Thus far we have considered fields over U^{N} as point-dependant multivectors in U_{N} rather than point-dependant
points in U^{N}. If we consider ¦ to map points in U^{N} to points in U^{N}, we have a transformation.
The image under ¦ of a subset of U_{N} is known as a "manifold" in U^{N}.
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 U^{N} is a set of points in
U^{N} locally representable 1-1 by a system of M scalar coordinates drawn from a well defined parameter space
M_{ap}.
A 0-curve is a point.
A 1-curve is a Â®U^{N} mapping p(t) t Î [t_{0},t_{1}]
and is either closed (a "loop") , infinite, or "bounded" by two 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 C_{M} is either closed or has as boundary a closed (M-1)-curve
conventionally written dC_{M}. Clearly ddC_{M} = f (the empty set).
We will denote the "interior" or contents of an M-curve C_{M} by C_{M}^{·} , so that
(dC_{M})^{·} = C_{M} .
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 ¦(M_{ap}) model.
The k-curve "spanned" by k-blade a_{k} is the infinite k-plane { p : pÙa_{k}=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}(x^{1},x^{2},..,x^{m})
sending V^{M} ® U^{N} defined over
a fully bounded finite subvolume (interior of a unimapped (M-1)-curve) M_{ap}ÌV^{M} .
We refer to M_{ap} as the parameter space or mapspace (aka. a (local) map ) .
Typically V^{M} = Â^{M}, but we will retain a more general view to accomodate
the Minkowski parameter spaces of Relativity physics. We will associate x with an M-D point (1-vector) within M_{ap} and
p=¦_{q}(x) with the associated N-D point (1-vector) in C_{M} Ì U^{N} . We will here use upper case symbols
to distinguish structures and operators defined within M_{ap} from their within C_{M}
counterparts.
We say an M-curve is differentiable if ¦_{q} is differentiable in
each of the parameters and is sufficiently "smooth" for derivatives of all orders to exist.
We say an M-curve is unimapped if a common "global" map is locally applicable
everywhere; ie. there is a single function ¦ : M_{ap}®U^{N}
with ¦(M_{ap})=C_{M} that serves for every q Î C_{M} . A path is thus a unimapped 1-curve.
We will not assume a unimapped manifold, but we will assume local unimapping in that we will
assume that attention is restricted to a neighbourhood of q over which ¦_{q} applies.
q is thus fixed and plays no useful part in our discussion and we will accordingly drop most _{q} suffixes for brevity .
Extended Mapspace
Suppose now that we have ¦:V^{M} ® U^{N} . Any M-curve M_{ap} in V^{M} induces
an M-curve C_{M}=¦(M_{ap}) in U^{N}.
Typically we might have a flat M-plane M_{ap} in V^{M} embedding to a "bendy" C_{M} in U^{N}
seen as the M_{ap} specific "slice" of ¦ .
Submanifolds
Given a unimapped M-curve C_{M}=¦(M_{ap}) and a point pÎC_{M} we can construct
k-dimensional submanifolds of C_{M} at p as the images under ¦ of the subvolumes of M_{ap}
obtained by keeping M-k of the mapspace parameters held at their ¦^{-1}(p) values.
We say an M-curve is isomapped if ¦ can be extended over a subvolume of V^{N}
to define an invertible function ¦: V^{N}®U^{N} creating an N-curve of which
C_{M} is a submanifold.
Generator Functions
[under construction]
Embedded Frame
We have a particular M_{ap}-point-dependant embedded frame in U_{N} over our q neighbourhood
consisting of M U_{N} tangent vectors specified by the alternate notations
h_{kp } º
(d/dx^{k})¦_{q}(x)
= Ð^{x}_{ek}¦_{q}(x)
= (e_{k}¿Ñ_{x})¦_{q}(x)
= ¦_{q}^{Ñx}(e_{k})
where Ð^{x}_{ek} and Ñ_{x} are the standard directed and undirected
derivatives for directions within the mapspace M_{ap}.
The embedded frame is not orthogonal for general ¦_{q}. It is defined with regard to a patricular basis for the
mapspace.
