 Multivector Manifolds
10th August 1851: On Tuesday evening at Museum, at a ball in the gardens. The night was chill, I dropped too suddenly from the Differential Calculus into ladies' society, and could not give myself freely to the change. After an hour's attempt to do so, I returned, cursing the mode of life I was pursuing; next morning I had already shaken hands, however, with Diff. Calculus and forgot the ladies." --- Thomas Archer Hirst

Thus far we have considered fields over UN as point-dependant multivectors in UN rather than point-dependant points in UN. If we consider ¦ to map points in UN to points in UN, we have a transformation. The image under ¦ of a subset of UN is known as a "manifold" in UN.
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 UN is a set of points in targetspace UN locally representable 1-1 by a system of M scalar coordinates drawn from a well defined connected parameter space (aka. mapspace or (local) map ) MapÌVM .
A 0-curve is a point.
A 1-curve is a Â®UN mapping p(t) t Î [t0,t1] 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 CM is either closed or has as boundary a closed (M-1)-curve conventionally written dCM. Clearly ddCM is the empty set f . We will denote the "interior" or contents of an M-curve CM by d-1CM or CM· , so that   (dCM)· = CM .
We call an (N-1)-curve a hypercurve.

The closed k-curve "spanned" by k-blade ak is the infinite (unbounded) k-plane { p : pÙak=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 ¦(Map) 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(x1,x2,..,xm) sending VM ® UN defined over a fully bounded finite subvolume (interior of a unimapped (M-1)-curve) MapÌVM . Typically VM = ÂM, 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 Map and p=¦q(x) with the associated N-D point (1-vector) in CM Ì UN .   We will use the symbol _map or sometimes upper case symbols to distinguish structures and operators defined within Map from their within CM 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 ¦ : Map®UN with ¦(Map)=CM that serves for every q Î CM . 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 ¦:VM ® UN . Any M-curve Map in VM induces an M-curve CM=¦(Map) in UN. Typically we might have a flat M-plane segment Map in VM embedding to a "bendy" CM in UN seen as the Map specific "slice" of ¦ .

Submanifolds
Given a unimapped M-curve CM=¦(Map) and a point pÎCM we can construct k-dimensional submanifolds of CM at p as the images under ¦ of the subvolumes of Map 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 VN to define an invertible function ¦: VN®UN creating an N-curve of which CM is a submanifold.

Embedded Frame
We have a particular Map-point-dependant embedded frame in UN over our q neighbourhood consisting of M UN tangent vectors specified by the alternate notations
hkp  º (d/dxk)¦q(x) = Ðxeðk¦q(x) = (eðk¿Ñx)¦q(x) = ¦qÑx(ek)
where Ðxeðk and Ñx are the standard directed and undirected derivatives for directions within the mapspace Map.
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 UN with N-M orthonormal vectors lieing wholly outside Ip but in the absence of an extended N-D mapspace invertibly mapped into UN such an extension is only unique in the case M=N-1 when we can define hNp  = ( h1p Ù h2p Ù... hN-1p )-1i .

Example: Spherical Surface
We can formulate coventional spherical polar coordinates as ¦:VM=Â3 ® UN=Â3
¦(x)=¦(qeðq + feðf + reðr) = r sinq cosfe1 + r sinq sinfe2 + r cosqe3 = rRq,f§(e3)
where eðq,eðf,eðr are an orthonormal basis for VM and e1,e2,e3 are an orthonormal basis for UN. and   Rq,f = (-½e12f) (-½e31q) .

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 UN=VM=Â3  with eði=ei and regard ¦ as a transformation  then
¦Ñx(dx)) = (Ñx¿dx)¦(x)
= (dqr cosq cosf - dfr sinq sinf + dr sinq cosf)e1 + (dqrcos(q) sinf + dfr sinq cosf + dr sinq sinf) e2 + (-dqr sinq + dr cosq) e3  .

The embedded frame in UN is orthogonal.
hqp  = rRq,f§(e1) = req ;
hfp  = e3×¦(q,f,r) = (e3Ù¦(q,f,r))* = r sinqRq,f§(e2) = r sinqef;
hrp  = ¦(q,f,r)~ = Rq,f§(e3) = er
These have magnitudes  r;  r sinq ; and 1 respectively.

