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.
The following outline of the basics of the calculus of manifolds stresses the operator nature of derivatives more than is customary. 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 UN locally representable 1-1 by a system of M scalar coordinates drawn from a well defined parameter space Map.
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 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" 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 = f (the empty set). We will denote the "interior" or contents of an M-curve CM by CM· , so that   (dCM)· = CM .
We call an (N-1)-curve a hypercurve.

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 will assume the ¦(Map) model.

The k-curve "spanned" by k-blade ak is the infinite k-plane { p : pÙak=0 } and a k-sphere can be regarded as a particularly simple k-curve.

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 . We refer to Map as the parameter space or mapspace (aka. a (local) map ) . 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 here use 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 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.

Generator Functions
[under construction]

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) = Ðxek¦q(x) = (ek¿Ñx)¦q(x) = ¦qÑx(ek)
where Ðxek and Ñx are the standard directed and undirected derivatives for directions within the mapspace Map.
The embedded frame is not orthogonal for general ¦q. It 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 .

Spherical Surface Example
We can formulate coventional spherical polar coordinates as ¦:VM=Â3 ® UN=Â3
¦(x)=¦(qeq + fef + rer) = r sinq cosfe1 + r sinq sinfe2 + r cosqe3 = rRq,f§(e3)
where eq,ef,er are an orthonormal basis for VM and e1,e2,e3 are an orthonormal basis for UN. and   Rq,f = ee12f ee31q

Difficulties arise near the poles (q=p or 0) with our tangent vectors. 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) ;
hfp  = e3×¦(q,f,r) = (e3Ù¦(q,f,r))* = r sinqRq,f§(e2);
hrp  = ¦(q,f,r)~ = Rq,f§(e3)
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 qeq+fef whereas tangent vectors are expressed in terms of e1,e2,e3 but we will here consider the Map as existing in UN and set eq=e1 , ef=e2.

Inverse Embedded Frame
We can construct an inverse embedded M-frame   hip  of 1-vectors within the tangent space 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.

Spherical Surface Example

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)eq + r(-dp1 sinq sinf + dp2 sinq cosf)ef + (dp1 sinq cosf + dp2 sinq sinf + dp3 cosq)er     defined for dpÎÂ3 . This reduces to
¦Dp(dp) = (dp1 cosq cosf + dp2 cosq sinf)eq + (-dp1 sinq sinf + dp2 sinq cosf)ef     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)
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 a homegeneous higher dimensiunal embedding) with  . 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 vansihes 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 .

Spherical Surface Example
gx(dx) = r2dqeq + r2 sin2qdfef + er     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)eq + r(-dp1 sinx siny + dp2 sinq cosf)ef + (dp1 sinq cosf + dp2 sinq sinf + dp3 cosq)er
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]... = r2dqeq + r2 sin2qdfef + er  .]

Restricting r=1 for S2 gives gx(dx) = gx(dqeq+dfef) = dqeq + sin2qdfef

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 regard 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[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, 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) 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.

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¯Ip(a) = (¯[a]¿Ñ)  .
Directed tangential derivatives commute whenever directed ones do, so the integrability condition allows us to commute directed tangential derivatves.

Undirected Tangential 1-Derivative       Ñ

One can define the undirected tangential derivative 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)  .
But (b¿Ñ) = Ð¯(b) = 0 if ¯b=0 so we have Ñp = Ñb Ðp¯(b)  = Ñb Ðp¯(b) .

The map coordinate based definition of the undirected tangential derivative Ñ[CM] pF(p) of a multivector-valued field F(p) defined over a neighbourhood of point q within a M-curve CM is
Ñ º Ñ[CM] p º Ñ[Ip] p º åk=1M  hkp Ðp hkp  = åk=1M  hkp Ðxeðk
where hip  are a reciprocal embedded frame for CM ; Ð hkp  operaties in UN ; and Ðxeðk = (d/dxk) operates in the mapspace Map as
ÐeðkF(p) = ÐxeðkF(¦(x)) = (Ñp¿¦Ñx(eðk))F(p) = FÑp(¦Ñx(eðk)) = FÑp( hkp ) .
When the particular M-curve CM under discussion is unambiguos, we will abbreviate Ñ[CM] p to Ñp and thence to Ñ.

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

Moving out of the mapspace, suppose instead 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 i at every p, though it coincides with Ip only at the particular p of interest. Then we can write
Ñ = ¯Ip(Ñp) = åi=1N. ¯Ip(ei)Ðei = åi=1M. eiÐei = åi=1N. eiÐ¯Ip(ei) .
For a more general basis for UN, it is natural to define the tangential derivative by
Ñp º åi=1N eiÐ¯Ip(ei)  .
We can also define a orthotangential derivative Ñ^ º Ñ^p º Ñp - Ñp   =   åi=1N eiÐ^Ip(ei) .

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.
We can think of Ñ[CM] p as being Ñp "restricted" to act within M-curve CM ( The symbol Ñ is visually suggestive of a "portion" of Ñ ). It is the directed 1-derivative "splayed out" only over directions lieing within Ip . We clearly have Ñ[UN] p = Ñp and Ñ[1] = Ñ<0> . The use of hkp  in the defintion of Ñ[CM]p "counter scales the expansions" of ¦Ñ to ensure Ñp p º Ñ[CM]p p = M .

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(Ñp(ap)). Nonetheless, it is important to recognise that in essence Ñ = ¯(Ñ) = ¯(Ñ)     abbreviating (¯(Ñp))(apÑ) = Ñp(ap)     " ap .

The projection ¯ 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)  .]

Perhaps the best symbolic definition is Ñp º Ñb Ðp¯(b) .

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.

Suppose now that Fp=F(p) is a multivector valued field defined over CM with Fp not necessarily within Ip. Fp induces a field in the mapspace Fx = F¦(x) for which ÑxFx need not lie in Ip since 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 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.

Alternate Ñ  Definition
The 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 the fundamental theorem of calculus and generalise complex residue theory.

Basic Form
The basic theorem (aka. Fundamental Theorem of Calculus [Basic]) relates derivatives within an M-curve to evaluations on it's boundary.

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

The proof [ommitted here, see Hestenes] is an almost trivial consequence of the coordinate-independant definition Ñ[CM]F . It has the following consequences.

• if dCM = f, ie. if CM is closed, then òCM dMp Ñ[CM]F(p) = 0 for any F(p) .
Since d2CM = 0 for any CM it follows that òCM dMp p Ñ[CM]Ñ[dCM] F(p) = 0 for any Fp and CM
• A "volume" 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) = Fp, which we often can.
• In the case M=N we have Ñ[CM] = Ñ and hence
òCM i dNp Ñ¦(p) = òdCM in-1dN-1p ¦(p) Þ òCM dNp (Ñ¿¦(p) + ÑÙ¦(p) ) = òdCM dN-1p (n¿¦(p) + nÙ¦(p))
(where n is the (assumed postive signature) normal to hypercurve dCM). This provides (scalar part) the divergence theorem (aka. Gauss's theorem) and (bivector part when N=3) 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).

A more general form of interest is
òdCM Gp dM-1p Fp   =   (-1)M-1òCM (GpÑp) dMp Fp     +     òCM Gp dMp (Ñp Fp) [ 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/dx2 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, exploited to allow us 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 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
Included for completeness. Casual reader should skip.
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 òO L(p,dMp) / |O|     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" .

Further differentials and derivatives
To the extent that attention is focused on intrinsic properties of the manifold, Ñ is fragmented and various brands of derivative emerge. If one "renounces" the extrinsic orientation M-blade pseudoscalar, one is left with the intrinsic affine connection and associated Riemann and Ricci curvature tensors which tend to be mathematically and notationally fearsome. Nonetheless, we shall proceed some way along this route. The casual reader should skip to the next chapter.

Operator Notations
If f1(a) is an operator taking one parameter then f1×(a,b) º ½(f1(a)×f1(b))
If f2(a,b) is an operator taking two parameters then f2×(a,b) º ½(f2(b,a)-f2(a,b))
f1<k>(a) º f1(a<k>)

We define f1¨ by f1¨ (a) ¨ a = f1(a) where ¨ is a multivector product.
Thus, for example, if tensor fp(a) is scalar valued then ¦¿(a) is the 1-tensor f(a) = (f(a))a (a-2) satisfying f(a)¿a = f(a) .

There is a strong temptation to abbreviate ¦(a) as ¦a , "ommiting the brackets", but this is dangerous because ¦a more properly denotes the composite operator (¦(a))(b) = ¦(ab) .
Confusion can arise in particular in the case of an operator like Ñp which we also consider to be "like" a 1-vector geometrically. f(Ñ) and fÑ are then fundamentally different constructs.
We will insert a composition product symbol ° between two operators only when we wish to emphasise an "assumed" compositional product. The composition product symbol will normally be ommitted for brevity. Thus
f°g(a) º f(g(a)) º (fg)(a)   º fg(a)

How are we to interpret f1Ñg1 º (f1Ñ)g1 ? The "differentialiser" Ñ converts f1 from an operator or tensor taking one nonprimary argument to one taking two, it introduces a second non-primary 1-vector parameter. In nonabbreviated expressions we will add this parameter at the rightmost end of the parameter list with
f1Ñ(a,b) = (Ðbf1)(a) = Ðb(f1(a)) - f1(Ðba)  . We will interpret the g1 in the "composite product" f1Ñg1 as applying to the "newest" rightmost nonprimary parameter so that f1Ñg1(a,b) º f1Ñ(a,g1(b)) .
From a programmers' perspective, we can think of "parsing" our composite operator expressions from left to right pushing introduced parameters onto a stack. When a "compositional irregularity" such as f1Ñg1 is encoutered it is the most "recent" parameter to which they apply in accordance wth a stanard "last in first out" stack. We will accordingly refer to f1Ñg1(a,b) º f1Ñ(a,g1(b)) as the LIFO convention.

Tangential 1-differential  Ñ

We can define the tangential 1-differential of field Fp by
FÑ[CM]pp(a) º FpÑ(a) º (a*Ñ[CM]p)F(p)   =   (a*¯(Ñp))F(p)   =   (¯(a)*Ñp)F(p)   =   FpÑ(¯(a)) º FpÑ¯(a) which gives us
Ñ   =   Ñ¯     abbreviating FpÑ(a) = FpÑ¯(a) º FpÑ(¯(a)) .
; the a-directed tangential derivative is equivalent to the ¯(a)-directed derivative.

Consider now (ÑFp)Ñ(a) = ¯Ñ(Ñb,a)FpÑ(b) + ÑbFpÑ2(b,a)     [ HS 4-1.18 ]

More generally suppose Fp(a1,a2,...ak) is an extensor tensor of k multivector variables themselves p-dependant. We can take a 1-differential of the secondary differntial Ña1Fp(a1,a2,...ak) to give
(Ña1Fp)Ñ(a1,..,ak,b) = ¯Ñ(Ña1,b)Fp(a1,..,ak) + Ña1FpÑ(a1,..,ak,b) .
For 1-vector a1 = ¯a1 we can write this as
(b*Ñ)ÑÞ   =   ¯Ñ(ÑÞ, b)     + ÑÞ(b*Ñ)     which we will here refer to as the primary-secondary Ñ commutation rule     [ HS 4-1.19b ] .

Analagous to the gradifying substitution rule we have the tangential gradifying substituion rule that applying the operatior (Ñß¿ÑÞ) is equivalent to replacing the first nonprimary argument of a tensor with ÑÜ .
[ Proof :  (Ña¿Ñß)Fp(a,b,..) = åi=1M ei2 Ðaei(ÐpeiFp)(a,b,...) = åi=1M (ÐpeiFp)(ei2ei,b,...) = åi=1M (ÐpeiFp)(ei,b,...)
= åi=1M FpÑ(Ðpeiei,b,...) = FpÑ(Ñ,b,...)  .]

Hypercurve
For M=N-1 we have Ñ = ¯(Ñ) = Ñ - np(np¿Ñ) = Ñ - npÐnp .
Since np¿Ðanp = 0 " a , Ñpnp lies within iN-1p and ^[Ñpnp] = np-1ÙÐnpnp  .
More generally, let np be a unit 1-field. Ñp = np2Ñp = np(np¿Ñp) + np(npÙÑp) corresponding to tangential and orthogonal components of Ñp .

Directed Coderivative      Ð()

With regard to a multivector field Fp defined over CM (but not necessarily confined within Ip) we can most simply define the a-directed coderivative for aÎUN as the projection of the a-directed tangential derivative, ie. the projection of the ¯Ip(a)-directed dirivative
Ð = ¯ Ð¯() º ¯° Ð¯()     abbreviating ÐaFp º ¯IpÐ¯IpaFp

The symbol Ð can thus be thought of as an abbreviation
Ð   =   ¯ Ð¯()   =   Ð¯()¯ - ¯Ñ     abbreviating Ðb(ap) = ¯( Ð¯(b)ap) = Ð¯(b)(¯(ap)) - ¯Ñ(ap,b)     [ HS 5-1.1 ] . The underscore serves to remind us of a p and Ip (ie. CM) dependance.
[ Proof :   Ð¯(d)(¯(ap)) = ( Ð¯(d)¯)(ap) + ¯( Ð¯(ap)) Þ Ðd(ap) º ¯( Ð¯(d)(ap)) = Ð¯(d)(¯(ap)) - ( Ð¯(d)¯)(ap) = Ð¯(d)(¯(ap)) - ¯Ñ(ap)  .]