We can extend the embedded frame to a frame for U_{N} with N-M orthonormal vectors lieing wholly outside I_{p} but in the absence of an extended N-D mapspace invertibly mapped into U^{N} such an extension is only unique in the case M=N-1 when we can define h_{Np } = ( h_{1p }Ù h_{2p }Ù... h_{N-1p })^{-1}i .
Spherical Surface Example
We can formulate coventional spherical polar coordinates as ¦:V^{M}=Â^{3} ® U^{N}=Â^{3}
¦(x)=¦(qe_{q} + fe_{f} + re_{r})
= r sinq cosfe_{1}
+ r sinq sinfe_{2}
+ r cosqe_{3}
= rR_{q,f}_{§}(e_{3})
where e_{q},e_{f},e_{r} are an orthonormal basis for V^{M} and e_{1},e_{2},e_{3} are an orthonormal basis for U^{N}.
and
R_{q,f} =
e^{-½e12f}
e^{-½e31q}
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 U^{N}=V^{M}=Â^{3} with e_{ð}^{i}=e^{i} and regard ¦ as a transformation then
¦^{Ñ}_{x}(dx)) = (Ñ_{x}¿dx)¦(x)
= (dqr cosq cosf - dfr sinq sinf + dr sinq cosf)e_{1}
+ (dqrcos(q) sinf + dfr sinq cosf + dr sinq sinf) e_{2}
+ (-dqr sinq + dr cosq) e_{3} .
The embedded frame in U^{N} is orthogonal.
h_{qp } = rR_{q,f}_{§}(e_{1}) ;
h_{fp }
= e_{3}×¦(q,f,r)
= (e_{3}Ù¦(q,f,r))^{*} = r sinqR_{q,f}_{§}(e_{2});
h_{rp } = ¦(q,f,r)^{~}
= R_{q,f}_{§}(e_{3})
These have magnitudes r; r sinq ; and 1 respectively.
If we fix r=1, so restricing M_{ap} to a 2D parameterspace { q,f }
and obtaining C_{M}=S_{2} , the 2-curve boundary surface of a 3D unit sphere, then the derivative reduces to
¦^{Ñ}_{x}(dx)) = (Ñ_{x}¿dx)¦(x)
= (dq cosq cosf - df sinq sinf)e_{1}
+ (dq cosq sinf + df sinq cosf) e_{2}
- dq sinq e_{3} .
From a purist point of view, mapspace coordinates have the form qe_{q}+fe_{f}
whereas tangent vectors are expressed in terms of e_{1},e_{2},e_{3} but we will here consider the M_{ap} as existing in U_{N}
and set e_{q}=e_{1} , e_{f}=e_{2}.
Inverse Embedded Frame
We can construct an inverse embedded M-frame
h^{i}_{p } of 1-vectors within the tangent space satisfying
h^{i}_{p }¿ h_{jp } = d_{i j} via
h^{k}_{p } º
(-1)^{k-1}( h_{1p }Ù.. h_{k-1p }Ù h_{k+1p }Ù.. h_{Mp })( h_{1p }Ù.. h_{Mp })^{-1}
.
Note that the h^{i}_{p }ÎI_{p} and are a frame for precisely the same subspace as are the embedded frame.
We can invert ¦_{q} to express the x^{i}(p) = e^{i}¿¦_{q}^{-1}(p) as M scalar fields defined over
a local (M-curve) neighbourhood of q.
When M=N so that ¦_{q} : U^{N} ® U^{N} we have
h^{i}_{p } º Ñ_{p }(x^{i}(p))
= Ñ_{¦q(x) }x^{i}
= ¦_{x}^{-D}(e^{i}) .
[ Proof :
h_{ip }¿ h^{j}_{p }
= ¦_{qx}^{Ñ}(e^{i}) ¿ ¦_{qx}^{-D}(e_{j})
= e^{i}¿ ¦_{qx}^{D}(¦_{qx}^{-D}(e_{j}))
= e^{i}¿e_{j}
= d_{ij}
.]
h^{i}_{p } is the normal to the coordinate isosurface x^{i}(p)=x^{i}(q) at q.