If we fix r=1, so restricing Map to a 2D parameterspace { q,f } and obtaining CM=S2 ,  the 2-curve boundary surface of a 3D unit sphere, then the derivative reduces to
¦Ñx(dx)) = (Ñx¿dx)¦(x) = (dq cosq cosf - df sinq sinf)e1 + (dq cosq sinf + df sinq cosf) e2 - dq sinq e3  .

From a purist point of view, mapspace coordinates have the form qeðq+feðf whereas tangent vectors are expressed in terms of e1,e2,e3 but we will here consider the Map as existing in UN and set eðq=e1 , eðf=e2.

Inverse Embedded Frame
We can construct an inverse embedded M-frame   hip  of UN 1-vectors within the tangent space CM satisfying hip ¿ hjp  = di j via
hkp  º (-1)k-1( h1p Ù.. hk-1p Ù hk+1p Ù.. hMp )( h1p Ù.. hMp )-1 .
Note that the hip ÎIp and are a frame for precisely the same subspace as are the embedded frame.

We can invert ¦q to express the xi(p) = ei¿¦q-1(p) as M scalar fields defined over a local (M-curve) neighbourhood of q.
When M=N so that ¦q : UN ® UN we have
hip  º Ñp (xi(p)) = Ñ¦q(xxi = ¦x-D(ei) .
[ Proof :   hip ¿ hjp    = ¦qxÑ(ei) ¿ ¦qx-D(ej) = ei¿ ¦qxD(¦qx-D(ej)) = ei¿ej = dij  .]
hip  is the normal to the coordinate isosurface xi(p)=xi(q) at q. In a nonorthogonal embedded frame, we can have hip Ù hip  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 hip ~ = Rq,f§(ei).
¦Dp(dp) º Ñx(dp¿¦(x)) = Ñx(dp1r sinq cosf + dp2r sinq sinf + dp3r cosq)
= r(dp1 cosq cosf + dp2 cosq sinf - dp3 sinq)eðq + r(-dp1 sinq sinf + dp2 sinq cosf)eðf + (dp1 sinq cosf + dp2 sinq sinf + dp3 cosq)eðr     defined for dpÎÂ3 . This reduces to
¦Dp(dp) = (dp1 cosq cosf + dp2 cosq sinf)eðq + (-dp1 sinq sinf + dp2 sinq cosf)eðf     when pÎS2 and dp is in the tangent space at p.

We have M scalar fields defined over CM by xi(p) = eði¿¦-1(p) = eði¿x .
For spherical mapping these are
q(p1e1 + p2e2 + p3e3) = cos-1(p3(p12+p22+p32))     [ p12 = (p1)2 ]
f(p1e1 + p2e2 + p3e3) = tan-1(p2/p1) [  With the usual caveats specifying quadrant ]
r(p1e1 + p2e2 + p3e3) = (p12+p22+p32)½

We have Ñp q(p) = -(1-(p32(p12+p22+p32)-1) ( e1(-p3x1(p12+p22+p32)-3/2 + e2(-p3x2(p12+p22+p32)-3/2 + e3((p12+p22)(p12+p22+p32)-3/2
= -((p12+x22)(p12+p22+p32)-1) (p12+p22+p32)-_3/2( e1(-p3p1) + e2(-p3p2) + e3(p12+p22) )
=?= r-2 hqp  = r-1 hqp ~
Ñp f(p) = (r sinq)-2 hfp  = (r sinq)-1 hfp ~
Ñp r(p) = p~ = hrp  = hrp ~ .

Local Orientation
The hkp  define a CM-point-dependant nondegenerate UN M-blade Jp º h1p Ù..Ù hMp  that spans the tangent space at p=¦q(x). CM has tangent M-plane (1+q)ÙJq at q , or (e0+q)ÙJq in GHC. The invertibility of ¦q ensures that Jq is nondegenerate.
The unit pseudoscalar for the tangent space given by Ip º Jp~ 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 IM-1p then it is conventional to specify the orientation ("handedness") of Ip by defining Ip = IM-1p np where np = IM-1p-1 Ip is the spur, the unit outward normal to the boundary at p.
The normalisation condition Ip2 = ±1 gives (ÐaIp)Ip + Ip(ÐaIp)=0 and taking the scalar part yields (ÐaIp)¿Ip=0 Þ ¯Ip(ÐaIp)=0.
Hence  ¯Ip(ÑpIp) = 0 .
[ Proof :   Choosing an orthonormal frame with e1Ùe2Ù...eM =Ip we obtain ÑpIp = åi=1N ei ÐeiIp = åi=M+1N ei ÐeiIp all terms of which lie outside Ip.  .]
Note that Ñp¿Ip ¹  0 in general.