The directed coderivative of an extended field is defined by
(ÐdFp)(a1,a2,...) º Ðd(Fp(a1,a2,..)) - Fp(Ðda1,a2,...)- Fp(a1,Ðda2,...) = ¯( Ð¯(d)(Fp(a1,a2,..)) ) - Fp(¯( Ð¯(d)a1),a2,...)- Fp(a1,¯( Ð¯(d)a2),...)
Note that this differs from [ HS 4-3.3 ] which inserts subtracts - Fp(Ðd¯(a1),a2,...)- Fp(a1,¯( Ð¯(d)a2),...)

Coderivative     Ñ
The covariant derivative Ða within an M-curve can be approached in a number of creative ways. One is to simply write down all the properties we would like a derivative to have ( such as ÑpFp Î Ip ; Ðpa1p + Ðpa2p = Ðp(a1p+a2p) ; and so forth ) and then intone "as defined so mote it be" three times at midnight. Another is based on projections [see General Relativity ].    Many appeal to a notion of "parallel transport" which is defined (or not) in a variety of ways.
We define the undirected coderivative 1-vector operator as the projected tangential derivative
ÑFp º Ñ[Ip]pFp º ¯Ip(Ñ[Ip]pFp)     which we can express operationally as Ñ º  ¯Ñ     noting carefully that this denotes operator composition ¯°Ñ º ¯(Ñ(ap)) rather than ¯(Ñ)(ap) = ¯(¯(Ñ))(ap) = ¯(Ñ)(ap) = Ñ(ap) . The differentiating scope of the Ñ is to be thought of as extending rightwards only in the usual manner, and not effecting the ¯ .
We have the operator identity Ða = (a¿Ñp) .
Since ÑFp Î Ip for a scalar field Fp the coderivative operator Ñp is equivalent to the tangential derivative Ñp when acting on scalar fields, and for aÎIp the a-directed coderivative Ða is equivalent to the a-directed derivative Ða when acting on scalar fields.

Ñp(ap) º ¯(Ñp(ap)) = Ñp(¯(apÑ))     by the projected product rule .
This is particularly clear when expressed in coordinate terms with a fortuitous basis as Ñp(¯(apÑ)) = åi=1N ei Ð¯(ei)(¯(apÑ)) = åi=1M ei(¯( Ð¯(ei)ap)) = åi=1M ¯(ei)(¯( Ð¯(ei)ap)) = åi=1M ¯(ei( Ð¯(ei)ap)) = ¯(ei(åi=1M  Ð¯(ei)ap)) = ¯(Ñp(Ap)) º Ñp(Ap) .

For an extended field Fp(a1p,a2p,...akp) with aipÎUN we have a-directed primary coderivative
Ð¯aFp(a1p,a2p,...akp) º (ÐaFp)(a1p,a2p,...akp)
= Ða(Fp(a1p,a2p,...akp)) - Fp(Ða(¯Ip(a1p)),a2p,...akp) - Fp(aip,Ða(¯Ip(a2p)),,...akp) - Fp(aip,a2p,...,Ða(¯Ip(akp)))
and an ith exterior  coderivative
Ñi®¿p Fp(a1p,a2p,...akp) º ¯Ip(Ñi®¿p Fp(a1p,a2p,...akp)) , the projection of the ith conveyed divergence.

The cogradifying substitution rule that applying the operator (Ñß¿ÑÞ) = (Ñß¿_csgrad) º ¯°(Ñß¿ÑÞ) is equivalent to replacing the first nonprimary parameter with ÑÜ defined by follows immediately as the projection of tne tangential gradifying substitution rule.

We will see that ÑÙÑÙap = 0     " path-independant ap , so that Ñ2 , while not a scalar operator, does not increase grade .

Hypercurve
For M=N-1 we have Ñnp = Ñnp .

Projection Differential (1.2)-tensor     ¯Ñ

¯Ñ(ap,d) º ¯IpÑp(ap,d) º Ðd(¯Ip(ap)) - ¯Ip(Ðdap)     (informally) the rate of change of ¯Ip(a) in direction d , is of of less interest than ¯Ñ,  the primary tangential 1-differential of ¯Ip (ie. the second tangential 1-differential of the identity function) ,
¯Ñp(ap,d)   =   ¯Ñp(ap,¯d)   =   1Ñ2(ap,d)   =   ( Ð¯(d)¯Ip)(ap)     =   Ð¯(d)(¯(ap)) -  ¯( Ð¯(d)ap)   =   2( Ð¯(d)×¯)(ap)  .
The symmetry ¯Ñ(a,b) =¯Ñ(b,a) for a,b Î Ip follows from the symmetry of 1Ñ2 in UN.
Tangentially differentiating the outtermorphism result ¯Ip(aÙb)=¯Ip(a) Ù ¯Ip(b) yields
¯Ñ(aÙb,d) =  ¯Ñ(a,d) Ù ¯(b) +  ¯(a) Ù ¯Ñ(b,d)     [ HS 4-2.6 ] and hence
¯Ñ(a1Ùa2Ù....ak,d) = åi=1k (-1)i+1 ¯Ñ(ai,d)Ù¯Ip(a1)Ù... ¯Ip(ai-1)Ù ¯Ip(ai+1)Ù... ¯Ip(ak) . [ HS 4-2.31 ]

¯¯Ñ   =   ¯Ñ^       ;     ^¯Ñ¯   =   ¯Ñ¯       [ HS 4-2.11 ] and so
¯=¯Ñ   =   ^=¯Ñ   =   0     abbreviating ¯(¯Ñ(¯(a),b)) = ^(¯Ñ(^(a),b)) = 0     " a,b.
[ Proof :  Tangentially differentiating  ¯2(a)  =¯(a) gives ¯Ñ(a,d) = ¯Ñ(¯(a),d) + ¯(¯Ñ(a,d)) Þ ¯¯Ñ(a,d) = ¯Ñ((1-¯)(a),d) = ¯Ñ(^(a),d) .
Thus ¯=¯Ñ = ¯Ñ ^¯  = ¯Ñ 0 = 0  .]

Trivially therefore, ¯¯Ñ(Ip,d) = 0 and hence ¯Ñ(Ip,d)¿Ip = 0     " d .
ÑÞ¿(¯Ñ¯) = 0 abbreviating Ñb¿¯Ñ(¯a,b) = 0 " a,b        [ HS 4-2.17a ] .
[ Proof :  ÑÞ¿¯Ñ(¯a,b)  (¯(ÑÞ))¿¯Ñb) =  ÑÞ¿(¯¯Ñb)) = ÑÞ¿0 .  .]

ÑÞÙ(¯Ñ^) = 0
[ Proof : See [ HS 4-2.17b ]  .]

¯Ñ enables us to express the tangential differential of the undirected derivative in terms of the first and second tangential differential as
(ÑFp)Ñ(a) = ¯Ñ(Ña,b)FpÑ(a)) + ÑaFpÑ2(a,b)     provided aÎIp .
[ Proof : aÎIp Þ FpÑ(a) = (a¿Ñ)Fp = (¯(a)¿Ñ)Fp = (¯(a)¿Ñ)Fp = FpÑ(¯(a))  .]

Hypercurve
For M=N-1 we have ¯Ñ(ap,b) = (Ðb¯)(ap)   = -np2((ap¿npÑ(b))np + (ap¿np)npÑ(b)) .
Thus ¯Ñ(¯(ap),b) = -np2 (¯(ap)¿npÑ(b))np = -np2 (ap¿npÑ(b))np ;
Thus ¯Ñ(np,b) = -npÑ(b) .     , the normalisation condition on np providing np¿npÑ(b)=0 .

Projection Second Differential (1;3)-tensor     ¯Ñ2
We can take the second tangential differential of ¯, ie. the third tangential differntial of 1 obtaining
¯Ñ2(a,b,c) º Ð¯(c)(¯Ñ(a,b)) - ¯Ñ( Ð¯(c)a,b)) - ¯Ñ(a, Ð¯(c)b)) .

Squared Projection Differential (1;3)-tensor     (¯Ñ)2
(¯Ñ)2(a,b,c) º ¯Ñ(¯Ñ(a,b),c)

(¯Ñ)2×¯ = Ð ×¯     abbreviating (¯Ñ)2(¯Fp,b,a) - (¯Ñ)2(¯Fp,a,b) = Ða(Ðb(¯Fp))-Ðb(Ða(¯Fp))
[ Proof :  Recalling ¯=¯Ñ=0 we have:
ÐaÐb(¯Fp) = Ða( Ð¯(b)(¯Fp) - ¯Ñ(¯Fp,b) ) = ¯( Ð¯(a)( Ð¯(b)(¯Fp) - ¯Ñ(¯Fp,b) )) = ¯=( Ð¯(a) Ð¯(b))(Fp) - ¯( Ð¯(a)(¯Ñ(¯Fp,b)) )
= ¯=( Ð¯(a) Ð¯(b))(Fp) - ¯[ ( Ð¯(a)¯Ñ)(¯Fp,b) +  ¯Ñ( Ð¯(a)(¯Fp),b) + ¯Ñ(¯Fp, Ð¯(a)b) ]
= ¯=( Ð¯(a) Ð¯(b))(Fp) - ¯¯Ñ2(¯Fp,b,a) - ¯¯Ñ( Ð¯(a)(¯Fp),b)
= ¯=( Ð¯(a) Ð¯(b))(Fp) - ¯=(¯Ñ2)(Fp,b,a) - ¯¯Ñ( ( Ð¯(a)¯)Fp)+¯( Ð¯(a)Fp),b)
= ¯=( Ð¯(a) Ð¯(b))(Fp) - ¯=(¯Ñ2)(Fp,b,a) - ¯(¯Ñ)2(Fp,a,b)
Þ (Ða×Ðb)Fp = (¯Ñ)2×(Fp,a,b) by symmetry of ¯Ñ2 and integrability condition Ð¯(a)× Ð¯(b)=0  .]

Shape (<0.2>;1)-multitensor     [Ñ¯]
We define the shape [Ñ¯] of CM to be the undirected tangential 1-derivative of the projector , an abbreviation of
[Ñ¯](ap) º Ñß[CM]p¯Ip(ap) º Ñß ¯Ip(ap) = (Ñb Ðp¯(b) ¯Ip)(ap) = Ñb ¯Ñ(ap,b) = Ñb ¯Ñ(ap,b) = Ñb ¯Ñ(ap,b) .

[Ñ¯](ap) = 0 so [Ñ¯] annihilates scalars.

[Ñ¯](ap) decomposes as [Ñ¯](^Ip(ap)) + [Ñ¯](¯Ip(ap)) and, joy of joys, this corresponds precisely to its grade decomposition into scalar and bivector parts. We can express this immense good fortune operationally with regard to general operands as
[Ñ¯] ^   =   ¯ [Ñ¯]   =   [Ñ¿¯]     [ HS 4-2.35b ] which stems from [ÑÙ¯]^ = [Ñ¿¯]¯ = 0 holding generally [ HS 4-2.37 ] ; and
[Ñ¯]¯   =   ^[Ñ¯]    =   [ÑÙ¯]       [ HS 4-2.35a ] since ^[Ñ¯] = [Ñ¯]-¯[Ñ¯] = [Ñ¯]-[Ñ¯]^ = [Ñ¯]¯ .

For general operands we have   [Ñ¯](a)<k+1> = [Ñ¯](¯Ip(a<k>))    ;     [Ñ¯](a)<k-1> = ¯Ip([Ñ¯](a<k>))     and so
Ñ¯ = Ñ¯ + [Ñ¯]¯ = Ñ¯ + [ÑÙ¯]     [ HS 4-3.7a ] with Ñ¿¯ = Ñ¿¯ . Hence
^Ñ¯ = [Ñ¯]¯ = [ÑÙ¯]¯
[ Proof :  Ñ(¯(ap)) = Ñ(¯(ap)) + ^Ñ(¯(ap)) = Ñ¯(ap) + ^( [Ñ¯](ap) + Ñ¯(apÑ) ) = Ñ¯(ap) + ^([Ñ¯](ap) + Ñ(ap))
= Ñ¯(ap) + ^[Ñ¯](ap) = Ñ¯(ap) + [Ñ¯]¯(ap) = (Ñ+[Ñ¯])¯(ap) .  .]

With tangential operands ¯(ap)=ap understood we can write this as
Ñ = Ñ + [Ñ¯] = Ñ + [ÑÙ¯]       [ HS 4-3.6a ] and   ^Ñ = [Ñ¯] = [ÑÙ¯] .

[ÑÙ¯](a) º ([Ñ¯](a))<2> = [Ñ¯]¯(a) is known as the curl (2;1)-tensor .

Scalar ¯1 [Ñ¯](a) =[Ñ¯](^Ip(a)) = (Ñp¿¯Ip)(a) is expressible as (Ñb[ÑÙ¯](b))¿a where 1-vector
Ñb[ÑÙ¯](b) = º ÑÞ[ÑÙ¯](b) = ÑÞ¿[ÑÙ¯](b) º ÑÞ¿[ÑÙ¯](b)

[¯(Ñ)]   =   ¯Ñ(Ñ)   =   ÑÞ [ÑÙ¯]   =   ÑÞ¿[ÑÙ¯]   =   [Ñ¿¯]¿     [ HS 4-2.20 ]
is known as the spur 1-field of CM and lies outside Ip .