In a nonorthogonal embedded frame, we can have h^{i}_{p }Ù h_{ip } ¹ 0 so
the normal to the isosurface need not be parallel to the streamline tangent.
Spherical Surface Example
For spherical coordinates mapping, the normalised reciprocal frame is h^{i}_{p }^{~} = R_{q,f}_{§}(e_{i}).
¦^{D}_{p}(dp) º Ñ_{x}(dp¿¦(x))
= Ñ_{x}(dp^{1}r sinq cosf + dp^{2}r sinq sinf + dp^{3}r cosq)
= r(dp^{1} cosq cosf + dp^{2} cosq sinf - dp^{3} sinq)e_{q}
+ r(-dp^{1} sinq sinf + dp^{2} sinq cosf)e_{f}
+ (dp^{1} sinq cosf + dp^{2} sinq sinf + dp^{3} cosq)e_{r}
defined for dpÎÂ^{3} .
This reduces to
¦^{D}_{p}(dp) =
(dp^{1} cosq cosf + dp^{2} cosq sinf)e_{q}
+ (-dp^{1} sinq sinf + dp^{2} sinq cosf)e_{f}
when pÎS_{2} and dp is in the tangent space at p.
We have M scalar fields defined over C_{M} by x^{i}(p) = e_{ð}^{i}¿¦^{-1}(p) = e_{ð}^{i}¿x .
For spherical mapping these are
q(p^{1}e_{1} + p^{2}e_{2} + p^{3}e_{3}) =
cos^{-1}(p^{3}(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-½}) [ p^{1}^{2} = (p^{1})^{2} ]
f(p^{1}e_{1} + p^{2}e_{2} + p^{3}e_{3}) = tan^{-1}(p^{2}/p^{1})
r(p^{1}e_{1} + p^{2}e_{2} + p^{3}e_{3}) = (p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{½}
We have
Ñ_{p} q(p) =
-(1-(p^{3}^{2}(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-1})^{-½}
( e^{1}(-p^{3}x^{1}(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-3/2}
+ e^{2}(-p^{3}x^{2}(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-3/2}
+ e^{3}((p^{1}^{2}+p^{2}^{2})(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-3/2}
= -((p^{1}^{2}+x^{2}^{2})(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-1})^{-½}
(p^{1}^{2}+p^{2}^{2}+p^{3}^{2})^{-_3/2}( e^{1}(-p^{3}p^{1})
+ e^{2}(-p^{3}p^{2})
+ e^{3}(p^{1}^{2}+p^{2}^{2}) )
=?= r^{-2} h_{qp } = r^{-1} h_{qp }^{~}
Ñ_{p} f(p) = (r sinq)^{-2} h_{fp } = (r sinq)^{-1} h_{fp }^{~}
Ñ_{p} r(p) = p^{~} = h_{rp } = h_{rp }^{~} .
Local Orientation
The h_{kp } define a C_{M}-point-dependant nondegenerate U_{N} M-blade
J_{p} º h_{1p }Ù..Ù h_{Mp }
that spans the tangent space at p=¦_{q}(x).
C_{M} has tangent M-plane (1+q)ÙJ_{q} at q
(or (e_{0}+q)ÙJ_{q} in a homegeneous higher dimensiunal embedding)
with .
The invertibility of ¦_{q} ensures that J_{q} is nondegenerate.
The unit pseudoscalar for the tangent space given by
I_{p} º J_{p}^{~}
is called the orientation
of the M-curve at p .
For a 1-curve, the orientation is the unit tangent 1-vector.
An M-curve is orientable if a continuous unit-valued M-tangent blade can be defined over it.
The classic example of a nonorientable 2-curve is a Moebius strip.
A manifold is flat if it has the same orientation everywhere.
We will be concerned with orientable nonflat manifolds here.
If the boundary of an M-curve has orientation I_{M-1p} then it is conventional to specify the orientation ("handedness")
of I_{p} by defining I_{p} = I_{M-1p} _np where
_np = I_{M-1p}^{-1} I_{p} is the spur, the unit outward normal to the boundary at p.