Spherical Surface Example
For spherical coordinates S2 mapping, Jp = hqp Ù hfp  = hqp  hfp  = sinqRq,f§(e1)Rq,f§(e2) = sinqRq,f§(e12) .
Although Jp vanishes at the poles, it is natural to define Ip=Rq,f§(e12)     " p=Rq,f§(e3) ÎS2 .

The Metric
Let a,b be two 1-vectors in an M-curve's mapspace MapÌVM. Whereas aÙb = ¦-Dx(¦Ñx(a)Ù¦Ñx(b)) is geometrically meaningful , a¿b is not. In particular, it does not equal ¦Ñx(a)¿¦Ñx(b). Rather, we have the cocontraction a¿b = a¿xb º ¦Ñx(a)¿¦Ñx(b) and hence a variant geometric "coproduct" with point-dependant inner product ¿ replacing ¿ .

We postulate a symmetric point-dependant metric 1-tensor gx : VM ® VM such that a¿gx(b) = gx(a)¿b = ¦Ñx(a)¿¦Ñx(b) = a¿b .
For M=N the obvious candidate is gx º ¦Dx¦Ñx .     If ¦Ñx is symmetric, gx = (¦Ñx)2 .

Example: Spherical Surface
gx(dx) = r2dqeðq + r2 sin2qdfeðf + eðr     with associated line element length dx¿gx(dx) = r2dq2 + r2 sin2qdf2 + dr2
[ Proof : ¦Ñx(dx)) = (Ñx¿dx)¦(x)
= (dqr cosq cosf-dfr sinq sinf+dr sinq cosf)e1 + (dqrcos(q) sinf+dfr sinq cosf+dr sinq sinf)e2 + (-dqr sinq+dr cosq)e3  .
¦Dp(dp) º Ñx(dp¿¦(x)) = Ñx(dp1r sinq cosf + dp2r sinq sinf + dp3r cosq)
= r(dp1 cosq cosf + dp2 cosq sinf - dp3 sinq)eðq + r(-dp1 sinx siny + dp2 sinq cosf)eðf + (dp1 sinq cosf + dp2 sinq sinf + dp3 cosq)eðr
Thus gx(dx) = ¦Dp(¦Ñx(dx))
= ¦Dp( (dqr cosq cosf -dfr sinq sinf +dr sinq cosf)e1 + (dqrcos(q) sinf +dfr sinq cosf +dr sinq sinf) e2 + (-dqr sinq +dr cosq) e3 )
= ...[Tedious manipulations]... = r2dqeðq + r2 sin2qdfeðf + eðr  .]

Restricting r=1 for S2 gives gx(dx) = gx(dqeðq+dfeðf) = dqeðq + sin2qdfeðf

In much of the literature, notably with regard to Genral Relativity, the metric gx is regarded as profoundly fundamental. Indeed, it is sometimes known as the fundamental tensor. However, we will here regard gx as less fundamental than ¦Ñx which we will also make little subsequent use of in this chapter, considering orientation M-blade Ip described below as the defining "property" of an M-curve.

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 CM. This provides a k-foliation of k-curves over CM. Every p Î CM is contained within a k-curve having the given k-frame as embedded frame at p. These k-curves are not contained within CM in general, but will be if the k-frame at p is within Ip " pÎCM .

Suppose now that instead of a k-frame for each p we merely have a unit M-blade valued field Ip defined over pÎUN.
If p0ÎUN is known to lie in CM then another UN point p1 will lie in CM iff there exists a path (1-curve)   { p(t) Î UN : t Î [t0,t1] } with p(t0)=p0 ; p(t1)=p1 ; and p'(t)ÎIp(t) " tÎ[t0,t1]. Establishing whether this is, in fact, the case for a given p0 and p1 may be far from easy. If all we can say about ¦ with regard to p=¦(x) is that ¦Ñx(i)=Ip then things can get tricky. ( If ¦-1(p1) is available, of course, then p1ÎCM Û ¦-1(p1)ÎMap.)
However, we can nonetheless think of a non-degenerate-M-blade-valued field over UN (ie. one that nowhere vanishes) as defining a foliation of M-curves over UN, just one M-curve containing any given pÎUN. If Ip=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 UN 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 UN as determined by a unit M-blade field Ip defined over UN.

