Thus far we have considered fields over U^{N} as point-dependant multivectors in U_{N} rather than point-dependant
points in U^{N}. If we consider ¦ to map points in U^{N} to points in U^{N}, we have a transformation.
The image under ¦ of a subset of U_{N} is known as a "manifold" in U^{N}.
Computationally speaking, we assume all functions are smooth enough that
differentials are intrinsically approximatable to great accuracy by adaptive refinement methods,
and so are, from the programmer's perspective, just another "function", albeit a computationally intensive
one perhaps requiring further derivatives to evaluate.
We tackle the subject in some depth here (primarily because of the importance of the Einstein tensor in general Relativity)
in what is essentially a reformulation of a subset of the material in
CAtoGC.
Curves and Manifolds
We now generalise the concepts of "curves" and "surfaces".
An M-curve (also known as a M-dimensional manifold) in U^{N} is a set of points in
targetspace U^{N} locally representable 1-1 by a system of M scalar coordinates drawn from a well defined connected
parameter space (aka. mapspace or (local) map )
M_{ap}ÌV^{M} .
A 0-curve is a point.
A 1-curve is a Â®U^{N} mapping p(t) t Î [t_{0},t_{1}]
and is either closed (a "loop") , infinite, or "bounded" by two end points.
We will call a 1-curve with a specific parameterisation a path.
A 2-curve is a surface and is either closed (eg. a 2-sphere), infinite, or "bounded"
by a 1-curve.
A 3-curve is a "solid" and is either closed or bounded by a surface (2-curve).
In general, M-curve C_{M} is either closed or has as boundary a closed (M-1)-curve
conventionally written dC_{M}. Clearly ddC_{M} is the empty set f .
We will denote the "interior" or contents of an M-curve C_{M} by d^{-1}C_{M} or C_{M}^{·} , so that
(dC_{M})^{·} = C_{M} .
We call an (N-1)-curve a hypercurve.
The closed k-curve "spanned" by k-blade a_{k} is the infinite (unbounded) k-plane { p : pÙa_{k}=0 } and a k-sphere can be regarded as a particularly simple closed k-curve.
Defining M-curves via M intrinsic coordinates and a mapping function in this manner is but one approach. We can also create an N-D M-curve by "sweeping" an (M-1)-curve through N-D space, but for now we assume the ¦(M_{ap}) model.
We can locally parameterise an M-curve in a neighbourhood of a given fixed point q within it
as the range of an invertible (but generally nonlinear) local map function ¦_{q}(x^{1},x^{2},..,x^{m})
sending V^{M} ® U^{N} defined over
a fully bounded finite subvolume (interior of a unimapped (M-1)-curve)
M_{ap}ÌV^{M} .
Typically V^{M} = Â^{M}, but we will retain a more general view to accomodate
the Minkowski parameter spaces of Relativity physics. We will associate x with an M-D point (1-vector) within M_{ap} and
p=¦_{q}(x) with the associated N-D point (1-vector) in C_{M} Ì U^{N} . We will use the symbol _map
or sometimes upper case symbols
to distinguish structures and operators defined within M_{ap} from their within C_{M}
counterparts.
We say an M-curve is differentiable if ¦_{q} is differentiable in
each of the parameters and is sufficiently "smooth" for derivatives of all orders to exist.
We say an M-curve is unimapped if a common "global" map is locally applicable
everywhere; ie. there is a single function ¦ : M_{ap}®U^{N}
with ¦(M_{ap})=C_{M} that serves for every q Î C_{M} . A path is thus a unimapped 1-curve.
We will not assume a unimapped manifold, but we will assume local unimapping in that we will
assume that attention is restricted to a neighbourhood of q over which ¦_{q} applies.
q is thus fixed and plays no useful part in our discussion and we will accordingly drop most _{q} suffixes for brevity .
Extended Mapspace
Suppose now that we have ¦:V^{M} ® U^{N} . Any M-curve M_{ap} in V^{M} induces
an M-curve C_{M}=¦(M_{ap}) in U^{N}.
Typically we might have a flat M-plane segment M_{ap} in V^{M} embedding to a "bendy" C_{M} in U^{N}
seen as the M_{ap} specific "slice" of ¦ .
Submanifolds
Given a unimapped M-curve C_{M}=¦(M_{ap}) and a point pÎC_{M} we can construct
k-dimensional submanifolds of C_{M} at p as the images under ¦ of the subvolumes of M_{ap}
obtained by keeping M-k of the mapspace parameters held at their ¦^{-1}(p) values.
We say an M-curve is isomapped if ¦ can be extended over a subvolume of V^{N}
to define an invertible function ¦: V^{N}®U^{N} creating an N-curve of which
C_{M} is a submanifold.
Embedded Frame
We have a particular M_{ap}-point-dependant embedded frame in U_{N} over our q neighbourhood
consisting of M U_{N} tangent vectors specified by the alternate notations
h_{kp } º
(d/dx^{k})¦_{q}(x)
= Ð^{x}_{eðk}¦_{q}(x)
= (e_{ðk}¿Ñ_{x})¦_{q}(x)
= ¦_{q}^{Ñx}(e_{k})
where Ð^{x}_{eðk} and Ñ_{x} are the standard directed and undirected
derivatives for directions within the mapspace M_{ap}.
The embedded frame is not orthogonal for general ¦_{q} and
is defined with regard to a patricular basis for the mapspace.
We can extend the embedded frame to a frame for U_{N} with N-M orthonormal vectors lieing wholly outside I_{p} but in the absence of an extended N-D mapspace invertibly mapped into U^{N} such an extension is only unique in the case M=N-1 when we can define h_{Np } = ( h_{1p }Ù h_{2p }Ù... h_{N-1p })^{-1}i .
Example: Spherical Surface
We can formulate coventional spherical polar coordinates as ¦:V^{M}=Â^{3} ® U^{N}=Â^{3}
¦(x)=¦(qe_{ðq} + fe_{ðf} + re_{ðr})
= r sinq cosfe_{1}
+ r sinq sinfe_{2}
+ r cosqe_{3}
= rR_{q,f}_{§}(e_{3})
where e_{ðq},e_{ðf},e_{ðr} are an orthonormal basis for V^{M} and e_{1},e_{2},e_{3} are an orthonormal basis for U^{N}.
and
R_{q,f} =
(-½e_{12}f)^{↑}
(-½e_{31}q)^{↑} .