The normalisation condition I_{p}^{2} = ±1
gives (Ð_{a}I_{p})I_{p} + I_{p}(Ð_{a}I_{p})=0
and taking the scalar part yields (Ð_{a}I_{p})¿I_{p}=0 Þ ¯_{Ip}(Ð_{a}I_{p})=0.
Hence ¯_{Ip}(Ñ_{p}I_{p}) = 0 .
[ Proof : Choosing an orthonormal frame with e_{1}Ùe_{2}Ù...e_{M} =I_{p} we obtain
Ñ_{p}I_{p} = å_{i=1}^{N} e^{i} Ð_{ei}I_{p}
= å_{i=M+1}^{N} e^{i} Ð_{ei}I_{p}
all terms of which lie outside I_{p}.
.]
Note that Ñ_{p}¿I_{p} ¹ 0 in general.
Spherical Surface Example
For spherical coordinates S_{2} mapping,
J_{p} = h_{qp }Ù h_{fp }
= h_{qp } h_{fp } =
sinqR_{q,f}_{§}(e_{1})R_{q,f}_{§}(e_{2})
= sinqR_{q,f}_{§}(e_{12}) .
Although J_{p} vansihes at the poles,
it is natural to define
I_{p}=R_{q,f}_{§}(e_{12})
" p=R_{q,f}_{§}(e_{3}) ÎS_{2} .
The Metric
Let a,b be two 1-vectors in an M-curve's mapspace M_{ap}ÌV_{M}.
Whereas aÙb = ¦^{-D}_{x}(¦^{Ñ}_{x}(a)Ù¦^{Ñ}_{x}(b)) is
geometrically meaningful , a¿b is not. In particular, it does not equal
¦^{Ñ}_{x}(a)¿¦^{Ñ}_{x}(b).
Rather, we have the cocontraction
a¿b = a¿_{x}b º ¦^{Ñ}_{x}(a)¿¦^{Ñ}_{x}(b)
and hence a variant geometric "coproduct" with point-dependant inner product
¿ replacing ¿ .
We postulate a symmetric point-dependant metric 1-tensor
g_{x} : V_{M} ® V_{M} such that
a¿g_{x}(b)
= g_{x}(a)¿b
= ¦^{Ñ}_{x}(a)¿¦^{Ñ}_{x}(b) = a¿b.
For M=N the obvious candidate is g_{x} º
¦^{D}_{x}¦^{Ñ}_{x} .
If ¦^{Ñ}_{x} is symmetric, g_{x} = (¦^{Ñ}_{x})^{2} .
Spherical Surface Example
g_{x}(dx) = r^{2}dqe_{q}
+ r^{2} sin^{2}qdfe_{f}
+ e_{r}
with associated line element length
dx¿g_{x}(dx) = r^{2}dq^{2} + r^{2} sin^{2}qdf^{2} + dr^{2}
[ Proof :
¦^{Ñ}_{x}(dx)) = (Ñ_{x}¿dx)¦(x)
= (dqr cosq cosf-dfr sinq sinf+dr sinq cosf)e_{1}
+ (dqrcos(q) sinf+dfr sinq cosf+dr sinq sinf)e_{2}
+ (-dqr sinq+dr cosq)e_{3} .
¦^{D}_{p}(dp) º Ñ_{x}(dp¿¦(x))
= Ñ_{x}(dp^{1}r sinq cosf + dp^{2}r sinq sinf + dp^{3}r cosq)
= r(dp^{1} cosq cosf + dp^{2} cosq sinf - dp^{3} sinq)e_{q}
+ r(-dp^{1} sinx siny + dp^{2} sinq cosf)e_{f}
+ (dp^{1} sinq cosf + dp^{2} sinq sinf + dp^{3} cosq)e_{r}
Thus g_{x}(dx) = ¦^{D}_{p}(¦^{Ñ}_{x}(dx))
= ¦^{D}_{p}(
(dqr cosq cosf -dfr sinq sinf +dr sinq cosf)e_{1}
+ (dqrcos(q) sinf +dfr sinq cosf +dr sinq sinf) e_{2}
+ (-dqr sinq +dr cosq) e_{3}
)
= ...[Tedious manipulations]... =
r^{2}dqe_{q}
+ r^{2} sin^{2}qdfe_{f}
+ e_{r}
.]