[Ñ¿¯](a) = [¯(Ñ)].a [ HS 4-2.18 ] so the shape (<0,2>;1)-tensor is "recoverable" from the curl (2;1)-tensor as [Ñ¯](a) = [ÑÙ¯](a) + (Ñb[ÑÙ¯](b)).a .

1-vector ¯Ip(b)¿[Ñ¯](a) = ¯Ip(b)¿[ÑÙ¯](a) is normal to CM , ie. ¯Ip ( ¯Ip(b)¿[Ñ¯](a)) = 0 .

[Ñ¯](a1Ùa2Ù...ak) = åi=1k.  (-1)i+1 [ [Ñ¯](ai) Ù ¯Ip(a1Ù..Ùai-1Ùai+1..Ùak) + ¯Ip([Ñ¯](ai) × (a1Ù..Ùai-1Ùai+1..Ùak) ]     [ HS 2.41c ] provides the extension of [Ñ¯] to general multivectors.

[Ñ¯]  = ÑÞ2 ¯Ñ     abbreviating [Ñ¯](ap) = Ñb¯Ñb) expressing the shape as the secondary tangential derivative of ¯Ñ [ HS 4-2.14 ].
[ Proof :  Ðbd ¯Ñ(ap,b) = Ðbd( Ð¯(b)(¯(ap)) -  ¯( Ð¯(b)ap)) = Lime ® 0 e-1[ Ð¯(b+ed)(¯(ap)) -  ¯( Ð¯(b+ed)ap)) - Ð¯(b)(¯(ap)) -  ¯( Ð¯(b)ap))]
=   Ð¯(d)(¯(ap)) -  ¯( Ð¯(d)ap)) =  ( Ð¯(d)¯)(ap) so we have both Ñß ¯(ap) º (Ñ¯)(ap) = Ñb¯Ñ(ap,b) º ÑÞ2¯Ñ(ap,b)     and Ñ¯ß ¯(ap) º [Ñ¯](ap) = Ñb¯Ñ(ap,b) º ÑÞ2¯Ñ(ap,b)
Since ¯Ñ(a,b)=¯Ñ(b,a) the result follows.  .]

The grade-tangency associations of [Ñ¯] immediately provide
[Ñ¯]^ = [Ñ¿¯] = ÑÞ2¿¯Ñ     ;     [Ñ¯]¯ =  [ÑÙ¯] = ÑÞ2Ù¯Ñ       [ HS 4-2.16 ].
[ Proof :  [Ñ¯](a) º (Ñp¯))(a) = Ñb¯Ñp(a,b) so .... ???  .]

Ñ2 = [Ñ¯](Ñ) + ¯Ñ2
[ Proof :  Ñ2(ap) =Ñ(¯Ñ(ap)) = [Ñ¯](Ñ) + ¯Ñ(Ñ(ap)) = [Ñ¯](Ñ)(ap) + ¯Ñ(Ñ(ap)) = [Ñ¯](Ñ)(ap) + ¯ÑÑ(ap)
= [Ñ¯](Ñ)(ap) + ¯¯ÑÑ(ap) = [Ñ¯](Ñ)(ap) + ¯Ñ2(ap) = [Ñ¯](Ñ)(ap) + ¯(Ñ¿Ñ)ap  .]

Hence ÑÙÑ = [Ñ¯](Ñ) = [ÑÙ¯](Ñ) with regard to path-independant functions. [ HS 4-3.10b ] .

We have the following Shape Properties:

1.   [ÑÙ¯]^ = 0   ie. [ÑÙ¯](^Ip(ap)) = 0  " ap       [ HS 4-2.37a ]
2.   [Ñ¿¯]¯ = 0 . ie. [Ñ¿¯](¯Ip(ap)) º (Ñ¿¯)(¯(ap)) = 0   " ap     [ HS 4-2.37b ]
3. ¯[ÑÙ¯] = 0     when acting on 1-vectors ie. ¯[ÑÙ¯](a)=0  " a . [ HS 4-2.30 ]
[ Proof : ¯(a)¿[ÑÙ¯](b) = ¯Ñ(¯(a),b) [ HS 4-2.26 ] Þ ¯( ¯(a)¿[ÑÙ¯](b) ) = ¯¯Ñ(¯(a),b)   = ¯=¯Ñ(a,b) = 0 .
Þ ??? ¯( a¿¯([ÑÙ¯](b)) )=0 ????  .]

^[Ñ¿¯] = 0     follows immediately from ^¯1=0 with regard to 1-vectors.
4. I [Ñ¯] = I ×[Ñ¯] = I ×[ÑÙ¯]     abbreviating Ip×[ÑÙ¯](a) = Ip×[Ñ¯](a) = -[Ñ¯](a)×Ip = -[ÑÙ¯](a)×Ip = -[ÑÙ¯](a)Ip    (4-4.5)
[ Proof :  3 Þ [ÑÙ¯](a)¿Ip = 0 and since [ÑÙ¯](a)ÙIp=0 the result follows from bivector nature of [ÑÙ¯]  .]
Hence [ÑÙ¯](a) = ([Ñ¯](a)×Ip)Ip-1 = -Ip-1([ÑÙ¯](a))
Also  Ip.[ÑÙ¯] = IpÙ[ÑÙ¯] = 0     for 1-vector arguments     [ HS 4-4.3 ] which yields
[ÑÙ¯]2 = 0 .
[ Proof :   ¯Ip([ÑÙ¯](a)) º ¯ [ÑÙ¯](a) = 0 Þ [ÑÙ¯](a)=^([ÑÙ¯](a)) Þ [ÑÙ¯]([ÑÙ¯](a)) = 0 by 1.  .]
5. ¯=[Ñ¯]     º     ¯[Ñ¯]¯     =     0        [ HS 4-2.45 ]
[ Proof : ¯([Ñ¯](¯(a))) = [Ñ¯](^(¯(a))) = [Ñ¯](0) = 0  .]
^=[Ñ¯]     º     ^[Ñ¯]^     =     0
[ Proof : ^[Ñ¯]^ = ^¯[Ñ¯] = 0  .]
6. ¯Ñ(a,b) = ¯(a)×[ÑÙ¯](b) - ¯( a×[ÑÙ¯](b) ) = ¯(a)×[Ñ¯](b) - ¯( a×[Ñ¯](b) )
is crucially important because it expresses ¯Ñ geometrically in terms of [ÑÙ¯] and ¯ .
[ Proof : See [ HS 4-2.33 ].  .]
In particular, it gives ¯Ñ(¯(a),b) = ¯(a)×[ÑÙ¯](b) = ¯(a)×[Ñ¯](b) .
[ Proof : The projected product rule gives ¯(¯(a)[ÑÙ¯](b)) = ¯(a)¯([ÑÙ¯](b)) = 0 and as ¯(¯(a)Ù[ÑÙ¯](b)) = ¯(a)Ù¯([ÑÙ¯](b)) = 0 we must have ¯( ¯(a)×[ÑÙ¯](b) ) = 0  .]
7. [ÑÙ¯](b) = Ip-1¯Ñ(Ip,b) = Ip(ÐbIp-1 = -¯Ñ(Ip,b)Ip-1       expresses [ÑÙ¯] geometrically in terms of ¯Ñ and Ip without further differention and provides perhaps the simplest mental picture of the curl tensor as the bivector which acting geoemtrically on Ip returns -ÐpaIp , the negated rate of change of Ip in direction a. .
[ Proof :  a=Ip in 6 gives ¯Ñ(Ip,b) = Ip×[ÑÙ¯](b) = - [ÑÙ¯](b)Ip by 4.  .]
8. ¯Ñ(a,b)=¯(a)¿[ÑÙ¯](b) - ¯(a¿[ÑÙ¯](b))     [ HS 4-2.27 ]
[ Proof :  a×b<2> = a¿b<2> combined with 5  .]
9. (¯Ñ)2(Fp,a,b) =   ¯( [Ñ¯](b)×(¯(Fp)×[Ñ¯](a)) ) - ¯( Fp×[Ñ¯](a) )×[Ñ¯](b)
[ Proof :  Applying Shape Property 6 twice while exploiting Shape Property 3
¯Ñ(¯Ñ(Fp,a),b) =   ¯(¯Ñ(Fp,a))×[Ñ¯](b) - ¯( ¯Ñ(Fp,a)×[Ñ¯](b) )
= ¯(¯(Fp)×[Ñ¯](a) - ¯( Fp×[Ñ¯](a) ))×[Ñ¯](b)     -     ¯( (¯(Fp)×[Ñ¯](a) - ¯( Fp×[Ñ¯](a) ))×[Ñ¯](b) )
= 0 - ¯2( Fp×[Ñ¯](a) ))×[Ñ¯](b) - ¯( (¯(Fp)×[Ñ¯](a) )×[Ñ¯](b) )   + 0     by the projected bivector commutation rule  .]
If ¯(Fp)=Fp this reduces to
¯Ñ(¯Ñ(Fp,a),b) =   ¯( [Ñ¯](b)×(Fp×[Ñ¯](a)) )     [ HS 5-1.3 ]   which we can abbrviate to
¯Ñ2¯ = - ¯[Ñ¯]×[Ñ¯]×¯ .
10. [Ñ¯](Ip) = ÑpÙIp = ÑpIp     [ HS 4-2.44 ]
12. ¯Ip(b)¿[ÑÙ¯](a) = ¯Ip(a)¿[ÑÙ¯](b)     [ HS 4-2.28 ]
13. ¯ÑIp(a,d) = ¯Ip(b) ¿ [ÑÙ¯](d) - ¯Ip(a¿[ÑÙ¯](d))     [ HS 4-2.27 ]
14. ¯ÑIp([ÑÙ¯](a),d) = - ¯(Ip)([ÑÙ¯](a)×[ÑÙ¯](d))
16. a*(¯IpÑp(d,b)) = b*(¯IpÑp(d,a))
17. ¯IpÑ(d,aÙb) = ¯IpÑ(d,a) Ù ¯Ip(b)   +   ¯Ip(b) Ù ¯IpÑp(d,b)
18. ¯IpÑp(b,¯Ip(a)) = ¯IpÑp(a,¯Ip(b))
19. --- end list

[Ñ¯]  = ÑÞ ¯Ñ enables us to grade extend [Ñ¯] from the grade extension of_prl0g() as
[Ñ¯](aÙb) = Ñc¯Ñ(aÙb,c) = Ñc(¯Ñ(a,c)Ù¯(b)+¯(a)Ù¯Ñ(b,c))     so we immediately have [Ñ¯](^(a)Ù^(b))=0     [ HS 4-2.40c ].  Furthermore
[Ñ¯](¯(aj)Ù¯(b)) = ([Ñ¯](¯(aj))Ù¯(b) + (-1)j¯(aj)Ù([Ñ¯]¯(b))     [ HS 4-2.40a ]
[ Proof :  [Ñ¯](¯(aj)Ù¯(bk)) =  [Ñ¯]¯(ajÙbk) =  [ÑÙ¯](ajÙbk) = ÑcÙ( ¯Ñ(¯(aj),c)Ù¯(bk) + (¯(aj)Ù¯Ñ(¯(bk),c) )
= ÑcÙ( ¯Ñ(¯(aj),c)Ù¯(bk) + (-1)jk¯Ñ(¯(bk),c)Ù¯(aj) ) = (ÑcÙ¯Ñ(¯(aj),c))Ù¯(bk) + (-1)_jk(ÑcÙ¯Ñ(¯(b),c))Ù¯(aj)
= ([ÑÙ¯](¯(aj))Ù¯(bk) + (-1)jk([ÑÙ¯]¯(b))Ù¯(aj) = ([ÑÙ¯](¯(aj))Ù¯(bk) + (-1)jk+j(k+1)¯(aj)Ù[ÑÙ¯]¯(bk)
= ([Ñ¯](¯(aj))Ù¯(bk) + (-1)j¯(aj)Ù([Ñ¯]¯(bk))  .]

In particular, [Ñ¯]¯(aÙb)  = [Ñ¯]¯(a)Ù¯(b)-[Ñ¯]¯(b)Ù¯(a) .

[Ñ¯](^(aj)Ùb) =   [Ñ¯](^(aj)Ù¯(b)) =   [Ñ¯](^(aj))Ù¯(b) + (-1)j¯Ñ(^(aj))Ù(Ñ¿b)     [ HS 4-2.40b ]
[ Proof :  [Ñ¯](^(aj)Ù¯(bk)) = Ñc¯Ñ(^(aj)Ù¯(bk) ,c) = Ñc(¯Ñ(^(aj),c)Ù¯(bk)) = Ñc¿(¯Ñ(^(aj),c)Ù¯(bk)) + (ÑcÙ¯Ñ(^(aj),c))Ù¯(bk)
= (Ñc.¯Ñ(^(aj),c))Ù¯(bk) + (-1)j ¯Ñ(^(aj),c)(Ñc.¯(bk)) + [ÑÙ¯](^(aj))Ù¯(bk)     by the expanded inner product rule
= ([Ñ¯]^(aj) + (-1)j ¯Ñ(^(aj),c)(Ñc.¯(bk)) + 0  .]

