cos(n) +Sin(n).Sin(n))).dr/r
x.dx +y.dy +z.dz
= r.dr

from which we obtain

∂/∂x
= (∂r/∂x).∂/∂r +(∂m/∂x).∂/∂m +(∂n/∂x).∂/∂n
∂/∂y
= (∂r/∂y).∂/∂r +(∂m/∂y).∂/∂m +(∂n/∂y).∂/∂n
∂/∂z
= (∂r/∂z).∂/∂r +(∂m/∂z).∂/∂m +(∂n/∂z).∂/∂n

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:

g
= dx×dx +dy×dy +dz×dz
= dr×dr.(x.x+y.y+z.z)/r/r

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)

det(g)
= (g(s)^g(w)^g(u))×(dr^dm^dn)

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

dx^dy^dz

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 apply the usual analysis to obtain the basis-dependent tensor

Δ
= g\( ( τ[0,1,2] −τ[1,0,2] −τ[2,1,0] )( sum(: d(p(i)·g·p(j))×b(i)×b(j) ←[i,j] :) ) )/2

but g is diagonal and d(q·g·q) = d(1) = 0, so we only have to consider the cases i = 1 = j and i = 2 = j;

r.dr×dm×dm −r.dm×dr×dm −r.dm×dm×dr
+r.Cos(m).Cos(m).dr×dn×dn −r.Cos(m).Cos(m).dn×dr×dn −r.Cos(m).Cos(m).dn×dn×dr
=

for which b(i)·Δ = D(b(i)), i.e.

D(dr)
D(dm)
D(dn)

from which we may infer derivatives of the dual basis, via

• D(p(i)) = −sum(: (D(b(j))·p(i))×p(j) ←j :)

as

D(q)
= (dm×w +dn×u)/r
D(w)
D(u)

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.

## Expressing ∇2 in terms of polar co-ordinates

We can now compute

∂∂/∂x/∂x
= Cos(m).Cos(n).∂/∂r.(Cos(m).Cos(n).∂/∂r)
= Cos(m).Cos(m).Cos(n).Cos(n).∂∂/∂r/∂r
= Cos(m).Cos(m).Cos(n).Cos(n).∂∂/∂r/∂r
+(Sin(m).Sin(m).Cos(n).Cos(n) +Sin(n).Sin(n)).∂/∂r/r
∂∂/∂y/∂y
= Cos(m).Sin(n).∂/∂r.(Cos(m).Sin(n).∂/∂r)
= Cos(m).Cos(m).Sin(n).Sin(n).∂∂/∂r/∂r
= Cos(m).Cos(m).Sin(n).Sin(n).∂∂/∂r/∂r
+(Sin(m).Sin(m).Sin(n).Sin(n) +Cos(n).Cos(n)).∂/∂r/r
∂∂/∂z/∂z
= Sin(m).Sin(m).∂∂/∂r/∂r
= Sin(m).Sin(m).∂∂/∂r/∂r +Cos(m).Cos(m).∂/∂r/r

as the three terms in ∇2. Adding those all together we get

2
= ∂∂/∂x/∂x +∂∂/∂y/∂y +∂∂/∂z/∂z
= Cos(m).Cos(m).Cos(n).Cos(n).∂∂/∂r/∂r
+(Sin(m).Sin(m).Cos(n).Cos(n) +Sin(n).Sin(n)).∂/∂r/r
+ Cos(m).Cos(m).Sin(n).Sin(n).∂∂/∂r/∂r
+(Sin(m).Sin(m).Sin(n).Sin(n) +Cos(n).Cos(n)).∂/∂r/r
+ Sin(m).Sin(m).∂∂/∂r/∂r +Cos(m).Cos(m).∂/∂r/r
= (Cos(m).Cos(m).Cos(n).Cos(n) +Cos(m).Cos(m).Sin(n).Sin(n) +Sin(m).Sin(m)).∂∂/∂r/∂r
+(Sin(m).Sin(m).Cos(n).Cos(n) +Sin(n).Sin(n) +Sin(m).Sin(m).Sin(n).Sin(n) +Cos(n).Cos(n) +Cos(m).Cos(m)).∂/∂r/r

## The Spin Operator

Introduce a vector operator S defined to have co-ordinates Sx, Sy and Sz given by

[Sx, Sy, Sz]
= [x,y,z]^(−i.∇)
= −i.[ y.∂/∂z −z.∂/∂y, z.∂/∂x −x.∂/∂z, x.∂/∂y −y.∂/∂x ]
= −i.[
]
= −i.[ Sin(n).∂/∂m −Tan(m).Cos(n).∂/∂n, −Cos(n).∂/∂m −Tan(m).Sin(n).∂/∂n, ∂/∂n ].radian

and compute the sum of squares of these co-ordinates:

S2
= Sx.Sx +Sy.Sy +Sz.Sz
+(Sin(n).∂/∂m −Tan(m).Cos(n).∂/∂n).(Sin(n).∂/∂m −Tan(m).Cos(n).∂/∂n)
+(Cos(n).∂/∂m +Tan(m).Sin(n).∂/∂n).(Cos(n).∂/∂m +Tan(m).Sin(n).∂/∂n) )
+Sin(n).Sin(n).∂∂/∂m/∂m −Sin(n).Cos(n).∂/∂m.(Tan(m).∂/∂n)
−Tan(m).Cos(n).∂/∂n.(Sin(n).∂/∂m) +Tan(m).Tan(m).Cos(n).∂/∂n.(Cos(n).∂/∂n)
+Cos(n).Cos(n).∂∂/∂m/∂m +Sin(n).Cos(n).∂/∂m.(Tan(m).∂/∂n)
+Tan(m).Sin(n).∂/∂n.(Cos(n).∂/∂m) +Tan(m).Tan(m).Sin(n).∂/∂n.(Sin(n).∂/∂n) )
+Sin(n).Sin(n).∂∂/∂m/∂m +Cos(n).Cos(n).∂∂/∂m/∂m