Restricting r=1 for S_{2} gives
g_{x}(dx)
= g_{x}(dqe_{q}+dfe_{f}) = dqe_{q} + sin^{2}qdfe_{f}
In much of the literature, notably with regard to Genral Relativity, the metric g_{x} is regarded
as profoundly fundamental. Indeed, it is sometimes known as the fundamental tensor.
However, we will here regard g_{x} as less fundamental than ¦^{Ñ}_{x} which we will also make little
subsequent use of in this chapter, considering orientation M-blade I_{p} described below
as the defining "property" of an M-curve.
M-Curve as an M-blade-valued field
We have seen how the local mapping function ¦ defines a tangent M-frame everywhere on an M-curve.
Suppose alternatively we provide a k-frame at every point of an M-curve C_{M}.
This provides a k-foliation of k-curves over C_{M}. Every p Î C_{M} is contained within
a k-curve having the given k-frame as embedded frame at p. These k-curves are not
contained within C_{M} in general, but will be if the k-frame at p is within I_{p} " pÎC_{M} .
Suppose now that instead of a k-frame for each p we merely have a unit M-blade valued field I_{p}
defined over pÎU^{N}.
If p_{0}ÎU^{N} is known to lie in C_{M} then another U^{N} point p_{1} will lie in C_{M} iff there
exists a path (1-curve)
{ p(t) Î U^{N} : t Î [t_{0},t_{1}] } with p(t_{0})=p_{0} ; p(t_{1})=p_{1} ;
and p'(t)ÎI_{p(t)} " tÎ[t_{0},t_{1}]. Establishing whether this is, in fact,
the case for a given p_{0} and p_{1} may be far from easy.
If all we can say about
¦ with regard to p=¦(x) is that ¦^{Ñ}_{x}(i)=I_{p} then things can get tricky.
( If ¦^{-1}(p_{1}) is available, of course, then p_{1}ÎC_{M} Û ¦^{-1}(p_{1})ÎM_{ap}.)
However, we can nonetheless think of a non-degenerate-M-blade-valued field over U^{N} (ie. one that nowhere vanishes) as defining a foliation
of M-curves over U_{N}, just one M-curve containing any given pÎU^{N}.
If I_{p}=0 at some p then there is no M-curve passing through p and we say the foliation is partial.
A multivector-valued field over U^{N} can thus be 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 U^{N} as determined by a unit M-blade field I_{p} defined over U^{N}.
Hypercurves
An (N-1)-curve is known as a hypercurve and we can define a unit 1-vector
normal by
n_{p} º i_{N-1p}^{*} = i_{N-1p}i^{-1}
= (i)^{2} i_{N-1p}¿i .
n_{p} = (-1)^{N-1}e_{N} in the fortutious basis and
n_{p}^{2} = (-1)^{N-1} i_{N-1p}^{2} i^{2} = ±1
where M=N-1 .
.
Projector 1-multitensor ¯
The projector of M-curve C_{M} is the multitensor
¯(a) º ¯_{Ip}(a) º (a¿I_{p})I_{p}^{-1} .
We also have the rejector multitensor ^ º 1 - ¯ .
From our original discussion of projection into blades we know that ¯^{2}=¯ and ¯(aÙb)=¯(a)Ù¯(b) and that we can regard ¯ as a grade-preserving "idempotent outtermorphic operator" satisfying the "product rule" ¯(¯(a)b) = ¯(a)¯(b). We have defined ¯ explicitly using ¿ here, but much of the following applies to more generally hypothesised projectors satisfying such basic properties.
Hypercurve
For M=N-1 we have ¯_{iN-1p}(a_{p}) = a_{p} - (n_{p}^{-1}¿a_{p})n_{p} .
Integration over an M-curve
We assume here than the concept of scalar integration is well understood. That is,
that ò_{t0}^{t1}F(t) dt is defined for a multivector-valued function F of a scalar t
as the limit of a sum of elemental contributors.