Difficulties arise with our tangent vectors near the poles (q=p or 0). This is inevitable when operating on a closed surface and follows from a famous mathematical result known as the "hairy ball theorem".
If we set U^{N}=V^{M}=Â^{3} with e_{ð}^{i}=e^{i} and regard ¦ as a transformation then
¦^{Ñ}_{x}(dx)) = (Ñ_{x}¿dx)¦(x)
= (dqr cosq cosf - dfr sinq sinf + dr sinq cosf)e_{1}
+ (dqrcos(q) sinf + dfr sinq cosf + dr sinq sinf) e_{2}
+ (-dqr sinq + dr cosq) e_{3} .
The embedded frame in U_{N} is orthogonal.
h_{qp } = rR_{q,f}_{§}(e_{1})
= re_{q}
;
h_{fp }
= e_{3}×¦(q,f,r)
= (e_{3}Ù¦(q,f,r))^{*} = r sinqR_{q,f}_{§}(e_{2})
= r sinqe_{f};
h_{rp } = ¦(q,f,r)^{~}
= R_{q,f}_{§}(e_{3}) = e_{r}
These have magnitudes r; r sinq ; and 1 respectively.
If we fix r=1, so restricing M_{ap} to a 2D parameterspace { q,f }
and obtaining C_{M}=S_{2} , the 2-curve boundary surface of a 3D unit sphere, then the derivative reduces to
¦^{Ñ}_{x}(dx)) = (Ñ_{x}¿dx)¦(x)
= (dq cosq cosf - df sinq sinf)e_{1}
+ (dq cosq sinf + df sinq cosf) e_{2}
- dq sinq e_{3} .
From a purist point of view, mapspace coordinates have the form qe_{ðq}+fe_{ðf}
whereas tangent vectors are expressed in terms of e_{1},e_{2},e_{3} but we will here consider the M_{ap} as existing in U_{N}
and set e_{ðq}=e_{1} , e_{ðf}=e_{2}.
Inverse Embedded Frame
We can construct an inverse embedded M-frame
h^{i}_{p } of U^{N} 1-vectors within the tangent space C_{M} satisfying
h^{i}_{p }¿ h_{jp } = d_{i j} via
h^{k}_{p } º
(-1)^{k-1}( h_{1p }Ù.. h_{k-1p }Ù h_{k+1p }Ù.. h_{Mp })( h_{1p }Ù.. h_{Mp })^{-1}
.
Note that the h^{i}_{p }ÎI_{p} and are a frame for precisely the same subspace as are the embedded frame.
We can invert ¦_{q} to express the x^{i}(p) = e^{i}¿¦_{q}^{-1}(p) as M scalar fields defined over
a local (M-curve) neighbourhood of q.
When M=N so that ¦_{q} : U^{N} ® U^{N} we have
h^{i}_{p } º Ñ_{p }(x^{i}(p))
= Ñ_{¦q(x) }x^{i}
= ¦_{x}^{-D}(e^{i}) .
[ Proof :
h_{ip }¿ h^{j}_{p }
= ¦_{qx}^{Ñ}(e^{i}) ¿ ¦_{qx}^{-D}(e_{j})
= e^{i}¿ ¦_{qx}^{D}(¦_{qx}^{-D}(e_{j}))
= e^{i}¿e_{j}
= d_{ij}
.]
h^{i}_{p } is the normal to the coordinate isosurface x^{i}(p)=x^{i}(q) at q.
In a nonorthogonal embedded frame, we can have h^{i}_{p }Ù h_{ip } ¹ 0 so
the normal to the isosurface need not be parallel to the streamline tangent.
Example: Spherical Surface
For spherical coordinates mapping, the normalised reciprocal frame is h^{i}_{p }^{~} = R_{q,f}_{§}(e_{i}).
¦^{D}_{p}(dp) º Ñ_{x}(dp¿¦(x))
= Ñ_{x}(dp^{1}r sinq cosf + dp^{2}r sinq sinf + dp^{3}r cosq)
= r(dp^{1} cosq cosf + dp^{2} cosq sinf - dp^{3} sinq)e_{ðq}
+ r(-dp^{1} sinq sinf + dp^{2} sinq cosf)e_{ðf}
+ (dp^{1} sinq cosf + dp^{2} sinq sinf + dp^{3} cosq)e_{ðr}
defined for dpÎÂ^{3} .
This reduces to
¦^{D}_{p}(dp) =
(dp^{1} cosq cosf + dp^{2} cosq sinf)e_{ðq}
+ (-dp^{1} sinq sinf + dp^{2} sinq cosf)e_{ðf}
when pÎS_{2} and dp is in the tangent space at p.
We have M scalar fields defined over C_{M} by x^{i}(p) = e_{ð}^{i}¿¦^{-1}(p) = e_{ð}^{i}¿x .
For spherical mapping these are
q(p^{1}e_{1} + p^{2}e_{2} + p^{3}e_{3}) =
cos^{-1}(p^{3}(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-½}) [ p^{1}^{2} = (p^{1})^{2} ]
f(p^{1}e_{1} + p^{2}e_{2} + p^{3}e_{3}) = tan^{-1}(p^{2}/p^{1})
[ With the usual caveats specifying quadrant ]
r(p^{1}e_{1} + p^{2}e_{2} + p^{3}e_{3}) = (p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{½}
We have
Ñ_{p} q(p) =
-(1-(p^{3}^{2}(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-1})^{-½}
( e^{1}(-p^{3}x^{1}(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-3/2}
+ e^{2}(-p^{3}x^{2}(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-3/2}
+ e^{3}((p^{1}^{2}+p^{2}^{2})(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-3/2}
= -((p^{1}^{2}+x^{2}^{2})(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-1})^{-½}
(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-_3/2}( e^{1}(-p^{3}p^{1})
+ e^{2}(-p^{3}p^{2})
+ e^{3}(p^{1}^{2}+p^{2}^{2}) )
=?= r^{-2} h_{qp } = r^{-1} h_{qp }^{~}
Ñ_{p} f(p) = (r sinq)^{-2} h_{fp } = (r sinq)^{-1} h_{fp }^{~}
Ñ_{p} r(p) = p^{~} = h_{rp } = h_{rp }^{~} .