Hypercurves

An (N-1)-curve is known as a hypercurve and we can define a unit 1-vector normal by
np º iN-1p* = iN-1pi-1 = (i)2 iN-1p¿i . np = (-1)N-1eN in the fortutious basis and np2 = (-1)N-1 iN-1p2 i2 = ±1 where M=N-1 . .

Projector 1-multitensor     ¯
The projector of M-curve CM is the multitensor
¯(a) º ¯Ip(a) º (a¿Ip)Ip-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(ap) = ap - (np-1¿ap)np .

Integration over an M-curve
We assume here than the concept of scalar integration is well understood. That is, that òt0t1F(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 [t0,t1] by defining
òC F(p) dp º òt0t1F(p(t))p'(t) dt .
More generally   we have òC F(p) dp defined to be the limit of a summation at n-1 of n "samplepoints" p[i] along the 1-curve . p 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, òC F(p)dp ¹ òC 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 IMp[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(x1,x2) then the directed area element has form d2p = ( h1p Ù h2p )dx1dx2 .

For integrating over an M-curve CM=¦(Map) parametrised as p(x1,x2,..,_xuM) we have contributary (M-1)-simplex elements
dMp   =   ( h1p Ù h2p .. hMp )dx1dx2..dxM   =   |¦Ñ|Ipdx1dx2..dxM and we can think of M succesive scalar integrals of multivector F(p)Ip via
òCM F(pdmp = òCM F(p)( h1p Ù h2p ..Ù hMp ) dx1dx2...dxM .
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 |CM| º òCM Ip-1dp . This is the conventional arc length, surface area, and volume of CM 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 ax º a(x) is the field
F(ax)(k) º (2p)N ò dNx i-1 (-i(x¿k)) ax = (2p)N ò |dNx| (-i(x¿k)) ax     where the x integration is usually taken over all UN rather than a particular subspace of interest, and i commutes with ax and has i2=-1.
The inverse transform is
F-1(bk)(x) º (2p)N ò dNk i-1 (+i(x¿k)) bk where we are now integrating over k ÎUN .
For N=1 we have the scalar unitary Fourier transform F(a(x)(k) = (2p) ò-¥¥ dx (-ixk) a(x) . and its inverse F-1(b(var(k))(x) = (2p) ò-¥¥ dk (+ixk) b(k) .

F((i(x.b))(k) = (2p)N ò dNx i-1 (i(x¿(b-k)) = (2p)N Pj=1N  ò-¥¥ dxj (ixj(bj-kj))
The scalar integrals vanish except when bj=kj when we (informally) obtain ¥.
Since ò-KK dk ò-¥¥ dx cos(xk) = ò-¥¥ dx ò-KK dk cos(xk) = ò-¥¥ dx 2x-1 sin(Kx) = 4 ò0¥ 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(x1)d(x2)...d(xN) is an N-D Dirac delta function.
Of course F-1(d(b-k))(y) = (2p)N(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)N factors in the definitions of F and F-1 serve to ensure that F-1F = 1 .

An obvious geometric generalisation is F(ax)(k) º ò (2p)N ò dNx i-1 (xÙk) ax 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 ax.

Example: F(Ha) in Â

Let Ha(x) = (2a3)-1 for |x|£a and 0 elsewhere in Â . (2p) ò-¥¥ dx Ha(x) (-i kx) = (2a)-1 (2p) ò-aa dx (-i kx) = (2a)-1 (2p) (-ik)-1 [ (-i kx) ]-aa = i(2ak)-1 (2p) ( -2i sin(ak) ) = (2p) Sin(ak)       where Sin(x) º x-1 sin(x) .