====

The primary 1-differential of the extended curl (2;1)-tensor is
[ÑÙ¯]Ñ(a,b) º Ð¯(b) [ÑÙ¯](ap) - [ÑÙ¯]( Ð¯(b)(ap))  .

Hypercurve
For M=N-1 we have [ÑÙ¯](a) = np2 npnpÑ(a) = np2 npÙnpÑ(a)
[ Proof :  ÐbiN-1p = Ðb(npi) = (Ðbnp)i = npÑ(b)i so Shape Property 7 gives
[ÑÙ¯](a) = iN-1p-1( Ð¯(a)iN-1p) = i-1np-1npÑ(a)i = np2 _psiinvd(npnpÑ(a)) = np2 _psiinvd(npÙnpÑ(a)) = np2 npÙnpÑ(a)     since i commutes with all bivectors.  .]

[¯(Ñ)] = np2 np(Ñp¿np) = np2 np(Ñp¿np)
[ Proof :   Ña[ÑÙ¯](a) = -np2 Ña[npÑ(a)np] = -np2 (ÑanpÑ(a))np = -np2 (ÑaÐanp)np = -np2 (Ñpnp)np = -np2 np(Ñp¿np)  .]
Thus [Ñ¿¯](a) = [¯(Ñ)].a = -np2 (Ñ¿np)(ap¿np)

Squape 1-multitensor     [Ñ¯]2

The shape tensor [Ñ¯] raises the grade of ¯Ip(a) while lowering the grade of ^Ip(a), preserving neither grade nor tangency (containment within Ip) but the squared shape or squape 1-tensor [Ñ¯]2 preserves both with [Ñ¯]2  = ^=([Ñ¯]2)  + ¯=([Ñ¯]2) .
[ Proof :   Follows from ¯=[Ñ¯] = ^=[Ñ¯] = 0 and [Ñ¯]^ = ¯[Ñ¯] since
[Ñ¯]2 = (¯+^)[Ñ¯](¯+^)[Ñ¯](¯+^) = ^[Ñ¯]¯[Ñ¯]^  + ¯[Ñ¯]^[Ñ¯]¯ = ^[Ñ¯]2^2  + ¯[Ñ¯]2¯  .]

We also have [Ñ¯]2^ = ^[Ñ¯]2   = [Ñ¯]¯[Ñ¯] º [Ñ¯]=¯ . [ HS 4-2.46b ]

The squape tensor decomposes as [Ñ¯]<2> + [Ñ¯]<1> + [Ñ¯]<0>

Ricci 1-tensor       ([Ñ¯]2¯)
The intrinsic squape 1-multitensor is the projection of the squape ¯[Ñ¯]2   = [Ñ¯]2¯ =   ¯=([Ñ¯]2) , acting entirely upon and within Ip .
We refer to the instrinsic squape acting only on 1-vectors as the Ricci 1-tensor.

We will show eventually that
[Ñ¯]2 ¯   =   (ÑÙÑ) ¯   =   (ÑÙÑ) ×¯     [ HS 5-1.28 5-1.29 ] abbreviating
[Ñ¯]2(¯(ap)) = (ÑÙÑ)(¯(ap)) = (ÑÙÑ)×(¯(ap))     " ap .

[Ñ¯]2¯(aÙb) = [Ñ¯]2(a) Ùb + 2¯( [Ñ¯]Ñ(b)×[Ñ¯]Ñ(a) ) + aÙ [Ñ¯]2(b)     [ HS 4-2.48 ]
[ Proof :  ... ???  .]

¯[Ñ¯]2 = -Ñ[ÑÙ¯] = -Ñ¿[ÑÙ¯]     abbreviating ¯[Ñ¯]2(b) = -Ñp[ÑÙ¯](b) = -Ñp¿[ÑÙ¯](b) . [ HS 5-1.19 ]
[ Proof : Ña   ¯[Ñ¯]×(a,b) = -ÑaÐa[ÑÙ¯](b) = -Ñp[ÑÙ¯](b)     by Curvature Identity 2   below  .]

[ Proof : ???  .]

Curvature 2-tensor     [Ñ¯]×
Recall that the commutator product of two bivectors is itself a bivector.
If Ip satisfies the integrability condition then the antisymmetric full curvature 2-tensor
cpaÙb º [Ñ¯]×(a,b) º [Ñ¯](a)×[Ñ¯](b) = [ÑÙ¯](a)×[ÑÙ¯](b)

satisfies [Ñ¯]×(a,b) = ½([ÑÙ¯]Ñ(a,b) - [ÑÙ¯]Ñ(b,a))     [ HS 4-4.17 ] where [ÑÙ¯]Ñ(ap,b) º Ð¯(b)([ÑÙ¯](ap)) - [ÑÙ¯]( Ð¯(b)ap) .
[ Proof :  ?????    See HS  .]
We can write this with regard to 1-vector arguments as
[Ñ¯]×   =   [ÑÙ¯]×   =   ([ÑÙ¯]Ñ)×     º     [ÑÙ¯]Ñ×
Thus we can categorise some second dervivative properties of an M-curve geometrically from the first derivative shape.

Curvature differential (2;3)-tensor     [Ñ¯]×Ñ

This enables us to express the primary differential of the curvature:
[Ñ¯]×Ñ(a,b,c) º ([Ñ¯]×(a,b))Ñ(c) º Ðßc([Ñ¯]×(a,b)) º (Ðc([Ñ¯]×))(a,b)
as ½([ÑÙ¯]Ñ(a,b,c)-[ÑÙ¯]Ñ2(b,a,c)) = ½([ÑÙ¯]Ñ(a,b,c)-[ÑÙ¯]Ñ2(b,c,a)) .
Cyclically permuting the a,b,c and summing we obtain the generalised Bianchi identity
S¬abc ([ÑÙ¯](a)×[ÑÙ¯](b))Ñ(c)  = 0
S¬abc ([Ñ¯]×)Ñ(a,b,c)  º  S¬abc Ðßc([Ñ¯]×)(a,b)   =   0     [ HS 5-1.39 ] which we can also write as ([Ñ¯]×)Ñ®¿ = 0 . The curvature thus has vanishing exterior 1-differential.

Setting a=[Ñ¯](a) in Projection Property 6 [ [ HS 4-2.33 ]] gives
¯Ñ([Ñ¯](a),b) = ¯([Ñ¯](a))×[Ñ¯](b) - ¯([Ñ¯](a)×[Ñ¯](b)) = - ¯([Ñ¯](a)×[Ñ¯](b)) º  - ¯([Ñ¯]×(a,b))
which we can abbreviate (with 1-vector arguments understood) to
¯ [Ñ¯]×   =   ¯ [ÑÙ¯]×   =   -¯Ñ [ÑÙ¯]
ie.   1Ñ2 [ÑÙ¯] = - 1Ñ [ÑÙ¯]×  = - 1Ñ [Ñ¯]× .

Since the curvature 2-tensor is skewsymmetric and bilinear in its two nonprimary arquments it defines a bivector-valued curvature (*1)-multitensor] of a single bivector argument
[Ñ¯]×(aÙb) º [Ñ¯]×(a,b) . It is natural to extend this to a multivector argument via [Ñ¯]×(a) = [Ñ¯]×(aÙ1) = [Ñ¯]×(a,1) = 0 . º [Ñ¯]×(a,b) . It is natural to extend this to a multivector argument via [Ñ¯]×(a)=[Ñ¯]×(a)=0

Intrinsic curvature 2-tensor       ¯[Ñ¯]×

The intrinsic curvature of an M-curve can be defined in a number of ultimately equivalent ways. We will here regard it primarily as the projection of the full 2-curvature
¯( [Ñ¯](a)×[Ñ¯](b)) = ¯([ÑÙ¯](a)×[ÑÙ¯](b) )     and denote it by
¯[Ñ¯]×     =     ¯[ÑÙ¯]×
with ¯[Ñ¯]×(a,b) º ¯( [Ñ¯](a)×[Ñ¯](b) ) .
It is bivector-valued and bilinear in a,b and is thus an antisymmetric 2-tensor, ie. a 2-form. The remaining, rejected, component of the full curvature existing outside Ip is known as the extrinsic curvature 2-tensor.

Riemann curvature 2-tensor       ¯=[Ñ¯]×

The Riemann curvature of an M-curve is the intrinsic curvature restricted to the M-curve
¯=[Ñ¯]×(a,b) º ¯[Ñ¯]×(¯(a),¯(b)) = ¯([Ñ¯](¯a)×[Ñ¯](¯ |b)) = ¯([ÑÙ¯](a)×[ÑÙ¯](b)) = ¯[Ñ¯]×(¯(aÙb))
so that nonprimary arguments outside Ip are mapped to 0.

(¯[Ñ¯]×)Ñ®¿(a,b,c) = S¬abc ¯([Ñ¯]×^(aÙb))×[Ñ¯]¯(c)     with the immediate consequence (¯=[Ñ¯]×)Ñ®¿(a,b,c) = 0 , ie. the Riemann curvature has vanishing exterior 1-codifferential, which is the traditional second Bianchi identity.
[ Proof :  Ðßc ¯[Ñ¯]×(aÙb) º (Ðc ¯[Ñ¯]×)(aÙb) = ¯Ñ , c)) + ¯( Ðßc[Ñ¯]×(aÙb))
Þ S¬abc (Ðc ¯[Ñ¯]×)(aÙb) = S¬abc ¯([Ñ¯]×(aÙb))×[Ñ¯](c) .     Hence
S¬abc (Ðc ¯[Ñ¯]×)(aÙb) = S¬abc ¯Ñ , c)  + ¯( S¬abc Ðßc[Ñ¯]×(aÙb)) = S¬abc ¯Ñ , c)  + ¯( 0)
= S¬abc (¯([Ñ¯]×(aÙb))×[Ñ¯](c)   - ¯([Ñ¯]×(aÙb) ×[Ñ¯](c) )       by Shape Proprety 6
= S¬abc ¯([Ñ¯]×(aÙb))×[Ñ¯](c)   - ¯(S¬abc (([Ñ¯](a)×[Ñ¯](b))×[Ñ¯](c) ) = S¬abc ¯([Ñ¯]×(aÙb))×[Ñ¯](c)       by Jacobi Identity .  .]

Hence S¬abc (Ðc ¯[Ñ¯]×)(aÙb) = S¬abc ¯( Ð¯(c) ¯[Ñ¯]×)(aÙb) ) = S¬abc ¯( ¯([Ñ¯]×(aÙb))×[Ñ¯](¯(c)) )
= S¬abc ¯( [Ñ¯]×(aÙb))×[Ñ¯](¯(c)) )        by the projected bivector commutation rule
Replacing (aÙb) with ¯(aÙb) gives a vanishing result by the generalised Bianchi identity.  .]

We can express the Riemann curvature  ¯( [Ñ¯](a)×[Ñ¯](b)) solely in terms of the intrinsic squape multitensor via
¯[Ñ¯]×(aÙb) = ½( [Ñ¯]2(a)Ùb + aÙ[Ñ¯]2(b) - [Ñ¯]2(aÙb) )     " a,b Î Ip     [ HS 4-2.48 ].

Since [Ñ¯]2 preserves grade we must have
Ña¿[Ñ¯]2(aÙb) = -2Ña¿(¯[Ñ¯]×(a,b)) = -2¯(Ña)¿([Ñ¯]×(a,b)) = -2Ña¿([Ñ¯]×(a,b))     and similarly
Ña¿[Ñ¯]2(aÙb) = -2Ña¿(¯[Ñ¯]×(a,b)) = [Ñ¯]2(b) .

The Riemann curvature is protractionless
ÑaÙ¯[Ñ¯]×(¯(aÙb))   =   0     [ HS 5-1.11 ] .

We also have
ÑßÙ¯[Ñ¯]×¯   =   0     [ HS 5-1.13a ]
[ Proof : See [ HS p191 ]  .]

There are various alternate and equivalent definitions for the intrinsic 2-curvature one can adopt based on the following Curvature Identites.