Local Orientation
The h_{kp } define a C_{M}-point-dependant nondegenerate U_{N} M-blade
J_{p} º h_{1p }Ù..Ù h_{Mp }
that spans the tangent space at p=¦_{q}(x).
C_{M} has tangent M-plane (1+q)ÙJ_{q} at q ,
or (e_{0}+q)ÙJ_{q} in GHC.
The invertibility of ¦_{q} ensures that J_{q} is nondegenerate.
The unit pseudoscalar for the tangent space given by
I_{p} º J_{p}^{~}
is called the orientation
of the M-curve at p .
For a 1-curve, the orientation is the unit tangent 1-vector.
An M-curve is orientable if a continuous unit-valued M-tangent blade can be defined over it.
The classic example of a nonorientable 2-curve is a Moebius strip.
A manifold is flat if it has the same orientation everywhere.
We will be concerned with orientable nonflat manifolds here.
If the boundary of an M-curve has orientation I_{M-1p} then it is conventional to specify the orientation ("handedness")
of I_{p} by defining I_{p} = I_{M-1p} n_{p} where
n_{p} = I_{M-1p}^{-1} I_{p} is the spur, the unit outward normal to the boundary at p.
The normalisation condition I_{p}^{2} = ±1
gives (Ð_{a}I_{p})I_{p} + I_{p}(Ð_{a}I_{p})=0
and taking the scalar part yields (Ð_{a}I_{p})¿I_{p}=0 Þ ¯_{Ip}(Ð_{a}I_{p})=0.
Hence ¯_{Ip}(Ñ_{p}I_{p}) = 0 .
[ Proof : Choosing an orthonormal frame with e_{1}Ùe_{2}Ù...e_{M} =I_{p} we obtain
Ñ_{p}I_{p} = å_{i=1}^{N} e^{i} Ð_{ei}I_{p}
= å_{i=M+1}^{N} e^{i} Ð_{ei}I_{p}
all terms of which lie outside I_{p}.
.]
Note that Ñ_{p}¿I_{p} ¹ 0 in general.
Spherical Surface Example
For spherical coordinates S_{2} mapping,
J_{p} = h_{qp }Ù h_{fp }
= h_{qp } h_{fp } =
sinqR_{q,f}_{§}(e_{1})R_{q,f}_{§}(e_{2})
= sinqR_{q,f}_{§}(e_{12}) .
Although J_{p} vanishes at the poles,
it is natural to define
I_{p}=R_{q,f}_{§}(e_{12})
" p=R_{q,f}_{§}(e_{3}) ÎS_{2} .
The Metric
Let a,b be two 1-vectors in an M-curve's mapspace M_{ap}ÌV_{M}.
Whereas aÙb = ¦^{-D}_{x}(¦^{Ñ}_{x}(a)Ù¦^{Ñ}_{x}(b)) is
geometrically meaningful , a¿b is not. In particular, it does not equal
¦^{Ñ}_{x}(a)¿¦^{Ñ}_{x}(b).
Rather, we have the cocontraction
a¿b = a¿_{x}b º ¦^{Ñ}_{x}(a)¿¦^{Ñ}_{x}(b)
and hence a variant geometric "coproduct" with point-dependant inner product
¿ replacing ¿ .
We postulate a symmetric point-dependant metric 1-tensor
g_{x} : V_{M} ® V_{M} such that
a¿g_{x}(b)
= g_{x}(a)¿b
= ¦^{Ñ}_{x}(a)¿¦^{Ñ}_{x}(b) = a¿b
.
For M=N the obvious candidate is g_{x} º
¦^{D}_{x}¦^{Ñ}_{x} .
If ¦^{Ñ}_{x} is symmetric, g_{x} = (¦^{Ñ}_{x})^{2} .
Example: Spherical Surface
g_{x}(dx) = r^{2}dqe_{ðq}
+ r^{2} sin^{2}qdfe_{ðf}
+ e_{ðr}
with associated line element length
dx¿g_{x}(dx) = r^{2}dq^{2} + r^{2} sin^{2}qdf^{2} + dr^{2}
[ Proof :
¦^{Ñ}_{x}(dx)) = (Ñ_{x}¿dx)¦(x)
= (dqr cosq cosf-dfr sinq sinf+dr sinq cosf)e_{1}
+ (dqrcos(q) sinf+dfr sinq cosf+dr sinq sinf)e_{2}
+ (-dqr sinq+dr cosq)e_{3} .
¦^{D}_{p}(dp) º Ñ_{x}(dp¿¦(x))
= Ñ_{x}(dp^{1}r sinq cosf + dp^{2}r sinq sinf + dp^{3}r cosq)
= r(dp^{1} cosq cosf + dp^{2} cosq sinf - dp^{3} sinq)e_{ðq}
+ r(-dp^{1} sinx siny + dp^{2} sinq cosf)e_{ðf}
+ (dp^{1} sinq cosf + dp^{2} sinq sinf + dp^{3} cosq)e_{ðr}
Thus g_{x}(dx) = ¦^{D}_{p}(¦^{Ñ}_{x}(dx))
= ¦^{D}_{p}(
(dqr cosq cosf -dfr sinq sinf +dr sinq cosf)e_{1}
+ (dqrcos(q) sinf +dfr sinq cosf +dr sinq sinf) e_{2}
+ (-dqr sinq +dr cosq) e_{3}
)
= ...[Tedious manipulations]... =
r^{2}dqe_{ðq}
+ r^{2} sin^{2}qdfe_{ðf}
+ e_{ðr}
.]
Restricting r=1 for S_{2} gives
g_{x}(dx)
= g_{x}(dqe_{ðq}+dfe_{ðf}) = dqe_{ðq} + sin^{2}qdfe_{ðf}
In much of the literature, notably with regard to Genral Relativity, the metric g_{x} is regarded
as profoundly fundamental. Indeed, it is sometimes known as the fundamental tensor.
However, we will here regard g_{x} as less fundamental than ¦^{Ñ}_{x} which we will also make little
subsequent use of in this chapter, considering orientation M-blade I_{p} described below
as the defining "property" of an M-curve.