Example: F(Ha) in Â3

As an example let Ha(x) = (3-14pa3)-1 for |x|£a and 0 elsewhere in Â3 and impose axies so that k=ke3. We have
(2p)-3/2 ò d3x Ha(x)(-i(k.x))   =   3(2p)-3/2a-2 k-2 ( Sin(ka) - cos(ka)) .
[ Proof : (2p)-3/2 ò d3x Ha(x)(-i(k.x)) = (2p)-3/2 ò0¥ dr ò0p dq ò02p df r2 sin(q) Ha(x)(-i(rk cos(q))
= (2p)-3/2 (3-14pa3)-1 ò0a dr ò0p dq ò02p df r2 sin(q) (-ir|k| cos(q))   =   (2p)-3/2 (3-12a3)-1 ò0a drr2 ò0p dq sin(q) (-ir|k| cos(q))
= (2p)-3/2 (3-12a3)-1 ò0a drr2 [(ir|k|)-1 (-ir|k| cos(q))]0p   =   -(2p)-3/2(3-12a3)-1 i |k|-1 ò0a drr [ (-ir|k| cos(q))]0p
= -(2p)-3/2(3-12a3)-1 i |k|-1 ò0a drr ( (ir|k|) - (-ir|k|) )   =   (2p)-3/2(3-1a3)-1 k-1 ò0a drr    sin(r|k|)
= (3-1a3)-1 k-1 [   k-2 sin(kr) - k-1r cos(kr) ] ]0a   =   (2p)-3/2(3-1a3)-1 k-1 (   k-2 sin(ka) - ak-1 cos(ka))   =   3(2p)-3/2a-2k-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 CM even when dÎIp . Strictly speaking we should define a 1-curve p(t) within CM such that p(0)=p and p'(t)=¯Ip(d) and form Ð[¯]d º Limt ® 0 t-1(F(p(t))-F(p(0))) [ HS 4-1.1 ] so we can restrict attention to points within CM 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)) + tÐp¦q-Ñx(¯Ip(d)F(p) + O(t2) and define Ð[¯]d º Ðx¦q-Ñx(¯Ip(d)) .

However if F(p) is continously defined throughout a UN neighbourhood of p then for dÎIp we have F(p(t)) = F(p+td+ O(t2)) = F(p) + tÐd+ O(t)F(p)ïp + O(t2) = F(p) + tÐdF(p)ïp + O(t2) so that t-1(F(p(t))-F(p(0))) = ÐdF(p)ïp + O(t).
Thus for general d, sufficiently smooth CM, and continuos F(p) defined near to as well as over CM, the ÐIp)¯(d definition is equivalent, and usually far easier to work with.

Suppose now that Fp=F(p) is a multivector valued field defined only over CM but not necessarily confined within Ip.
Fp induces a field in the mapspace Fx = F¦(x) for which ÐeðkFx need not lie in Ip . Even if we insist that Fp Î Ip so that ¯Ip(Fp)=Fp, then ÐeðkFx still need not lie in Ip.
Similarly, assuming Fp to be defined for p near to as well as within CM, ÐdFp need not lie in Ip even if dÎIp and FpÎIp " p. Thus the tangential derivative "acts within" CM but is not "confined to" CM in that it can (at p) "return" multivectors not contained in CM at p. It differentiates along tangents but is not itself tangent. For a derivative entirely "within" CM 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 Ñp = Ñb (b¿Ñ) = Ñb Ðp↓(b) and since (b¿Ñ) = Ð¯(b) = 0 if ¯(b)=0 we have
Ñp º Ñb Ðp↓(b) = Ñb Ðp↓(b)     providing perhaps the best symbolic definition of Ñ .

The mapspace coordinate based definition of the undirected tangential derivative Ñ[CM] pF(p) of a continuous multivector-valued field F(p) defined over a within-CM neighbourhood of point q on M-curve CM is
Ñ º Ñ[CM] p º åk=1M  hkp Ðxeðk     where hip  are a reciprocal embedded frame for CM and Ðxeðk = (d/dxk) operates in the mapspace Map as
ÐeðkF(p) = ÐxeðkF(¦(x)) = (Ñp¿¦Ñx(eðk))F(p) = FÑp(¦Ñx(eðk)) = FÑp( hkp ) . [  We have droppped the q suffix on ¦ for brevity. ]
The use of reciprocal tangent frame hkp  in the defintion of Ñ[CM]p "counter scales the expansion" of ¦Ñ to ensure Ñp p º Ñ[CM]p p = M .