1. ¯[Ñ¯]×   =   -¯Ñ[ÑÙ¯]     abbreviating ¯( [ÑÙ¯]×(a,b) ) = -¯Ñ([ÑÙ¯](a),b) = ¯Ñ([ÑÙ¯](b),a)     [ HS 5-1.5b ]
[ Proof :  Substituting ap=[ÑÙ¯](a) into  Shape Property 6   and exploiting Shape Property 3 :
¯Ñ([ÑÙ¯](a),b) = ¯([ÑÙ¯](a))×[ÑÙ¯](b) - ¯( [ÑÙ¯](a)×[ÑÙ¯](b) ) = 0 -¯( [Ñ¯](a)×[Ñ¯](b) )  .]
2. ¯[Ñ¯]×   =   [ÑÙ¯]Ñ = (Ð [ÑÙ¯])     abbreviating ¯[Ñ¯]×(a,b) = (Ðb[ÑÙ¯])(a) º Ðb([ÑÙ¯](a))-[ÑÙ¯](Ðba)     [ HS 5-1.5c ]
[ Proof :  (Ðb[ÑÙ¯])(a) = Ð¯(b)(¯([ÑÙ¯](a))) - ¯Ñ([ÑÙ¯](a),b) = Ð¯(b)0 - ¯Ñ([ÑÙ¯](a),b) and result follows from 1.  .]
3. ¯[Ñ¯]××¯    =   -2(¯Ñ)2×¯     abbreviating ¯([Ñ¯]×(a,b))×¯(Fp) =  2(¯Ñ(¯Ñ(¯Fp,a),b)-¯Ñ(¯Ñ(¯Fp,b,a))     [ HS 5-1.4 ]
[ Proof : Shape Property 9 gives
¯Ñ(¯Ñ(¯Fp,a),b) - ¯Ñ(¯Ñ(¯Fp,b),a) =   ¯( [Ñ¯](b)×(¯(Fp)×[Ñ¯](a)) ) -   ¯( [Ñ¯](a)×(¯(Fp)×[Ñ¯](b)) )
=   ¯( ¯(Fp)×([Ñ¯](b)×[Ñ¯](a)))     by Jacobi identity
=    ¯(Fp)×¯([Ñ¯](b)×[Ñ¯](a)))     by projected bivector commutation rule  .]
4. ¯[Ñ¯]××¯   =   -2Ð×¯     abbreviating ¯([Ñ¯]×(a,b))×¯(Fp) = - 2(Ða×Ðb)¯(Fp)     [ HS 5-1.4 ]
[ Proof :   Follows immediately from (¯Ñ)2×¯ = Ð ×¯ and the previous identity.  .]
cpab)×Fp º -(Ða×Ðb)Fp with FpÎIp understood is the traditional approach in much of the literature. For functions satisfying the integrability condition, directed derivatives commute but directed coderivatives may not. In restricting Ñp to act "within and withon" Ip, we "break" the commutivity of directed derivatives, but the commutativity "survives" both in the form of the antisymmetry of the Riemann 2-curvature so defined and also in the Ricci Identity
ÑÙÑÙ b2p = 0 for any bivector field b2p
This enables us to also regard the Riemann curvature as a pointer-dependant bivector-specific 1-tensor
cpaÙbc) º cpaÙb×c which we shall see later has the geometric interpretation of the 1-vector change in c if it is "paralell transported" around (the projection into CM of) a tiny planar loop through p in aÙb ,  divided by the content of that loop.

It is easy to fail to appreciate its true significance of this. In consequence of the integrability condition of Ip, it is possible to evaluate (Ða×Ðb)Fp without having to differentiate Fp by applying a particular linear function independant of Fp to Fp. This is why ÑÙÑ is essentially geometric, with no differentiating component.

5. ¯[Ñ¯]2   =   ÑÞ ¯[Ñ¯]×   =   ÑÞ ¿ ¯[Ñ¯]×   =   ÑÞ ¿ ¯[Ñ¯]×     provides the more traditional definition of the Rici 1-tensor as the contraction of the Riemann curvature 2-tensor . [ HS 5-1.17 ]
[ Proof :  ¯[Ñ¯]([Ñ¯](b)) = [Ñ¯]([Ñ¯](¯b) ) = [Ñ¯]([ÑÙ¯](b)) = (ÑaÐpa¯)([ÑÙ¯](b)) = Ña¯Ñ([ÑÙ¯](b),a) = Ña¿¯Ñ([ÑÙ¯](b),a) combined with Curvature Identity 1 . Finally, ÑÞ¿¦p(ap) = ¯(ÑÞ¿¦p(ap)) = ÑÞ¿¦p(ap) for any 1-vector-valued ¦p(ap) .  .]
6. ¯=[Ñ¯]×(aÙb)   =   (b¿Ñp) [ÑÙ¯](a)   =   Ðb[ÑÙ¯](a)     [ HS 5-1.5c ]
expresses ¯=[Ñ¯]×(aÙb) as the b-directed primary coderivative of [ÑÙ¯](a), the geometric operator that sends Ip to ÐpaIp .
[ Proof :  ???  .]
7. ¯=[Ñ¯]×(aÙb)   =   ¯ÑIp([ÑÙ¯](b),a)   =   ÑuÙ¯ÑaÑv ¯Ñ(¯Ñ(v,b),a)     [ HS 5-1.5b ] .
[ Proof :  ???  .]
8. ¯=[Ñ¯]×(aÙb)   =    (ÑuÙÑv)(¯Ña(u)¿¯Ñb(v))     [ HS 5-1.5d ] .
[ Proof :  ???  .]
9. ÑßÙ¯=[Ñ¯]×    =   0 abbreviating (ÑÙ¯=[Ñ¯]×)(aÙb)=0     [ HS 5-1.13a ] .
[ Proof :  ???  .]
10. ¯=[Ñ¯]×(ÑÜÙb)   =   ÑÙ([Ñ¯]2¯)(b)     [ HS 5-1.22 ].
[ Proof : ¯=[Ñ¯]×Ñ(ÑÙb) º ¯(¯=[Ñ¯]×Ñ(ÑÙb)) = ¯((ÑÞ¿Ñ)¯=[Ñ¯]×(aÙb))     by the tangential gradifying substition rule
= ¯( ÑÞ¿(ÑÙ¯=[Ñ¯]×(aÙb) + ÑÙ(ÑÞ.¯=[Ñ¯]×(aÙb)) )     via a¿(bÙc)=(a.b)c-bÙ(a.c)
= ¯( Ñ¿0 + ÑÙ([Ñ¯]2¯)(b)) ) º ÑÙ([Ñ¯]2¯)(b)     by the Bianchi identity.  .]

We thus have ÑßÙ[Ñ¯]×(aÙb) º (ÑÙ[Ñ¯]×¯)(aÙb) = ÑaÙ¯ÑIp([ÑÙ¯](b),a) = 0     which we can write as the Bianchi identity
ÑßÙ¯=[Ñ¯]× = 0     [ HS 5-1.13a ].
[ Proof :   (Ðc¯=[Ñ¯]×)(aÙb) = Ðc( Ð¯(v)Ù The gradifying substitutuin rule gives _..... ?????  .]

Hypercurve
For M=N-1 we have [Ñ¯]×(a,b) = ¯[Ñ¯]×(a,b) = - npÑ(aÙb) .
[ Proof :  [Ñ¯](a)×[Ñ¯](b) = [ÑÙ¯](a)×[ÑÙ¯](b) = np2 (npnpÑ(a))×(npnpÑ(b)) = -np4 (npÑ(a)×npÑ(b)) = - npÑ(a)ÙnpÑ(b) º - npÑ(aÙb)  .]
Hence [Ñ¯]2¯(b) = Ña¿¯[Ñ¯]×(a,b) = -Ña¿(npÑ(a)ÙnpÑ(b)) = ???

Scalar Curvature (0;1)-tensor         R = ÑÞ [Ñ¯]2¯

The scalar curvature (aka. total curvature) is traditionally presented as the second contraction of the Rieman curvature     R º (ÑÞ¿)2 ¯[ÑÙ¯]×     =   ÑÞ2 ¯[ÑÙ¯]×     =   ÑÞ [Ñ¯]2¯ ;
but is also, perhaps more fundamentally, the tangential point divergence of the spur     Ñ¿[¯(Ñ)]     [ HS 5-1.21 ] .

Ñp¿[Ñ¯]2¯(a) =  Ñp¿[Ñ¯]2¯(a) = Ñp¿[Ñ¯]2¯(a)

2 [Ñ¯]2Ñ(Ñ)   =   ÑR     [ HS 5-1.23 ]
[ Proof : Contracting the contracted Bianchi identity yields
Ñb¿((ÑÙ([Ñ¯]2¯))(b)) = Ñb¿(¯=[Ñ¯]×Ñ(ÑÙb)) Þ (Ñb¿Ñ)([Ñ¯]2¯)Ñ(b) - Ñ(Ñb¿([Ñ¯]2¯)Ñ(b)) = - Ñb¿(¯=[Ñ¯]×Ñ(bÙÑ))
Þ ([Ñ¯]2¯)Ñ(Ñ)) - Ñ(RÑ) = -([Ñ¯]2¯)Ñ(Ñ))     Þ 2([Ñ¯]2¯)Ñ(Ñ)) = Ñ(RÑ) .  .]

Einstein 1-tensor         (1-½ÑÞ)[Ñ¯]2¯
2([Ñ¯]2¯)Ñ(Ñ)) = Ñ(RÑ)  means that the symmetric (self-adjoint) Einstein 1-tensor
(1-½ÑÞ)[Ñ¯]2¯(a)   =   ([Ñ¯]2¯)(a) - ½Ra   =   [Ñ¯]2¯(a) - (Ñ¿[¯(Ñ)])a
has (1-½ÑÞ)[Ñ¯]2¯Ñ(Ñ) = ¯(1-½ÑÞ)[Ñ¯]2¯Ñ(Ñ) = 0 and so is zero (point) codivergent, ie.
[Ñ¿(1-½ÑÞ)[Ñ¯]2¯](a)   =   [Ñ¿(1-½ÑÞ)[Ñ¯]2¯](a)   =   0     " a Î Ip . [ HS 5-1.23 ]
It's (directional) cocontraction is Ña¿((1-½ÑÞ)[Ñ¯]2¯(a)) = Ña¿((1-½ÑÞ)[Ñ¯]2¯(a)) = ½(2-M)R .

(1-½Ña)(ÑpÙÑp) = 0

The Coordinate based approach
By defining an M-curve by means of Ip and ¯ we have presented the standard tensors of manifold calculus and verified the vanishing point-divergence of the Einstein tensor all without refernce to coordinates other than to simplify the occassional proof.
We will now run through the standard noncovariant coordinate coordinate representations of these entities since from a programmers' perspective these may be relevant. We will touch on the somewhat arcane metric-focussed covariant coordinate representations in the section on General Relativity.

M-field Ip can be expressed in UN via NCM scalar fields iij..mp = eij...m¿Ip though the normalisation condition renders one such field redundnat apart from sign. In the mapspace ¦Ñ-1(Ip) is everywhere the map pseudoscalar.
In UN the projector 1-tensor ¯ can be expressed with regard to a universal basis as N2 sclar fields although with regard to an extended tangential frame at p it can be regarded as an identity matrix having the last N-M terms in the lead diagonal zeroed.

The Riemann curvature tensor cpaÙb, as a bivector valued function of bivectors, constarined within and upon Ip requires MC22 = 1/4M2(M-1)2 scalar fields in theory although the Bianchi symmetries reduce the generality considerably. At a fixed given p Î CM,  mapping bivectors in Ip to bivectors in Ip induces the obvious Cx(aÙb) = ¦-Ñp(cp¦Ñp(aÙb)) 2-tensor acting in the mapspace which we can represent with 1/4M2(M-1)2 scalar fields Cklij º ekl¿Cx(eij)

The alternate view of the Riemann curvature as a bivector specific 1-tensor , or a (1;3)-tensor antisymmetric in two of its arguments, provides the alternate coordinate form Ckijl º ek¿(Cx(eij)×el)

cpaÙb and Cx(aÙb) are alternate representations of the same "thing", although cp defined over subspace Ip of UN can more readily be "extended" out of CM.     We can contract the curvature tensor to obtain the Ricci 1-tensor)
cpb º Ña¿cpaÙb = åi=14 ei¿cpeiÙb = åi=14 ei¿ (( Ð¯(b)wei) - ( Ð¯(ei)wb) + (wb×wei))    , a point-dependant symmetric 1-vector-valued function of 1-vectors . Note that the divergence is with respect to direction rather than a point and is computable entirely "at" x.
The mapspace representation of the Ricci 1-tensor consists of M 2 scalars Ckj º ek¿ Cx(ei)   = åj=1M Ckjij or with Cij º åk=1M Ckijk which itself contracts to the scalar curvature R = åi=14 Cii .
The mapspace Einstein tensor is given by M2 scalar fields   Gkj º Ckj - ½R .

Streamline Coordinates

Introduction
The following is a more traditional approach to manifold calculus which we include for completeness. This section can safely be skipped by readers interested only in multivector methods.

Suppose we have a 1-field vp defined over M-curve CM such that vp Î Ip . It follows from a mathematical result concerning the unique solvability of ordinary differential equations that, provided vp is sufficiently smooth and is nowhere 0-valued, for every p0ÎCM there is a unique "tangent matcher" path in UN for vp :
Ci p = { p(t) : t Î [t0,t1] }  [ where t0<0<t1 ] with p(0)=p0 ; p(t) Î CM and  (/t)p(t) = vp(t) " t Î [t0,t1] . ( p(0)=p0 ensures that the parameterisation is unique up to choice of domain range [t0,t1])
Such paths are called integral curves or streamlines for vp. We will use both terms interchangeably here. Streamlines cannot intercept eachother (other than at points with vp=0 which we have disallowed) . Since every pÎCM has such a curve, the set of all integral curves for each vp (known as a congruence for vp) "fills" and effectively defines CM.
It is important to note that the 1-curve streamlines of field vp are independant of the magnitude of vp, (given vp2 ¹ 0) since it is the unit tangent that defines a 1-curve.

Suppose now we have M not-necessarily mutually orthogonal unit 1-fields ui p defined over M-curve CM satisfying
u1(p)Ùu2(p)Ù..uM(p) = apIp     " p Î CM where scalar field ap > 0 " p .
An obvious example is ui p=¦Ñx(ei)= hip  where CM=¦(Map) is a point embedding specification for CM.