M-Curve as an M-blade-valued field
We have seen how the local mapping function ¦ defines a tangent M-frame everywhere on an M-curve.
Suppose alternatively we provide a k-frame at every point of an M-curve C_{M}.
This provides a k-foliation of k-curves over C_{M}. Every p Î C_{M} is contained within
a k-curve having the given k-frame as embedded frame at p. These k-curves are not
contained within C_{M} in general, but will be if the k-frame at p is within I_{p} " pÎC_{M} .
Suppose now that instead of a k-frame for each p we merely have a unit M-blade valued field I_{p}
defined over pÎU^{N}.
If p_{0}ÎU^{N} is known to lie in C_{M} then another U^{N} point p_{1} will lie in C_{M} iff there
exists a path (1-curve)
{ p(t) Î U^{N} : t Î [t_{0},t_{1}] } with p(t_{0})=p_{0} ; p(t_{1})=p_{1} ;
and p'(t)ÎI_{p(t)} " tÎ[t_{0},t_{1}]. Establishing whether this is, in fact,
the case for a given p_{0} and p_{1} may be far from easy.
If all we can say about
¦ with regard to p=¦(x) is that ¦^{Ñ}_{x}(i)=I_{p} then things can get tricky.
( If ¦^{-1}(p_{1}) is available, of course, then p_{1}ÎC_{M} Û ¦^{-1}(p_{1})ÎM_{ap}.)
However, we can nonetheless think of a non-degenerate-M-blade-valued field over U^{N} (ie. one that nowhere vanishes) as defining a foliation
of M-curves over U_{N}, just one M-curve containing any given pÎU^{N}.
If I_{p}=0 at some p then there is no M-curve passing through p and we say the foliation is partial.
A multivector-valued field over U^{N} can thus be regarded as a sum of partial foliations of various grade curves.
Our fundamental view of M-curves in the remainder of this chapter will be collections of points in U^{N} as determined by a unit M-blade field I_{p} defined over U^{N}.
Hypercurves
An (N-1)-curve is known as a hypercurve and we can define a unit 1-vector
normal by
n_{p} º i_{N-1p}^{*} = i_{N-1p}i^{-1}
= (i)^{2} i_{N-1p}¿i .
n_{p} = (-1)^{N-1}e_{N} in the fortutious basis and
n_{p}^{2} = (-1)^{N-1} i_{N-1p}^{2} i^{2} = ±1
where M=N-1 .
.
Projector 1-multitensor ¯
The projector of M-curve C_{M} is the multitensor
¯(a) º ¯_{Ip}(a) º (a¿I_{p})I_{p}^{-1} .
We also have the rejector multitensor ^ º 1 - ¯ .
From our original discussion of projection into blades we know that ¯^{2}=¯ and ¯(aÙb)=¯(a)Ù¯(b) and that we can regard ¯ as a grade-preserving "idempotent outtermorphic operator" satisfying the "product rule" ¯(¯(a)b) = ¯(a)¯(b). We have defined ¯ explicitly using ¿ here, but much of the following applies to more generally hypothesised projectors satisfying such basic properties.
Hypercurve
For M=N-1 we have ¯_{iN-1p}(a_{p}) = a_{p} - (n_{p}^{-1}¿a_{p})n_{p} .
Integration over an M-curve
We assume here than the concept of scalar integration is well understood. That is,
that ò_{t0}^{t1}F(t) dt is defined for a multivector-valued function F of a scalar t
as the limit of a sum of elemental contributors.
We can immediately adapt this for the line integral
of a multivector valued field F(p)
along a 1-curve p(t) t Î [t_{0},t_{1}]
by defining
ò_{C} F(p) dp º ò_{t0}^{t1}F(p(t))p'(t) dt .
More generally
we have ò_{C} F(p) dp
defined to be the limit of a summation at n-1 of n "samplepoints" p_{[i]} along the 1-curve
. p_{[0]} and p_{[n]} are the given curve endpoints (the same point if integrating round a loop) ]
.
At each sample we form the geometric product
F(p_{[i]}) (Dp_{[i]})
where Dp_{[i]} º (p_{[i+1]}-p_{[i]}) .
As n ® ¥, all the 1-vector Dp_{[i]} ® 0.
Because we have a geometric rather than a scalar multiplication,
ò_{C} F(p)dp ¹ ò_{C} dpF(p)
in general.
In the particular case F(p(t)) = p'(t)^{~} the line integral is a pure scalar,
the conventional arc length of the
curve.
For integration over a 2-curve (ie. a surface integral) we proceed similarly, sampling at n points and evaluating
at each a contributary directed flat triangular area "mesh" element
(ie. a 2-simplex) having orientation I_{Mp[p]} and magnitude the
conventional scalar measure of area (simplex content);
these triangles tesselating to approximate the surface.
If the 2-curve is parameterised as p(x^{1},x^{2}) then the directed area element has form
d^{2}p = ( h_{1p }Ù h_{2p })dx^{1}dx^{2} .
For integrating over an M-curve C_{M}=¦(M_{ap}) parametrised as p(x^{1},x^{2},..,_xuM) we have contributary (M-1)-simplex elements
d^{M}p
= ( h_{1p }Ù h_{2p }.. h_{Mp })dx^{1}dx^{2}..dx^{M}
= |¦^{Ñ}|I_{p}dx^{1}dx^{2}..dx^{M}
and we can think of M succesive scalar integrals of multivector
F(p)I_{p} via
ò_{CM }F(p) d^{m}p =
ò_{CM }F(p)( h_{1p }Ù h_{2p }..Ù h_{Mp })
dx^{1}dx^{2}...dx^{M} .
It can be shown that this limit is independant of the precise nature and geometry of the "mesh" used.
We define the scalar content of an M-curve by
|C_{M}| º ò_{CM} I_{p}^{-1}dp .
This is the conventional arc length, surface area, and volume of C_{M} for M=1,2, and 3 respectively.
Fourier Transform
Having defined integration we can define the Fourier transform.
The (unitary) Fourier transform of a multivector field a_{x} º a(x)
is the field
F(a_{x})(k) º
(2p)^{-½N}
ò d^{N}x i^{-1} (-i(x¿k))^{↑} a_{x}
= (2p)^{-½N}
ò |d^{N}x| (-i(x¿k))^{↑} a_{x}
where the x integration is usually taken over all U^{N} rather than a particular subspace of interest,
and i commutes with a_{x} and has i^{2}=-1.