If F(p) is continuously defined over a UN neighbourhood of q and CM is smooth we also have
Ñ º Ñ[CM] p = Ñ[Ip] p º åk=1M  hkp Ðp hkp      where Ð hkp  operaties in UN .

Ñp is essentially Ñp confined to act within CM (as visually suggested by symbol Ñ  being a "portion" of Ñ) though note that ÑpF(p) may not be confined to Ip even when F(p) is.
Ñp is the directed 1-derivative "splayed out" only over directions lieing within Ip ; we clearly have Ñ[UN] p = Ñ[i] p = Ñp and Ñ = Ñ<0>.
When the particular M-curve CM under discussion is unambiguos, we will abbreviate Ñ[CM] p to Ñp or Ñ.

Suppose that e1,e2,...eM are an orthonormal basis for Ip (at a given p only) which we extend by eM+1,..eN to a fortuous universal basis for UN at every p, though it coincides with Ip only at the particular given p of interest. Then we can write
¯Ip(Ñp) = åi=1N. ¯Ip(ei)Ðei = åi=1M. eiÐei = Ñp = åi=1N. eiÐ¯Ip(ei) .

For a more general UN basis we have ¯Ip(Ñp) = åi=1N ¯Ip(eiÐei) = åi=1N ¯Ip(ei)Ðei so that
ej¿¯Ip(Ñp)   = åi=1N (ej¿¯Ip(ei))Ðei = ¯Ip(ej)¿åi=1N eiÐei = ¯Ip(ej)¿Ñp = Ð¯Ip(ej) .
Hence Ñp = ¯Ip(Ñp) and our targetspace coordinate definition of Ñp is
Ñp F(p) º åi=1N ¯Ip(ei)ÐeiF(pÑ)     with the differentiating scope of the Ðei applying to F(p) only.

We say F(p)  is monogenic (aka. analytic) on CM if Ñp F(p) = 0 over CM.

We can also define a orthotangential derivative Ñ^ º Ñ^p º Ñp - Ñp   =   åi=1N ^Ip(ei)Ðei .

For 1-curve C= { p(t) } we have Ñ[C] p = h1p (dF(p(t))/dt) = p'(t)-1(dF(p(t))/dt) , effectively the derivative with respect to arc length.

Though ¯(Ñ) º  ¯Ip(Ñp) is a natural notation for the undirected tangential derivative, we favour Ñ  º Ñp here to minimise confusion with the composition operator (¯IpÑ)(ap) º ¯Ip(ÑpapÑ) which lies within Ip even when ap does not. Nonetheless, it is important to recognise that in essence Ñ = ¯(Ñ) = ¯(Ñ)     abbreviating (¯(Ñp))(apÑ) = Ñp(ap)     " ap .

The projection ¯ º ¯Ip is   equivalent to the tangential 1-differential of the "scalar" identity multitensor 1p(a)=1(a)=a and so we have the alternate notation ¯ = 1Ñ abbreviating ¯Ip(a) = 1Ñ[Ip](a)     " multivector a.
[ Proof : (a¿Ñðp)p =(a¿åi=1M  hip Ðp hip )p =åi=1M (a¿ hip ) hip  =¯Ip(a)  .]

The symmetry (self-adjointness) of projection gives Ñ*ap = ¯(Ñ)*ap = Ñ*(¯Ip(ap)) 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*ap = ¯Ip(Ñ)*ap = Ñ*(¯Ip(ap))  .]

Recall that with regard to its action on functions satisfying the integrability condition Ða×Ðb = 0 " a,b, we have ÑÙÑ=0 . This is not the case for Ñ, but we will later show that ¯(ÑÙÑ)=0 and ¯(ÑÙÑÙap)=0     " ap.

Normalisation condition Ip2=±1 gives ¯Ip(ÐaIp)=0 which in turn yields Ñ Ip = Ñ Ù Ip

For p2 ³ 0, Ñ[bk] |p|k   =   Ñ[bk]( (p2)½k   =   k |p|k-2 ¯bk(p)     for nondegenerate k-blade bk.