Given a general multivector-valued function F(p) defined over CM , ui p generates at each pÎCM a specific "streamlined" derivative of F which is the tangential derivative within the 1-curve ui p streamline. We will write this as
dli(F(p)) º  F(pi(li))/li . We will sometime ommit the brackets and write dliF(p).
If we so differentiate the identity map F(p)=1(p)=p we recover ui p and, indeed, one can think of 1-field ui p as being the streamlined derivative operator field dlip . We will accordingly use   the symbol Ð¯(ui p) interchangeably with dli .

We introduce the notation  Dli¬i p to indicate the point pi(Dli) Î CM reached by moving from p by arclength Dli along the li (ie. the ui p) streamline through p.
Dli¬i p º òCi p [0,Dli] 1 dp = ò0Dli pi'(li) dli
We can extend  Dli¬i  to act on general multivector fields by  Dli¬i F(p) º F( Dli¬i p) .
This defintion is integrative, good for all Dli within the neighbourhood of ¦ applicability. For small Dli=e we can apply Taylor's Theorem to the streamline li-parameterisation and obtain F( e¬i p) = eedliF(p) and in particular [  Set F(p)=p ] we have  e¬i p = eedli(p) which we can write as
e¬i  = eedli .

We might be tempted to try to parameterise CM using y:Map®CM defined by
y(Dl1, Dl2, ..Dlm, q0) º  Dlm¬M  Dlm-1¬M-1 ... Dl2¬ Dl1¬q0 =( )=  Dlm¬M ( Dlm-1¬M-1 (... Dl2¬( Dl1¬q0))..) .
[   That is, starting from a fixed reference point q0ÎCM we follow the v1(p) stream line for arclength Dl1 to a new point  Dl1¬q0 from which we follow the v2(p) streamline for arc length Dl2 to  Dl2¬ Dl1¬p0 and so forth. After M such steps we arrive a final point qMÎCM which we can associate with parameters (Dl1,..,Dlm). ]
Any M scalars (sufficiently small that we stay within the neighbourhood) interpreted in this way define a point in CM, but not necessarily uniquely. If it is possible to reach the same point by more than one such route, then the mapping is noninvertible and does not provide a true coordinate system for CM.
Only if the dli (and hence the  Dli¬i ) everywhere commute with eachother do the li provide a true coordinate system.
[ Proof : ...  .]

( e¬i  ×  e¬j ) is a 1-field defined over CM returning a UN 1-vector ( e¬i  ×  e¬j )(p) = (eedli × eedlj)(p) = ... = e2(dli×dlj)(p)   + O(e3) .
Taking e®0 we can visualise  (dli×dlj)(p) º [ ui p,ujp ](p) as the 1-vector difference at p between following i and then j as opposed to j and then i streamlines by small arclength e.

Lie Derivative
We make very little use of Lie derivatives in this work. They are described here only for completeness. This section can safely be skipped by readers interested only in multivector methods.

Lie Bracket

The Lie Bracket is a highly general mathematical construct. With regard to 1-fields in an M-curve we define it by
[ui p,ujp](v) º  dli(dlj[v]) -  dlj(dli[v]) = 2(dli×dlj)v
which we can write as [ui p,ujp] = dliujp - dljui p .
The use of × above is semantically legitimate if we extend the definition of commutator multivector product × to operators in the obvious appropriate way.

Lie Drag

Any one of our ui p generates a congruence for CM so let us fix on ui p as a "preferred" congruence with associated dervivative field dli . Consider two points p0 and p0 in CM with p0= Dli¬i p0 for small Dli>0. Clearly p0= -Dli¬i p0 .
For j¹i, the specific ujp streamline through p0 (i¹1) at p with associated directed derivative dlj generates a distinct path pj*(lj) =  -Dli¬i pj(lj) passing through point p0 =  -Dli¬i p0 known as the Lie drag of ujp . [  Here * indicates "new" or "modifed" rather than a multivector dual ] . This path need not be the streamline at p0 of any of our uk p 1-fields and, in particular, it need not be the lj streamline there .
It is  however, a streamline having at p0 a tangent vector written (iÝ-Dli(ujp))(p0) fully within Ip0 and an associated streamlined differential operator dlj* = (iÝ-Dli(dlj))(p0)  .
Though dlj* is not one of the dlk, it is not general. Because of its construction it commutes with dli (ie. dlj*(dli(F))) = dli(dlj*(F))) . [ Proof : ....  .]
We refer to     (iÝ-Dli(ujp))(p0)     ( or (iÝ-Dli(dlj))(p0) ) as the Lie drag of ujp ( or dlj ) from p0 to p0=  -Dli¬i p0 .
We define the Lie drag of a scalar field f(p) defined over CM by (iÝ-Dli(f))(p) º f( -Dli¬i p) so that (iÝe(f))(p) = eedlif .                                              7

Lie Derivative
The 1-vector Lie derivative of a scalar field is simply £dlif(p) = dlif(p) .

Since the Lie drag of ujp (or dlj) from p0 to p0 can be meaningfully subtracted from ui p (or dli) at p0 . The Lie dervivative at p0 of ujp with regard to 1-field ui p is an operator defined (by its action on a general 1-field F) as
( £dli(dlj))F(p) º LimDli ® 0 [ ( (iÝ-Dli(dlj))(p0)(F(p)) - dlj(F(p)) )  / Dli ] ïp0 = LimDli ® 0 [2(dli×dlj)F  ïp0 ]
[ Proof : dlj*F ïp0 = dlj*F ïp0 - Dli(dli(dlj*(F))))ïp0 + O(Dli2) = dlj*F ïp0 + Dli(dli(dlj(F))ïp0 + O(Dli2) - Dli(dlj*(dli(F))ïp0 + O(Dli2)
Hence LimDli ® 0 [ ( (iÝ-Dli(dlj))(p0) - dlj )F  / Dli ] ïp0 = LimDli ® 0 [((dlj* - dlj )F  ïp0 ) / Dli ] = LimDli ® 0 [(dli(dlj(F)) - dlj*(dli(F)) ) ïp0 ] which provides the result assuming dlj* can be safely replaced in the limit by dlj  .]

Omitting the brackets , we thus have the 1-vector operator identity
£ui pujp = £dlidlj = 2 dli×dlj = [ ui p,ujp ] .

Covariant Frame
Let us now identify the spaces UN and VN and adopt fixed frames {ei} and {ei} in both. With regard to ¿ within VN , {ei} is not a reciprocal frame for {ei}. Rather, we need the point-dependant (nonorthormal) covariant frame {e i º gx-1(ei)} which itself has ¿ reciprocal frame {ei º gx(ei)}
[ Proof :   e i¿ej = gx(gx-1(ei))ej = ei¿ej     Also e i¿ej = gx-1(ei)¿gx(ej) = gx(gx-1(ei))¿ej = ei¿ej  .]
For brevity we ommit an x subscript from ei , e i , and ¿ . The underline serves to remind us of the point-dependance.

Covectors

Any "inner product"  ¿ is a (0;2)-tensor (a potentially point-dependant bilinear scalar-valued function of two 1-vectors in VM) . For any 1-field ( (1;0)-tensor) vp , ¿ induces a scalar-valued directional function ( a (0;1)-tensor)     vp(a) º a¿vp     known as a covector field. We can recover vp at a given p from its covector as vp = Ñavp(a) . A vector and its covector are essentially alternate representations of the same "thing". If we "feed" vp "to" its covector we get "squared magnitude" 0-tensor vp(vp) = vp¿vp
For ¿ in ÂN , if 1-vector vp is regarded as a 1×N matrix "column vector" , then its covector is the N×1 matrix "row vector" (or transpose) vpT . For ¿ in Âp,q , we take the transpose of the conjugate vp .
With regard to ¿, we will here define the 1-covector of vp = åi=1N  viei as the 1-vector vp£ º  åi=1N viei so that vp£ ¿+ a =    vp¿a .
In a Euclidean space, vp£ = vp .
(1;0)-tensor vp has "contravariant" coordinate representation åiviei where vi º ei¿vp .
(0;1)-tensor vp has "covariant" coordinate representation åivie i where vi º vp(ei) = ei¿vp and e i is the 1-covector of ei which we associate with the 1-vector e i.

Parallel transport

Suppose we were to drag heavy iron girders around the (assumed perfectly spherical) Earth for a bet. We might start at the North Pole, with two girders girder (assumed straight and with ends clearly distinguished) lying "horizontal" on the ground ; perpendicularto a "vertical" flagpole we assume to mark the Pole and parallel to eachother.
Since the globe is spherical, the girder is actually balanced tangentially on a single point but the radius is so vast that the ground is for all girder-related purposes locally flat. Both gripping one end we start to drag the girder along a meridian, always pulling due South, that is, allowing no rotation in the local "horizontal" tangent plane.  When we reach the equator, or girder has retained its length and is still facing South (local coordinates) but, speaking 3-dimensionally, it is now parallel to the flagpole we leftbehind. If we then push the second girder sideways from the North Pole without rotation the girder maintains an East-West alignment. Once we reach the equator (at a different point to previously) we then drag the girder "lengthwards" 1/4 of the way round the equator to join the first girder, to which it is now perpendicular.

Moving a 1-vector along a curve while keeping it "held within" an M-curve in this way is known as parallel transport in the M-curve. The example shows that it preserves length but not direction. If we parallel transport a 1-vector from p to q within an M-curve then the result is dependant on the path taken.
Parallel transport is invoked to provide a notion of "parallelism" for 1-vectors in different tangent spaces (ie. 1-vectors "at" different points of an M-curve) that does not require a higher-dimensional embedding space. The "distance" between two points on a manifold can be meaningfully defined as the arclength of a minimial-arclength path joining the points (a geodesic), but to compare distant directions we need for every qÎCM a function ­qp C(a) Î Ip giving the tangent vector at pÎCM "parallel" to aÎIq at q ÎCM ; as obtained by "parallel transportation" of a "along" a given path C from q to p (eg. along a minimal length geodesic).

Let i­Dlip0(a) denote the parallel transport of a from p0 for arclength Dli along the ui p streamline.
We obviously want i­Dlip0(a)ÙI Dli¬i p0 = 0 and we might therefore think of defining i­Dlip(a) = ¯I Dli¬i p0(ujp)~ whenever ¯I Dli¬i p0(ujp) is nonnull but instead we simply postulate a 1-vector i­Dlip(a) within I Dli¬i p0 that satifies i­Dlip(a)2 = a2 .   We will henceforth assume a to be a unit vector.

The normalisation condition for i­Dlip(a) means that the right-connection ba,p(Dli) º a-1i­Dlip(a) for i­Dli at p satisfies ba,pba,p§ = 1 and is thus a 1-rotor expressable as a 2-spinor ewui p a p(Dli) , where wui p a p(Dli) = cos-1(a-1¿i­Dlip(a) (a-1Ùi­Dlip(a)~ is a UN 2-blade. [  we assume Dli is sufficiently small and CM sufficiently smooth that the subtended angle is comfortably below ½p ]

We would like the angle subtended by two 1-vectors parallelly transported along the same streamline to remain the same, ie. i­ep(a) ¿ i­ep(b) = a¿b and we can acheive this by insisting that ui p¿wui p a p(e) = O(e2) .
i­ep(a) º aewui p a p(e) = a + a¿wui p a p(e) + O(e2)
where for given scalar e, wui p a p(e) is a pure UN-bivector 2-field defined at every pÎCM for all aÎIp .
Parallel displacement along the ith streamline thus preserves orthonormality and corresponds to a Lorentz rotation having UN rotor Rp i(e) which we assume to tend to a well defined limit as e®0. M such rotor fields fully define parallel transport and effectively define CM.

Linear Connection

gp(ujp,dli,e) º ujp - i­ep(ujp) = -ujp¿wui p ujp p(e)  + O(e2)
is a 1-vector lying in general neither in Ip nor I e¬i p. It cannot be properly regarded as a tangent 1-field of the M-curve (traditionally speaking, it is not a "tensor").
We make the continuity assumption that Lime ® 0[ e¬i ujp - i­-e e¬i p( e¬i ujp)] = -gp(ujp,ui p). [  We use the symbol g here in accordance with traditional GR use of G for the gravitational connection acting in the mapspace. Conflict with the widespread use in multivector literature of g for a (typically Â1,3) fiducial frame is unfortunate. ]
Taking gp(ujp,ui p) º Lime ® 0[e-1gj p(ui p,e) ] we obtain the connection, the 1-vector component of dli ujp ïp that exists outside Ip.
gp(ujp,ui p) = -ujp¿wui p ujp = wui p ujp ×ujp .
gp(ujp,ui p) is orthogonal to both ui p and ujp at p .

A linear connection (aka. affine connection) derives from an assumed bilinearity of gp which can then can be fully defined by M2 1-fields   gp( hjp , hip ) which then effectively define ­qp C(a). Indeed, some authors define i­Dli as the consequence of a given linear connection.
We say the connection is symmetric if gp(ui p,ujp) = gp(ujp,ui p) .