The inverse transform is
F^{-1}(b_{k})(x) º
(2p)^{-½N}
ò d^{N}k i^{-1} (+i(x¿k))^{↑} b_{k}
where we are now integrating over k ÎU^{N} .
For N=1 we have the scalar unitary Fourier transform
F(a(x)(k) =
(2p)^{-½} ò_{-¥}^{¥ } dx (-ixk)^{↑} a(x) .
and its inverse
F^{-1}(b(var(k))(x) =
(2p)^{-½} ò_{-¥}^{¥ } dk (+ixk)^{↑} b(k) .
F((i(x.b)^{↑})(k)
= (2p)^{-½N}
ò d^{N}x i^{-1} (i(x¿(b-k))^{↑}
= (2p)^{-½N}
P_{j=1}^{N} ò_{-¥}^{¥} dx_{j} (ix_{j}(b_{j}-k_{j}))^{↑}
The scalar integrals vanish except when b_{j}=k_{j} when
we (informally) obtain ¥.
Since ò_{-K}^{K} dk ò_{-¥}^{¥} dx cos(xk)
= ò_{-¥}^{¥} dx ò_{-K}^{K} dk cos(xk)
= ò_{-¥}^{¥} dx 2x^{-1} sin(Kx)
= 4 ò_{0}^{¥} dx x^{-1} sin(Kx) = 2p we have
ò_{-¥}^{¥} dx cos(xk) = 2pd(k)
and hence
F((i(x.b)^{↑})(k) =
(2p)^{½N} d(b-k)
where d(x) = d(x_{1})d(x_{2})...d(x_{N})
is an N-D Dirac delta function.
Of course
F^{-1}(d(b-k))(y) =
(2p)^{-½N}(i(y¿b))^{↑}
and so
F^{-1}( F((i(x.b)^{↑})(k) )^{↑}) ) (y)
= F^{-1}( (2p)^{½N} d(b-k) ) (y)
= (i(y¿b))^{↑} , and more generally the (2p)^{-½N} factors
in the definitions of F and F^{-1} serve to ensure that
F^{-1}F = 1 .
An obvious geometric generalisation is F(a_{x})(k) º ò (2p)^{-½N} ò d^{N}x i^{-1} (xÙk)^{↑} a_{x} with pseudovector k=k^{*}=ki^{-1} providing the conventional 1-vector Fourier transform when i=i since then xÙ(k^{*}) = (x.k)^{*} = -i(x.k) . However for general grade k we must speak of left and right Fourier transforms since (xÙb)^{↑} may not commute with a_{x}.
Example: F(H_{a}) in Â
Let H_{a}(x) = (2a^{3})^{-1} for |x|£a and 0 elsewhere in Â . (2p)^{-½} ò_{-¥}^{¥} dx H_{a}(x) (-i kx)^{↑} = (2a)^{-1} (2p)^{-½} ò_{-a}^{a} dx (-i kx)^{↑} = (2a)^{-1} (2p)^{-½} (-ik)^{-1} [ (-i kx)^{↑} ]_{-a}^{a} = i(2ak)^{-1} (2p)^{-½} ( -2i sin(ak) ) = (2p)^{-½} Sin(ak) where Sin(x) º x^{-1} sin(x) .
Example: F(H_{a}) in Â^{3}
As an example let H_{a}(x) = (3^{-1}4pa^{3})^{-1} for |x|£a and 0 elsewhere in Â^{3}
and impose axies so that k=ke_{3}. We have
(2p)^{-3/2} ò d^{3}x H_{a}(x)(-i(k.x))^{↑}
= 3(2p)^{-3/2}a^{-2} k^{-2} ( Sin(ka) - cos(ka)) .
[ Proof :
(2p)^{-3/2} ò d^{3}x H_{a}(x)(-i(k.x))^{↑}
= (2p)^{-3/2}
ò_{0}^{¥} dr
ò_{0}^{p} dq
ò_{0}^{2p} df
r^{2} sin(q)
H_{a}(x)(-i(rk cos(q))^{↑}
= (2p)^{-3/2} (3^{-1}4pa^{3})^{-1}
ò_{0}^{a} dr
ò_{0}^{p} dq
ò_{0}^{2p} df
r^{2} sin(q)
(-ir|k| cos(q))^{↑}
= (2p)^{-3/2} (3^{-1}2a^{3})^{-1}
ò_{0}^{a} drr^{2}
ò_{0}^{p} dq
sin(q)
(-ir|k| cos(q))^{↑}
= (2p)^{-3/2} (3^{-1}2a^{3})^{-1}
ò_{0}^{a} drr^{2}
[(ir|k|)^{-1} (-ir|k| cos(q))^{↑}]_{0}^{p}
= -(2p)^{-3/2}(3^{-1}2a^{3})^{-1} i |k|^{-1}
ò_{0}^{a} drr
[ (-ir|k| cos(q))^{↑}]_{0}^{p}
= -(2p)^{-3/2}(3^{-1}2a^{3})^{-1} i |k|^{-1}
ò_{0}^{a} drr
( (ir|k|)^{↑} - (-ir|k|)^{↑} )
= (2p)^{-3/2}(3^{-1}a^{3})^{-1} k^{-1}
ò_{0}^{a} drr sin(r|k|)
= (3^{-1}a^{3})^{-1} k^{-1}
[ k^{-2} sin(kr) - k^{-1}r cos(kr)
]
]_{0}^{a}
= (2p)^{-3/2}(3^{-1}a^{3})^{-1} k^{-1}
( k^{-2} sin(ka) - ak^{-1} cos(ka))
= 3(2p)^{-3/2}a^{-2}k^{-2}( Sin(ka) - cos(ka))
.]
Differentiation within an M-curve
We state many results without proof in this section. Proofs may be found
in Hestenes & Sobczyk [ 4-4-2 and 4-4-4] and we include
here the numbers assigned to equivalent equations in that definitive work.
Directed Tangential Derivative Ð_{¯}
The a-directed tangential derivative is the
¯_{Ip}(a)-directed derivative.