Alternate Ñ  Definition
A coordinate-independant definition of the tangential derivative at q discussed at length in Hestenes NFMP is
Ñ[CM] p º Ip-1 Lim|O| ® 0 [ |O|-1 òdO  dM-1p F(p) ]     this being the geometric product of the inverted M-orientation Ip-1 at q and the limit (finite or otherwise) of the multivector-valued directed integral òdO  dm-1p F(p) taken over the boundary dO of a small M-curve O Ì CM 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, Ñp 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 Ñp as a "splaying" of multivector-directed derivatives. Our fundamental view of Ñ is essentially "that which satisfies" a*Ñ = Ða

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 CM to values on it's boundary dCM .

òdCM dM-1p F(p)   =   òCM dMp (ÑpF(p)) º òCM dMp Ñ[CM]F(p)   =   (-1)M+1òCM (Ñ¿dMp) F(pÑ)     where scaled M-blade dMp = IpdMp = Ip|dMp| is an elemental M-simplex for CM 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.

• if dCM = f, ie. if CM is closed, then òCM dMp Ñ[CM]F(p) = 0 for any F(p) defined everywhere in CM.
Since d2CM = 0 for any CM it follows that òCM dMp p Ñ[CM]Ñ[dCM] F(p) = 0 for any Fp and CM
• An interior integral òCM dMp F(p) can be replaced by a boundary integral provided we can find a "potential" H(p) with Ñ[CM]pH(p) = F(p).
• In the case M=N we have Ñ[CN] = Ñ and Ip=i and hence òCN i|dNp|Ñ¦(p) = òdCN in-1|dN-1p| ¦(p)
Þ òCN|dNp| (Ñ¿¦(p) + ÑÙ¦(p) ) = òdCN|dN-1p|(n¿¦(p) + nÙ¦(p))     (where n is the (assumed postive signature) unit normal to hypercurve dCN).
This provides (scalar part) the divergence theorem (aka. Gauss's theorem) and ((N-2)-vector part) the (dual of) a curl integral theorem.
• In the case M=1 we have òC1 dp Ñ[C1]F(p) = F(t1)-F(t0)     providing F(t) from by the path directed integration of the gradient of F given F(t0).
• For M=N-1 with Ip=np* we have òdCN-1 dN-2p F(p)   =   (-1)N òCN-1 (Ñ¿dN-1p) F(pÑ) which provides a generalised Kelvin-Stoke's theorem
òdCN-1 dN-2p ¯dN-2p(F(p)) = òCN-1 dN-1p ¯dN-1p(ÑpÙF(pÑ)) equating the integral over a hypercurve's boundary of F(p) projected into the boundary and the integration over the hypercurve of ÑpÙF(p) projected into the hypercurve, Ñp being the full N-D 1-derivative rather than its projection Ñp into dN-1p.
[ Proof : Consider k-vector Fk(p). For k>N-2 both projections vanish.  For 0£k£N-2 we have
òdCN-1 dN-2p ¯dN-2p(F(p)) = (-1)k(N-1) òdCN-1 ¯dN-2p(Fk(p)) dN-2p = (-1)k(N-1) òdCN-1 Fk(p)¿dN-2p
= (-1)k(N-1)+k(N-2-k) òdCN-1 dN-2p.Fk(p) = (-1)N(òCN-1 (Ñ¿dN-1p) Fk(pÑ))<N-2-k> = (-1)NòCN-1 (Ñ¿dN-1p) . Fk(pÑ)
= (-1)N+k(N-2-k)òCN-1 Fk(pÑ)¿(Ñ¿dN-1p) = (-1)N+k(N-k)òCN-1 (Fk(pÑ)ÙÑ)¿dN-1p = (-1)N+k(N-k+1)òCN-1 (ÑÙFk(pÑ))¿dN-1p
= (-1)N+k(N-k+1)òCN-1 ¯dN-1p(ÑÙFk(pÑ))dN-1p = (-1)N+k(N-k+1)+N(k+1)òCN-1 dN-1p ¯dN-1p(ÑÙFk(pÑ)) . = òCN-1 dN-1p ¯dN-1p(ÑÙFk(pÑ)) .  .]