Suppose now that the ui p are everywhere mutually orthogonal so that {ui p} provides an orthonormal frame for Ip with inverse frame {uip}. Because wui p ujp is a 2-blade perpendicular to Ip, we have wui p ujp ×ui p = 0     i¹j . Thus we can define M pure bivectors wui p p º åj=1M wui p ujp which satisfy gp(ujp,ui p) =  wui p p×ujp
The connection is symmetric if ujp¿wui p p = ui p¿wujp p     " 1£i,j£M .
A linear connection arises if wui p a is linear in a.
[ Proof : Since the ui p are orthogonal gp((åj=1M bjujp),(åi=1M aiui p)) = (åj=1M bjujp)¿wui p åi 1M= aiui p)}
= (åj=1M bjujp)¿(åi=1M aiwui p ui p ) = åi=1M åj=1M aibj(ujp¿wui p ui p ) = åi=1M åj=1M aibjgp(ujp,ui p)  .]

Within the mapspace
Parallel transport in the mapspace i­ex(b) º ¦Ñx+eei-1( i­e¦(x)(¦Ñp(b)) ) preserves ¿ instead of ¿.
[ Proof : i­ep(a) ¿x+eei i­ep(b) = i­ep(¦Ñx(a))¿i­ep(¦Ñx(b)) = ¦Ñp(a)¿¦Ñx(b) = a¿xb  .]

Let b be a map direction at xÎMap with corresponding CM direction a=¦Ñx(b) at p=¦(x). It may not make any geometrical "sense" to subtract a map 1-vector "at" x+eei from one "at" x but we can still do it!
Gx(b,ei,e) º b - i­ex(b) = b - ¦Ñx+eei-1(i­ep(¦Ñx(b))) = b - ¦Ñx+eei-1(¦Ñp(b) - gp(¦Ñp(b),dli,e))
is, for given ei and scalar e,  a map-1-vector-valued function of map-1-vector b, linear in b if the CM connection gp is linear. We then have Gx(ei,ej,e) = åk=1M  Gkijx(e)ek . Letting   Gx(b,ei) =   Lime ® 0(e-1Gx(b,ei,e)) we achieve an affine connection for the mapspace, symmetric if g is.
Gx(a,b) = ¦-Ñx(¦Ñ2x(a,b))
[ Proof :   See General Relativity Chapter  .]
We can think of G as a rule for moving map 1-vectors around. In the chaper on General Relativity we derive an expression for G in terms of the metric.

Directed Coderivative

The directed coderivative of b(p) with respect to streamline ui p at p is defined by parallel transporting b( e¬i p) back from  e¬i p to p and there subtracting b and dividing by e, in the limit as e®0. It is thus a 1-vector within Ip.
Ðui p(b) = dlib + gp(b,dli) = dlib + wui p ujp p×b
[ Proof : Ðui p(b(p)) º Lime ® 0[ i­-e e¬i p(b( e¬i p)) - vj(p) / e ]
= Lime ® 0[ ( i­-e e¬i p(b( e¬i p)) - b( e¬i p)) + ( b( e¬i p) - b(p)) / e ] = gp(b,dli) +dliujp  .]

In the limit, since li is  the natural (proper) parameterisation, we can consider dli to be equivalent to the N-D directional derivative Ðui p
Accordingly we can define a directed coderivative operator
Ða º Ða + wa×
for a given direction aÎIp which when applied at p to a 1-field over and within CM returns a 1-vector in Ip.
We then have
i­Dlip(ujp) = eDliÐli e¬i ujp = eDliÐli eDlidli ujp .
[ Proof : i­Dlip(ujp) = vj( Dli¬p) + DliÐlivj( Dli¬p) + Dli2Ð (ui p)2vj( Dli¬p) + ...  .]

Ða can be regarded as a differential over and "within" CM, although its component parts Ða and wa× cannot.

Ða generates a 1-vector codel-operator
Ñp º åk=1M ukpÐuk p = Ñ[CM] + åk=1M ukp(wuk p×) .

Within the mapspace
Ða induces a differential operator within M-D parameter space MapÌVM
Ðað(B(x)) = ¦-Ñp(Ð¦Ñp(a)(b(p)))     where b(p) º B(¦-1(p)) .
Ðað(vp) = Ða(vp) + G(a,vp)     where G is linear as a consequence of the linearity of ¦-Ñp .
In particular,
Gp(ej,ei) º Ðeiðej = åk=1M Gkjiek     where point-dependant scalar Gkji is known as a Christoffel symbol.

Geodesics
We say a 1-field ui p is geodesic if i­Dlip(ui p) = ui p( e¬i p)     " pÎCM, Dli
where Dli is assumed small enough to remain in the neighbourhood. Geodesic fields are thus invarient under parallel transport and satisfy Ðui pui p = 0.
The physical interpretation of geodesics are the potential trajectories of "free falling" particles and (assuming a natural streamline parameterisation pi'(li)2 = 1) we can then regard the
pi"(li) + g(pi'(li),pi'(li)) = 0
derived from Ðui p(ui p) = 0 as relating the "acceleration" of a "particle" to its velocity and the local "geometry" g.

An alternate definition of a geodesic is the "shortest subcurve between two points". If we fix two points p and q on a geodesic streamline then the arclength from p to q is stationary (minimal) for small changes in the path from p to q that keep it within CM. The equivalence of the two defintions is intuitively reasonable and the proof (ommitted here) arises from fairly straightforward calculus

Curvature
[Under Construction]

1-Curvature
The traditional curvature of a 2D 1-curve p(t) (ie. a 1-curve confined to a 2-plane ) is defined as
C(t) º Limd ® 0[ d-1 qÐ(p'(t+d) , p'(t)) ] where t is the natural parameterisation - ie. the instantaneous angular tangent change per unit arclenth. C(t)-1 is known as the raduius of curvature and coresponds to the radius of the circle matching the 1-curve to second order at p(t) . For an N-D 1-curve we have 1-curvature p"(t) which is a 1-vector orthogonal to p'(t) ("normal" to the 1-curve) as a consequence of the constancy of p'(t)2 . Its magnitude C(t) is the curvature of the 1-curve confined in the limit to 2-plane p(t).
For a 3D 2-curve we have tangent unit vectors h1,h2 and normal n=h1×h2  forming the SYMBOL">t) is the curvature of the 1-curve confined in trihedron) at p. Taking p as the origin, the induced coordinates (x1,x2,x3) satisfy
x3 = ½ (2x3/(x1)2) x12  + (2x3/x1x2) x1x2 + ½ (2x3/x2) x2 |2
By rotation about e3 we can diagnonalise this form as
x3 = ½ (k1(x1)2 + k2(x2)2)     where k1,k2 are known as principle curvatures.
If the surface  is parameterised as p(l1,l2) with h1p  = p/l1 , h2p  = p/l2 then
2p/lilj º hip /lj = åk=12 Gkij hkp  + bij h3p      i,j,k Î {1,2}
h3p /li = - åk l=12 gklbli hkp      i Î {1,2}

2-Curvature
The second directional coderivative operator ÐliÐlj is not symmetric in i,j.
We can take the skewsymmetrol of it to obtain the Rieman Curvature operator
cpui pÙujp(u)   =   cpui pÙujp×u   =   2(Ðli×Ðlj)u = ( Ð¯(li)wujp - Ð¯(lj)wui p + wui p×wujp) . u
[ Proof :
 ÐbÐau = ( Ð¯(b) + wb×)( Ð¯(a) + wa×)u = ( Ð¯(b) + wb×)( Ð¯(a)u + ½(wau-uwa)) = ( Ð¯(b) Ð¯(a)u + ½( Ð¯(b)wau- Ð¯(b)uwa) +wb×( Ð¯(a)u +   ½(wau-uwa)) = Ð¯(b) Ð¯(a)u + ½( Ð¯(b)wau - Ð¯(b)uwa + wb Ð¯(a)u - Ð¯(a)uwb ) + ½wb×(wau-uwa) = Ð¯(b) Ð¯(a)u + ½( Ð¯(b)(wau) + wb Ð¯(a)u - Ð¯(a)(uwb) - Ð¯(b)(uwa) ) + 1/4( wbwau - wbuwa - wauwb + uwawb ) Þ 2(¶b×¶a)u = 2( Ð¯(b)× Ð¯(a) + Ð¯(b)×(wa×) + (wb×)× Ð¯(a) + (wb×)×(wa×))u = Ð¯(b)(wau)- Ð¯(a)(wbu) + wb Ð¯(a)u -wa Ð¯(b)u + (wb×wa)×u = ( Ð¯(b)wa)u - ( Ð¯(a)wb)u + (wb×wa)×u = ( Ð¯(b)wa)×u - ( Ð¯(a)wb)×u + (wb×wa)×u
.]

Given a linear connection, the second-derivative curvature operator  cpaÙb(v) is thus a geometric commutatator product with a particular bivector cpaÙb .

The more general definition of the curvature operator    is 2(Ða×Ðb) - Ð[a,b] but we are most interested in the case when [a,b]=0 ( ie. Ða×Ðb =0 ) .

cpaÙb º Ð¯(b)wa - Ð¯(a)wb + wb×wa =( )= ( Ð¯(b)wa) - ( Ð¯(a)wb) + (wb×wa)     is known as the curvature 2-tensor (aka. Riemann-Christoffel tensor). It is a CM-point-dependant Ip-bivector-valued function of Ip-bivectors acting as a "second order correction" for the linear approximator ¦(p0) + ¦Ñp(dp) for ¦ near p0.

Geometric interpretation
If [ui p,ujp]=0 so that we have a loop  -Dli¬j  -Dli¬i  Dlj¬j  Dli¬i  p = p then
cpui pÙujp(uk p) = (DliDlj)-1  j­-Dlj(  i­-Dli(  j­Dlj(  i­Dli(uk p) ))))
which is clearly dependant solely on value of uk p and the geometry of CM . In particular, it is independant of any differential of uk p.
We thus have the geometric interpretation of cpui pujp)(uk p) as the 1-vector change in uk p resulting from parallel transport of uk p around (the projection into CM of) a tiny planar loop through p in ui pÙujp ,  divided by the "area" of that loop. This must lie in Ip so we have
cpui pÙujp(uk p) = ål=14 clijkulp     where scalar clijk º ulp¿cpui pÙujp(uk p) .
[ Proof : j­-Dlj i­-Dli j­Dlj i­Dli(uk p) = e-DljÐlj e-DliÐli eDljÐlj eDliÐliuk p = ...[tedious expansion]... = uk p + 2DliDlj(Ðli×Ðlj)uk p + O(Dl3)  .]

Symmetries and Bianci Relations
It is possible to express Cxeij in terms of the gij and their derivatives and doing so reveals some more symmetries:
Clkij º el¿(Cxeij(ek)) = ei¿(Cxelk(ej)) = -ek¿(Cxeij(el))   .
The last of these reflects the reversability of parallel transport.
We also have (from the G expression for Cijkl) the first Bianci identity
ek¿Cxeij + ej¿Cxeki + ei¿Cxejk = 0 .
which can also be expressed as     Cxeij ¿ ekl = Cxekl ¿ eij .
It follows that a¿Cx(bÙc)) + b¿Cx(cÙa) + c¿Cx(aÙb) = 0     " 1-vector a,b,c Î Ip.
As a result, the M 4 element matrix representation of Cx has only M 2(M 2-1)/12 independant elements (20 out of 256 for N=4).

ÐeiðCxejk + ÐejðCxeki + ÐekðCxeij = 02         is known as the second Bianci identity.

The codivergence of the Ricci tensor Ñðx¿ cpa º åiei¿Ðei cpa = ½ÐaCx
[ Proof : Ñðx¿ cpb = Ñðx¿(Ña¿cpab)) Þ ... ???  .]

Tortion
The differential operator Ða Ð¯(b) - Ðb Ð¯(a) - 2( Ð¯(a)× Ð¯(b)) is known as the tortion operator.   It is 0 for a symmetric connection when we thus have
£ui pujp = Ðui p Ð¯(ujp) - Ðujp Ð¯(ui p) .
Suppose now that ui p is a geodesic field and 1-vector bp is Lie dragged along geodesic ui p congruence so that £ui pbp = 0 . We must then have
Ðui p2 Ð¯(bp) = 2(Ðui p×Ðbp) Ð¯(ui p) .
[ Proof :  Ðui p Ð¯(bp) = Ðbp Ð¯(ui p) Þ Ðui p2 Ð¯(bp) = Ðui pÐbp Ð¯(ui p) = 2(Ðui p×Ðbp) Ð¯(ui p) + ÐbpÐui p Ð¯(ui p)   = 2(Ðui p×Ðbp) Ð¯(ui p) since ui p geodesic Þ Ðui p Ð¯(ui p) = 0  .]
Thus we have a second geometric interpretation of the curvature tensor as the second derivative of a vector dragged along a geodesic confluence, informally: a measure of geodesic "splay".