Ð_{[¯]a} º Ð_{¯(a)} º Ð_{¯Ip(a)}
= (¯[a]¿Ñ) .
Directed tangential derivatives commute whenever directed ones do, so the integrability condition
allows us to commute directed tangential derivatves.
We are being less than rigourous here because p+dd can lie outside M-curve C_{M}
even when dÎI_{p} . Strictly speaking we should define a 1-curve p(t) within C_{M} such that p(0)=p and
p'(t)=¯_{Ip}(d) and form
Ð_{[¯]d} º Lim_{t ® 0} t^{-1}(F(p(t))-F(p(0)))
_{[ HS 4-1.1 ]}
so we can restrict attention to points within C_{M} and only require F(p) be defined over the manifold.
We can do this via the mapspace by defining
p(t) = ¦_{q}(¦_{q}^{-1}(p)
+ t¦_{q}^{-Ñx}(¯_{Ip}(d)) )
so that F(p(t)) = F(p(0)) +
tÐ^{p}¦_{q}^{-Ñx}(¯_{Ip}(d)F(p)
+ _{O}(t^{2})
and define Ð_{[¯]d} º Ð^{x}_{¦q-Ñx(¯Ip(d))} .
However if F(p) is continously defined throughout a U^{N} neighbourhood of p
then for dÎI_{p} we have
F(p(t)) = F(p+td+ _{O}(t^{2}))
= F(p) + tÐ_{d+ O(t)}F(p)ï_{p} + _{O}(t^{2})
= F(p) + tÐ_{d}F(p)ï_{p} + _{O}(t^{2})
so that t^{-1}(F(p(t))-F(p(0))) =
Ð_{d}F(p)ï_{p} + _{O}(t).
Thus for general d, sufficiently smooth C_{M}, and continuos F(p)
defined near to as well as over C_{M}, the Ð^{Ip)}_{¯(d} definition is equivalent,
and usually far easier to work with.
Suppose now that F_{p}=F(p) is a multivector valued field defined only over C_{M} but
not necessarily confined within I_{p}.
F_{p} induces a field in the mapspace F_{x} = F_{¦(x)} for which
Ð_{eðk}F_{x} need not lie in I_{p} .
Even if we insist that F_{p} Î I_{p} so that ¯_{Ip}(F_{p})=F_{p}, then Ð_{eðk}F_{x} still need not lie in I_{p}.
Similarly, assuming F_{p} to be defined for p near to as well as within C_{M}, Ð_{d}F_{p} need not lie in I_{p} even if dÎI_{p} and F_{p}ÎI_{p} " p.
Thus the tangential derivative "acts within" C_{M} but is not "confined to" C_{M} in that it can (at p) "return" multivectors
not contained in C_{M} at p. It differentiates along tangents but is not itself tangent.
For a derivative entirely "within" C_{M} we must look to
the coderivative.
Undirected Tangential 1-Derivative Ñ
The undirected tangential derivative
can be defined somewhat abstractly as the 1-vector operator Ñ satisfying
Ð_{¯(a)} º (¯(a)¿Ñ) = (a¿¯(Ñ))
º (a¿Ñ) .
We then have Ñ_{p} = Ñ_{b} (b¿Ñ) = Ñ_{b} Ð^{p}_{↓(b)}
and since (b¿Ñ) = Ð_{¯(b)} = 0 if ¯(b)=0 we have
Ñ_{p} º Ñ_{b} Ð^{p}_{↓(b)}
= Ñ_{b} Ð^{p}_{↓(b)}
providing perhaps the best symbolic definition of Ñ .
The mapspace coordinate based definition of the undirected tangential derivative
Ñ_{[CM] p}F(p) of a continuous multivector-valued field
F(p) defined over a within-C_{M} neighbourhood of point q on M-curve C_{M} is
Ñ º
Ñ_{[CM] p} º
å_{k=1}^{M} h^{k}_{p }Ð^{x}_{eðk}
where h^{i}_{p } are a reciprocal embedded frame for C_{M}
and
Ð^{x}_{eðk} = (d/dx^{k}) operates in the mapspace M_{ap}
as
Ð_{eðk}F(p) = Ð^{x}_{eðk}F(¦(x))
= (Ñ_{p}¿¦^{Ñ}_{x}(e_{ðk}))F(p)
= F^{Ñp}(¦^{Ñ}_{x}(e_{ðk}))
= F^{Ñp}( h_{kp }) .
[ We have droppped the _{q} suffix on ¦ for brevity. ]
The use of reciprocal tangent frame
h^{k}_{p } in the defintion of Ñ_{[CM]p} "counter scales the expansion"
of ¦^{Ñ} to ensure Ñ_{p} p º Ñ_{[CM]p} p = M .
If F(p) is continuously defined over a U^{N} neighbourhood of q and C_{M} is smooth we also have
Ñ º
Ñ_{[CM] p} =
Ñ_{[Ip] p} º
å_{k=1}^{M} h^{k}_{p }Ð^{p}_{ hkp }
where Ð_{ hkp } operaties in U_{N} .
Ñ_{p} is essentially Ñ_{p} confined to act within C_{M}
(as visually suggested by symbol Ñ being a "portion" of Ñ)
though note that Ñ_{p}F(p) may not be confined to I_{p} even when F(p) is.
Ñ_{p}
is the directed 1-derivative "splayed out" only over directions lieing within I_{p} ;
we clearly have Ñ_{[UN] p}
= Ñ_{[i] p} = Ñ_{p}
and Ñ_{[1]} = Ñ_{<0>}.
When the particular M-curve C_{M} under discussion is unambiguos, we will abbreviate
Ñ_{[CM] p} to Ñ_{p} or Ñ.
Suppose that e_{1},e_{2},...e_{M} are an orthonormal basis for I_{p} (at a given p only) which we extend by e_{M+1},..e_{N} to
a fortuous universal basis for U_{N} at every p, though it coincides with I_{p}
only at the particular given p of interest.
Then we can write
¯_{Ip}(Ñ_{p}) = å_{i=1}^{N.} ¯_{Ip}(e^{i})Ð_{ei}
= å_{i=1}^{M.} e^{i}Ð_{ei}
= Ñ_{p}
= å_{i=1}^{N.} e^{i}Ð_{¯Ip(ei)} .