For scalar field F(p)=fp this is òdCN dN-1p fp = òCN dNp (Ñpfp)
For 1-vector F(p)=¦(p) and N=3 we recover the traditional Kelvin-Stoke's theorem òdc2 dp¿¦(p) = òc2 (Ñ×¦(pÑ))¿np|d2p|
[ Proof : òdc2 dp¿¦(p) = òdc2 dp ¯dp(¦(p)) = òc2 d2p ¯d2p(ÑÙ¦(pÑ)) = òc2 (ÑÙ¦(pÑ))¿d2p
= òc2 (ÑÙ¦(pÑ))¿(np*)|d2p| = -òc2 (((Ñ×¦(pÑ))*)¿(np*)|d2p| = òc2 ((Ñ×¦(pÑ))¿np|d2p|  .]

A more general form of the fundamental theorem is
òdCM Gp dM-1p Fp   =   (-1)M-1òCM GpÑÑp dMp Fp     +     òCM Gp dMp ÑpFpÑ . [ HS 7-3.10 ]
[ Proof :   òdCM Gp dM-1p Fp   =   òCM GpÑ dMpÑp Fp     +     òCM Gp dMpÑp FpÑ     and recall that dMp is a scaled M-blade pseudoscalar for the tangent space containing Ñ and so GpÑ dMpÑp = GpÑ(dMp.Ñp) = (-1)M-1GpÑ(Ñp.dMp) = (-1)M-1GpÑ(ÑpdMp) = (-1)M-1(GpÑp)dMp  .]

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)=y0 and y'(a)=y0' has solution y(x) = y0 + y0'(x-a) + òax dx" òax" dx' ¦(x',y(x')) which can be alternatively evaluated as
y(x) = y0 + y0'(x-a) + òax dx' (x-x')¦(x',y(x'))   =   y0 + y0'(x-a) + òab 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)=y0 ; y'(a)=y0'   . [  Hvsd(x) is the Heaviside step function zero for x£0 and 1 for x>0, here exploited to replace the indefinite òax dx' with the definite òab dx'   ]

If Gp,q is a 1-vector-valued Green's function having two primary CM point-valued arguments with Ñp Gp,q = Gp,q Ñq = 0 " p¹q and ÑpGp,q = -Gp,qÑq = 1 at p=q,  the Fundamental Theorem of Calculus provides
òdCM Gp,q dM-1p Fp   =   (-1)M-1IMq Fq + òCM Gp,q dMp (Ñp Fp)     which we can write as
Fq   =   (-1)MIMq-1 ( òdCM Gp,q dM-1p Fp     -     òCM Gp,q dMp (Ñp Fp) )       [ HS 7-4.7 ]
expressing an interior value Fq of Fp in terms of boundary values of Fp and interior values of ÑpFp . In essence, this provides a Ñ -1 in that we can reconstruct Fp from ÑFp provided we also have "boundary contraints" specifying Fp over an enclosing surface.
If Gp,q Î IMq " p Î dCM we can commute IMq across Gp,q , incurring a sign change if M is even, to obtain
Fq   =   - òdCM Gp,q IMq-1 dM-1p Fp     +      òCM Gp,q IMq-1 dMp (Ñp Fp)
If Fp is CM-monogenic so that òCM Gp,q dMp (Ñp Fp) vanishes, we have a geometric generalisation of Cauchy's Theorem
Fq   =   (-1)MIMq-1 òdCM Gp,q dM-1p Fp   =   - òdCM Gp,q IMq-1 dM-1p Fp [ HS 7-4.10 ] .

With M=N and Ip=i in a Euclidean space we have monogenic Gp,q = oN-1 (p-q) |p-q|-N as the Greens function for Ñp.

General Form
A differentiable M-form on an M-curve CM is a point-dependant multivector-valued linear function of the directed measure
L(p,dMp) = dp1dp2...dpML(x, h1p Ù h2p Ù.. hMp ) .
The exterior differential of an (M-1)-form is the M-form defined by dL(p,dM-1p) º LimO ® 0 |O|-1 òO L(p,dMp)     where O is a small volume enclosing x.
The general form of the fundamental theorem is òCM dL(p,dMp) = òdCM L(p,dM-1p) 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)   =   (IMaoN)-1 òdCM (p-a)|p-a|-N d2p F(p) where CM is a closed M-curve including a in its interior and IMa is the orientation for CM 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

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Copyright (c) Ian C G Bell 1998
Web Source: www.iancgbell.clara.net/maths or www.bigfoot.com/~iancgbell/maths
Latest Edit: 20 Jan 2008. 