[Under Construction]

Extremal M-Curves
A standard problem in classical mechanics is to determine the form of the curve assumed by a rope of length a hanging motionless between two fixed points a distance less than a apart under a uniform gravitational field, which is an extremal 1-curve in N=2 dimensions.
If we assume the rope to hold a curve shape p(s)=(x(s),y(x)) (the Y axis being gravitationally vertical) that does not loop "over" itself vertically, we have ds2 = dx2 + dy2 so that the length constraint is a = ò01 ds = òx0x1 dx(1+y'2)½ where y' º dy/dx.
The mechanical constraint is that total gravitational potential energy ò01 ds rgy(s)   =   rg òx0x1 dx y(1+y'2)½ be minimised, where g is uniform vertical gravitational acceleration and r is the mass of unit length of rope.
More generally, we might seek to minimise òx0x1 dx¦(x,y,y') subject to a constraint òx0x1 dxg(x,y,y') = a.
A standard approach is to form h(x,y,y') º ¦(x,y,y') + lg(x,y,y') where scalar l is known as a Lagrange multiplier. One then embodies freedom to vary the path by means of two z0 and z1 such that y(x0,z0,z1)=y(x0) ; y(x1,z0,z1)=y(x1) " z0,z1 (boundary condition) ; y(x,0,0)=y(x) ; and y(x,z0,z1) is twice continuosuly differentiable in all parameters.
Forming K(z0,z1) = ò+x0x1 dx h(x,y(x,z0,z1),y'(x,z0,z1)) the extremal condition requires K/z0 and K/z1 to vanish at z0=z1=0 and this leads (via integration by parts) to the Euler-Lagrange equation h/y = d/dx(h/y') .
In the case of the hanging rope, h(x,y,y') = (rgy-l)(1+y'2)½ and setting l = -rgy0 we have   h(x,y,y') = rg(y-y0)(1+y'2)½ = rg(y-y0)(1+(y-y0)'2)½ for some constant y0 .
Setting h = y-y0 the Euler-Lagrange equation is rg(1+h'2)½ = d/dx   rghh'(1+h'2) with first order simplification
h' rghh'(1+(y-y0)'2) - rgh(1+h'2)½ = b Þ rg(1+h'2)h = b having solution h = b cosh(x-c)/b) so the rope has shape y0 + b cosh((x-c)/b) where c,y0 and b are chosen to match the given endpoints.

We say an M-curve CM is M-extremal for an action functional ¦CM:UN ® UN if some particular magnitude measure (content, scalar part, square, modulus or whatever) of òCM dM-1p ¦CM(p) is maximised (or minimised) by CM in the sense that integrating other any M-curve which deviates only slightly from CM will produce a result no higher (or lower) than the CM integral. CM is a "locally optimal" M-curve for a particular integration ¦CM. We specify the dependancy of ¦ on CM to allow action functionals dependant on the geometric properties of the "sampling curve" CM , such as tangent or normal vectors , to allow "direction dependant sampling" or "velocity dependance".

Euler-Lagrange Equations
For M=1 we have a path p(s) 1-extremal for FC1 = F(s,p(s),p'(s)).

Hamilton's principle provides that the state of a system characterised at time t by k multivector variables x1(t),..,xk(t) varies from time t0 to t0 so as to 1-extremise a (usually real scalar) integral measure
S(t) º S(t0) + òt0t0 dt L(t,x1(t),..,xk(t),x'(t),...,xk'(t))     with the action functional L(t,x1(t),..,x1'(t)) known as the Lagrangian of the system.
L is usually assumed to be real-scalar valued and independant of p"(t) and higher derivatives and is traditionally assumed to seperate into "kinetic" and "potential" componenets independent of time and velocity respectively as L(t,p(t),p'(t)) = T(p(t),p'(t)) - V(t,p(t)) but more generally we might postulate a multivector-valued Lagrangian of k multivector-valued variables and their first temporal derivatives L(t,x1(t),..,xk(t),x1'(t),..,xk'(t)).
If the Lagrangian L is itself an integral over some spacial M-curve (typically an M=N-1 hypercurve reprenting a contemporal slice of a Base space) L = òBase(t) dN-1p L(t,p,p') that spacially integrated function is known as a Lagrangian density. Note that such requires a "velocity" p' be associated with every point p in Base(t)   .

Extremal Paths
A standard approach for M=N=1 to 1-extremise ¦(s,,') is to embody freedom to vary the path by means of two scalar parameters z0 and z1 such that (s0,z0,z1)=(s0) ; (s1,z0,z1)=(s1) " z0,z1 (boundary condition) ; (s,0,0)=(s) ; and (s,z0,z1) is twice continuosuly differentiable in all parameters. Forming K(z0,z1) = òs0s1 ds¦(s,(s,z0,z1),'(s,z0,z1)) the extremal condition requires K/z0 and K/z1 to vanish at z0=z1=0 and this leads (via integration by parts) to the Euler-Lagrange equation ¦/ = (d/ds)(¦/') .
In particular cases where ¦(s,,') = ¦(,') so that ¦/s=0 and there is no explicit s dependence,   we have d/ds('¦/' - ¦)= 0 giving a first order differential equation '¦/' - h = constant .
[ Proof :   y" h/y' + y'd/dxh/y' - h/-h/yy' - h/y'y"   =   y'(d/dxh/y' - h/y) - h/   =   0  .]

A nongeometric generalisation is to extremise òs0s1 ds¦(s,x1,x2,..xk,x1',x2',..xk') subject to M scalar constraints gi(s,x1,x2,...xk)=0 i=1,2,..,M (Note the absence of any g dependance on the xi'). To do so we form
h(s,x1,..,_xK,x1',..,xk') = ¦(s,x1,..,_xK,x1',..,xk') + åj=1M lj(s)gj(s,x1,..,xk,x1',..,xk') where lj(s) are M arbitary functions of s and obtain Euler-Lagrange formulae h/xi = (d/ds)(h/xi')     for i=1,2,..k .

Generalising geometrically, we have F(s,x1,x2,..,xk,x1',x2',..,+xk') with geometric Euler-Lagrange Equations F/xi =  (d/ds)(F/xi')     " i=1,2,..k and first order equation åi=1k  (xi'*Ñxi')F - F = constant (independant of s) if F/s = 0. Considering the xi as seperate grades of a single multivector argument x we can regard the Euler-Lagrange equations as individual coordinate terms of a single geometric equation
xF(s,x,x') = (d/ds)(x'F(s,x,x')) where x = åijk.. eijk../xijk.. over all blades comprising x space so that, for example, x = åi=1N ei(/xi).

Generalised Momentum

When F is a scalar-valued Lagrangian we have  (xi'*xi')L = xi'*(xi'L) and multivector m = Ñxi'L(x1,..xk,x1',..,xk) is known as generalised spacial momenta or cononical momenta. Typically xi' is spacial 1-vector valued and so momenutum is a spacial 1-vector.
If L is independant of xi so that L/xi=0 then the ith Euiler-Lagrange equation provides (d/dt) L/xi' = 0 so Mi = L/xi' is constant, ie. unchanging with t,

Thus the constants of a system are consequences of absent dependendencies (aka. symmetries) in the Lagrangian. Energy is conserved when _Lag depends on t only indirectly via x(t) and it derivatives. Momentum is conserved when _Lag depends on x only indirectly via x'.
For L(t,,') = ½m'2 - f(,t) we obtain classical momentum Ñ'½m'2 = m' , constant if f(,t)=f(t).
Adding an electromagnetic term qc-1ax¿' to L introduces qc-1ax to the momentum

If L(t,x1(t),..,xk(t),x1'(t),..,xk'(t))   =   L(x1(t),..,xk(t),x1'(t),..,xk'(t)) so that there is no explicit t dependance then the Hamiltonian H(x1(t),..,xk(t),x1'(t),..,xk'(t)) º åi=1k  (xi'*Ñxi')L - L     is a constant, the "energy" of the system.
We can regard the Hamiltonian as generalised temporal momentum. Conservation of energy but varying spacial momentum resulting from only non-relativistic potentials f(,t)=f() .

But H = - _piff[S,t] .

Even though, like the Lagrangian, S(t) cannot strictly be regarded as a function of position since it is only defined over our given extremal path, we can nontheless postualte an action field S(t,) an it can be shown that
m = (xi'*xi')L = ÑxiS(t,xi) and in particular we have the Hamilton-Jacobi equation H(xi,xi',t) = -S/t
[ Proof :     For any ¦(,')
d/dt ( åj=1k  xi' ¦/xj' - ¦)   =   åj=1k  xj" ¦/xj' + åi=1k  xj' d/dt ¦/xj' - ¦/t - åj=1k ¦/xjxj' - åj=1k ¦/xj'xj"   =   åj=1k  xj' d/dt ¦/xj' - ¦/t - åj=1k ¦/xjxj'   =   åj=1k  xi'( d/dt(¦/xj') - ¦/xj ) - ¦/t
=   åj=1k  xj'( d/dt(¦/xj') - ¦/xj )     if ¦/t = 0
=   0 if d/dt(¦/xj') = ¦/xj     j=1,2,..,k which are the Euler-Lagrange equations.

d/dx ( åi=1k  yi' ¦/yi' - ¦)   =   åi=1k  yi" ¦/yi' + åi=1k  yi' d/dx ¦/yi' - ¦/ - åi=1k ¦/yiyi' - åi=1k ¦/yi'yi"   =   åi=1k  yi' d/dx ¦/yi' - ¦/ - åi=1k ¦/yiyi'   =   åi=1k  yi'( d/dx(¦/yi') - ¦/yi ) - ¦/
=   åi=1k  yi'( d/dx(¦/yi') - ¦/yi )     if ¦/ = 0
=   0 if d/dx(¦/yi') = ¦/yi     i=1,2,..,k which are the Euler-Lagrange equations.  .]

Theoretical physics then becomes a quest for the One True Action Lagrangian density , usually assumed real scalar and generating kinematic equations involving zeroth, first,  and second differentials only. Geometric (multivector-valued) Lagrangians extermised in the sense that each multivector coordinate is stationary under variation. This is an intrinsically nonrelativistic approach in the case of multiple particulate systems since it requires a favoured temporal parametrisation t with L(t,p,p') = L(p,p')
As "locally shortest routes", Geodesics are 1-extremals for the scalar path length of ò0t ds|d1p¿g(d1p)|½ . Integrating a square root can be messy but fortunately such paths also extremise   ò0t ds ds p'(s)¿gp(s))(p'(s)) where p'(s) denotes the (d/ds)p(s) . _Be Thus we set L(p,p',s) = p'¿gp(p')|½ and minimise scalar integral   ò0t ds ds L(p(s),p'(s),s) by solving the N Euler-Lagrange equations d/ds(L/pi') = L/pi) for i=1,2,...N .
L(p,p',s) will generally depend on p via _gp but if _gp has a symmetry such that L/pi=0 for a particular coordinate i then L/pi' is constant along any geodesic path.
Geodesic flow can consequently be viewed as arising from a Lagrangian density L(p(t),p'(t)) = |p'(t)¿_gp(p'(t))|½ . with particles following timelike trajectories that extremise (minimally) their proper time (arclength).
If t is proper time parameterisation then L(p(t),p'(t)) = |p'(t)2|½ = 1 along a geodesic.
Electrodynamical forces due to a 1-vector four-potential ap (typically with Ñp¿ap=0) are introduced by adding scalar -q p'(t)¿ ap(t) to the Lagrangian density leading to the Lorentz force law mp"(t) = qfp.p'(t) where fp = ÑpÙap .
[ Proof : L = m(-p'2)½ - q p'¿ap Þ ÑpL = Ñp(m(-p'2)½ - q p'¿ ap) = -qÑp(p'¿ap) ; Ñp'L = -m(-p'(t)2) p'(t) - qap     so the Euler-Lagrange equation is qp'¿Ñpap = (d/dt)(m(-p'2) p' + qap ) = mp" + q(p'¿Ñp)ap Þ mp" = q(Ñp(ap¿p') - (p'¿Ñp)ap))   =   q(ÑpÙap)¿p'  .]

Extremal surfaces
A more geometric generalistaion is to consider the problem of finding the M-curve of given content maximising a particular boundary or interior integral. We might, for example, seek the loop starting and ending at a given point a and constrained to lie in a given 2-curve (surface) containing a that maximises enclosed content (area) for a given boundary content (pathlength)

Of fundamental importance is the fact that spacial Lagrangian density (Ñ[e123]yp)2 is extremised for integration over CM (subject to constraint of given boundary values over dCM) if yp satisfies the spacial Laplace equation Ñp [e123]2yp = 0 over CM .
In electrodynamics we have L = - 2-4p-1 fp2 + c-1jp¿ap where c is scalar lightspeed and fp = ÑpÙap extremised by solutions to Maxwell equations.

Next : 4D Spacetime

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References/Source Material

Bernard Schutz "Geometrical Methods of Mathematical Physics" Cambridge University Press 1980 [Amazon US UK]
[ Traditional presentation of manifold derivatives]

David Hestenes, Garret Sobczyk "Clifford Algebra to Geometric Calculus" D. Reidel Publishing 1984,1992 [Amazon US UK]
Since this document draws heavily from and frequently cites this work we clarify the notational differences between them.
Their P(a) is our ¯Ip(a) º ¯(a) ("projector").
Their Pb(a) is our ¯Ñ(a,b) ("differential of projector").
Their Sa is our [ÑÙ¯](a) ("curl").
Their Ñ is our Ñ  ("coderivative").
Their is predominantly our Ñ ("tangential derivative") but also our Ñ ("derivative") early in the work.
Their da is our Ðaß º ¯Ð¯aß ("extensor coderivative").
Their (a.) is our Ða=(a¿Ñ) ("directed derivative").
Their ei is our hip  (potentially nonorthonormal "tangent frame").
Their gi is our ei (orthonormal "fiducial (basis) frame").

Glossary   Contents   Author
Copyright (c) Ian C G Bell 1998
Web Source: www.iancgbell.clara.net/maths or www.bigfoot.com/~iancgbell/maths
Latest Edit: 31 Mar 2007.