For a more general U^{N} basis we have ¯_{Ip}(Ñ_{p})
= å_{i=1}^{N} ¯_{Ip}(e^{i}Ð_{ei})
= å_{i=1}^{N} ¯_{Ip}(e^{i})Ð_{ei}
so that
e_{j}¿¯_{Ip}(Ñ_{p})
= å_{i=1}^{N} (e_{j}¿¯_{Ip}(e^{i}))Ð_{ei}
= ¯_{Ip}(e_{j})¿å_{i=1}^{N} e^{i}Ð_{ei}
= ¯_{Ip}(e_{j})¿Ñ_{p}
= Ð_{¯Ip(ej)} .
Hence Ñ_{p} = ¯_{Ip}(Ñ_{p}) and our targetspace coordinate definition of Ñ_{p} is
Ñ_{p} F(p) º å_{i=1}^{N} ¯_{Ip}(e^{i})Ð_{ei}F(p_{Ñ})
with the differentiating scope of the Ð_{ei} applying to F(p) only.
We say F(p) is monogenic (aka. analytic) on C_{M} if Ñ_{p} F(p) = 0 over C_{M}.
We can also define a orthotangential derivative Ñ_{^} º Ñ_{^p} º Ñ_{p} - Ñ_{p} = å_{i=1}^{N} ^_{Ip}(e^{i})Ð_{ei} .
For 1-curve C= { p(t) } we have Ñ_{[C] p} = h^{1}_{p }(dF(p(t))/dt) = p'(t)^{-1}(dF(p(t))/dt) , effectively the derivative with respect to arc length.
Though ¯(Ñ) º ¯_{Ip}(Ñ_{p}) is a natural notation for the undirected tangential derivative, we favour Ñ º Ñ_{p} here to minimise confusion with the composition operator (¯_{Ip}Ñ)(a_{p}) º ¯_{Ip}(Ñ_{p}a_{p}_{Ñ}) which lies within I_{p} even when a_{p} does not. Nonetheless, it is important to recognise that in essence Ñ = ¯(Ñ) = ¯(Ñ) abbreviating (¯(Ñ_{p}))(a_{p}_{Ñ}) = Ñ_{p}(a_{p}) " a_{p} .
The projection ¯ º ¯_{Ip} is equivalent to the tangential 1-differential of the "scalar" identity
multitensor 1_{p}(a)=1(a)=a
and so we have the alternate notation
¯ = 1^{Ñ} abbreviating
¯_{Ip}(a) = 1^{Ñ[Ip]}(a) " multivector a.
[ Proof :
(a¿Ñ_{ðp})p
=(a¿å_{i=1}^{M} h^{i}_{p }Ð^{p}_{ hip })p
=å_{i=1}^{M} (a¿ h^{i}_{p }) h_{ip }
=¯_{Ip}(a)
.]
The symmetry (self-adjointness) of projection gives
Ñ_{*}a_{p} = ¯(Ñ)_{*}a_{p} = Ñ_{*}(¯_{Ip}(a_{p}))
which with 1-vector operands understood we can write as
Ñ¿ = Ñ¿¯ . The tangential divergence (aka. contraction)
is the divergence of the projection.
.
[ Proof :
Ñ_{*}a_{p} = ¯_{Ip}(Ñ)_{*}a_{p}
= Ñ_{*}(¯_{Ip}(a_{p}))
.]
Recall that with regard to its action on functions satisfying the integrability condition Ð_{a}×Ð_{b} = 0 " a,b, we have ÑÙÑ=0 . This is not the case for Ñ, but we will later show that ¯(ÑÙÑ)=0 and ¯(ÑÙÑÙa_{p})=0 " a_{p}.
Normalisation condition I_{p}^{2}=±1 gives ¯_{Ip}(Ð_{a}I_{p})=0 which in turn yields Ñ I_{p} = Ñ Ù I_{p}
For p^{2} ³ 0, Ñ_{[bk]} |p|^{k} = Ñ_{[bk]}( (p^{2})^{½k} = k |p|^{k-2} ¯_{bk}(p) for nondegenerate k-blade b_{k}.
Alternate Ñ Definition
A coordinate-independant definition of the
tangential derivative at q
discussed at length in Hestenes NFMP
is
Ñ_{[CM] p} º
I_{p}^{-1}
Lim_{|O| ® 0}
[ |O|^{-1} ò_{dO } d^{M-1}p F(p) ]
this being the geometric product of the inverted M-orientation I_{p}^{-1} at q and
the limit (finite or otherwise) of the multivector-valued directed integral
ò_{dO } d^{m-1}p F(p)
taken over the boundary dO of a small
M-curve O Ì C_{M}
enclosing q , divided by O content |O| as |O| ® 0 .
For M=N=3 and 1-field F(p) this is equivalent to the conventional integration-based definitions of
divergence and (the dual of) curl.
Indeed, for general N,
Ñ_{p} º Ñ_{[UN] p} provides a coordinate-independant definition of
the del-operator, suggesting a more fundamental consideration of differentiation as the
limit of the quotient of integrals.
One can continue to think in "coordinate" terms even when the symbolism is coordinate
independant and we will here retain the view of Ñ_{p} as a "splaying" of multivector-directed derivatives.
Our fundamental view of Ñ is essentially "that which satisfies" a_{*}Ñ = Ð_{a}
Fundamental Theorem of Calculus
Having defined integration over and differentaiation within an M-curve we can formulate a geometric
fundamental theorem of calculus and generalise complex residue theory.
Basic Form
The basic form of the Fundamental Theorem of Calculus
relates derivatives within an M-curve C_{M} to values on it's boundary dC_{M} .
ò_{dCM} d^{M-1}p F(p)
=
ò_{CM} d^{M}p (Ñ_{p}F(p))
º ò_{CM} d^{M}p Ñ_{[CM]}F(p)
= (-1)^{M+1}ò_{CM} (Ñ¿d^{M}p) F(p_{Ñ})
where scaled M-blade d^{M}p = I_{p}d^{M}p = I_{p}|d^{M}p| is an elemental M-simplex for C_{M} at p.
[ Proof :
Ommitted here,
but an almost trivial consequence of the
coordinate-independant definition of Ñ_{[CM]}F .
See Hestenes
.]
The theorem has the following important consequences.
A more general form of the fundamental theorem is
ò_{dCM} G_{p} d^{M-1}p F_{p}
= (-1)^{M-1}ò_{CM} G_{p}_{Ñ}Ñ_{p} d^{M}p F_{p}
+ ò_{CM} G_{p} d^{M}p Ñ_{p}F_{p}_{Ñ}
.
_{[ HS 7-3.10 ]}
[ Proof :
ò_{dCM} G_{p} d^{M-1}p F_{p}
=
ò_{CM} G_{p}_{Ñ} d^{M}pÑ_{p} F_{p}
+ ò_{CM} G_{p} d^{M}pÑ_{p} F_{p}_{Ñ}
and recall that d^{M}p is a scaled M-blade pseudoscalar for the tangent space containing Ñ
and so G_{p}_{Ñ} d^{M}pÑ_{p} =
G_{p}_{Ñ}(d^{M}p.Ñ_{p}) =
(-1)^{M-1}G_{p}_{Ñ}(Ñ_{p}.d^{M}p) =
(-1)^{M-1}G_{p}_{Ñ}(Ñ_{p}d^{M}p) =
(-1)^{M-1}(G_{p}Ñ_{p})d^{M}p
.]
Greens Functions
A Greens function is a function used to express a solution to a differential equation with particular boundary conditions
as a definite integral. For example,
(d/dx)^{2} y(x) = ¦(x, y(x)) subject to
y(a)=y_{0} and y'(a)=y_{0}' has solution
y(x) = y_{0} + y_{0}'(x-a) + ò_{a}^{x} dx"
ò_{a}^{x"} dx' ¦(x',y(x')) which can
be alternatively evaluated as
y(x) = y_{0} + y_{0}'(x-a) + ò_{a}^{x} dx' (x-x')¦(x',y(x'))
=
y_{0} + y_{0}'(x-a) + ò_{a}^{b} dx' G(x,x') ¦(x',y(x'))
where b³x and G(x,x') = (x-x') Hvsd(x-x')
is the Greens function for
(d/dx)^{2} for the boundary conditions y(a)=y_{0} ; y'(a)=y_{0}' .
[ Hvsd(x) is the Heaviside step function zero for x£0 and 1 for x>0,
here exploited to replace the indefinite ò_{a}^{x} dx' with the definite
ò_{a}^{b} dx'
]
If G_{p,q} is a 1-vector-valued Green's function having two primary C_{M} point-valued arguments with
Ñ_{p} G_{p,q}
= G_{p,q} Ñ_{q} = 0 " p¹q
and Ñ_{p}G_{p,q} = -G_{p,q}Ñ_{q} = 1 at p=q, the Fundamental Theorem of Calculus provides
ò_{dCM} G_{p,q} d^{M-1}p F_{p}
= (-1)^{M-1}I_{Mq} F_{q}
+ ò_{CM} G_{p,q} d^{M}p (Ñ_{p} F_{p})
which we can write as
F_{q} = (-1)^{M}I_{Mq}^{-1} (
ò_{dCM} G_{p,q} d^{M-1}p F_{p}
- ò_{CM} G_{p,q} d^{M}p (Ñ_{p} F_{p}) )
_{[ HS 7-4.7 ]}
expressing an interior value F_{q} of F_{p} in terms of boundary values of F_{p} and interior values of Ñ_{p}F_{p} .
In essence, this provides a Ñ ^{-1} in that we can reconstruct F_{p} from ÑF_{p} provided we also have
"boundary contraints" specifying F_{p} over an enclosing surface.
If G_{p,q} Î I_{Mq} " p Î dC_{M}
we can commute I_{Mq} across G_{p,q} , incurring a sign change if M is even, to obtain
F_{q} =
- ò_{dCM} G_{p,q} I_{Mq}^{-1} d^{M-1}p F_{p}
+ ò_{CM} G_{p,q} I_{Mq}^{-1} d^{M}p (Ñ_{p} F_{p})
If F_{p} is C_{M}-monogenic so that
ò_{CM} G_{p,q} d^{M}p (Ñ_{p} F_{p}) vanishes,
we have a geometric generalisation of Cauchy's Theorem
F_{q} = (-1)^{M}I_{Mq}^{-1} ò_{dCM} G_{p,q} d^{M-1}p F_{p}
= - ò_{dCM} G_{p,q} I_{Mq}^{-1} d^{M-1}p F_{p}
_{[ HS 7-4.10 ]} .
With M=N and I_{p}=i in a Euclidean space we have monogenic G_{p,q} = o_{N}^{-1} (p-q) |p-q|^{-N}
as the Greens function for Ñ_{p}.
General Form
A differentiable M-form on an M-curve C_{M} is a point-dependant multivector-valued
linear function of the directed measure
L(p,d^{M}p)
= dp^{1}dp^{2}...dp^{M}L(x, h_{1p }Ù h_{2p }Ù.. h_{Mp }) .
The exterior differential of an (M-1)-form is the M-form defined by
dL(p,d^{M-1}p) º
Lim_{O ® 0}
|O|^{-1}
ò_{O} L(p,d^{M}p)
where O is a small volume enclosing x.
The general form of the fundamental theorem is
ò_{CM} dL(p,d^{M}p)
= ò_{dCM} L(p,d^{M-1}p)
but we will not pursue this here.
Poles and Residues
Cauchy's Theorem
Relates the value of a monogenic function at a point to the surface integral over the boundary
of any M-curve including the point.
F(a)
= (I_{Ma}o_{N})^{-1}
ò_{dCM} (p-a)|p-a|^{-N} d^{2}p F(p)
where C_{M} is a closed M-curve including a in its interior and I_{Ma}
is the orientation for C_{M} at a.
Thus the surface integral over a closed surface of a monogenic function is independant of (and so extremal for)
the actual surface integrated over, insofar as we can vary the surface without effecting the result
provided we don't move the boundary across any poles of F.
The surface integral is zero unless the enclosed volume contains one or more poles of F, in which case
the integral is the sum of the "residues" at the poles , each independant of the surface used. Closed
surface integrals of monogenic functions are thus discrete-valued and tend to vary discontinuously.
They provide a "regional quantisation" .
[Under Construction]
Next : Manifold Restricted Tensors