We can immediately adapt this for the line integral
of a multivector valued field F(p)
along a 1-curve p(t) t Î [t_{0},t_{1}]
by defining
ò_{C} F(p) dp º ò_{t0}^{t1}F(p(t))p'(t) dt .
More generally
we have ò_{C} F(p) dp
defined to be the limit of a summation at n-1 of n "samplepoints" p_{[i]} along the 1-curve
. p_{[0]} and p_{[n]} are the given curve endpoints (the same point if integrating round a loop) ]
.
At each sample we form the geometric product
F(p_{[i]}) (Dp_{[i]})
where Dp_{[i]} º (p_{[i+1]}-p_{[i]}) .
As n ® ¥, all the 1-vector Dp_{[i]} ® 0.
Because we have a geometric rather than a scalar multiplication,
ò_{C} F(p)dp ¹ ò_{C} dpF(p)
in general.
In the particular case F(p(t)) = p'(t)^{~} the line integral is a pure scalar,
the conventional arc length of the
curve.
For integration over a 2-curve (ie. a surface integral) we proceed similarly, sampling at n points and evaluating
at each a contributary directed flat triangular area "mesh" element
(ie. a 2-simplex) having orientation I_{Mp[p]} and magnitude the
conventional scalar measure of area (simplex content);
these triangles tesselating to approximate the surface.
If the 2-curve is parameterised as p(x^{1},x^{2}) then the directed area element has form
d^{2}p = ( h_{1p }Ù h_{2p })dx^{1}dx^{2} .
For integrating over an M-curve C_{M}=¦(M_{ap}) we have contributary (M-1)-simplex elements
d^{M}p
= ( h_{1p }Ù h_{2p }.. h_{Mp })dx^{1}dx^{2}..dx^{M}
= |¦^{Ñ}|I_{p}dx^{1}dx^{2}..dx^{M}
and we can think of M succesive scalar integrals of multivector
F(p)I_{p} via
ò_{CM }F(p) d^{m}p =
ò_{CM }F(p)( h_{1p }Ù h_{2p }..Ù h_{Mp })
dx^{1}dx^{2}...dx^{M} .
It can be shown that this limit is independant of the precise nature and geometry of the "mesh" used.
We define the scalar content of an M-curve by
|C_{M}| º ò_{CM} I_{p}^{-1}dp .
This is the conventional arc length, surface area, and volume of C_{M} for M=1,2, and 3 respectively.
Fourier Transform
Having defined integration we can define the Fourier transform.
The (unitary) Fourier transform of a multivector field a_{x} º a(x)
is the field
F(a_{x})(k) º
(2p)^{-½N}
ò d^{N}x i^{-1} (-i(x¿k))^{} a_{x}
= (2p)^{-½N}
ò |d^{N}x| (-i(x¿k))^{} a_{x}
where the x integration is usually taken over all U^{N} rather than a particular subspace of interest,
and i commutes with a_{x} and has i^{2}=-1.
The inverse transform is
F^{-1}(b_{k})(x) º
(2p)^{-½N}
ò d^{N}k i^{-1} (+i(x¿k))^{} b_{k}
where we are now integrating over k ÎU^{N} .
For N=1 we have the scalar unitary Fourier transform
F(a(x)(k) =
(2p)^{-½} ò_{-¥}^{¥ } dx (-ixk)^{} a(x) .
and its inverse
F^{-1}(b(var(k))(x) =
(2p)^{-½} ò_{-¥}^{¥ } dk (+ixk)^{} b(k) .
F((i(x.b)(k)
= (2p)^{-½N}
ò d^{N}x i^{-1} (i(x¿(b-k))^{}
= (2p)^{-½N}
P_{j=1}^{N} ò_{-¥}^{¥} dx_{j} (ix_{j}(b_{j}-k_{j}))^{}
The scalar integrals vanish except when b_{j}=k_{j} when
we (informally) obtain ¥.
Since ò_{-K}^{K} dk ò_{-¥}^{¥} dx cos(xk)
= ò_{-¥}^{¥} dx ò_{-K}^{K} dk cos(xk)
= ò_{-¥}^{¥} dx 2x^{-1} sin(Kx)
= 4 ò_{0}^{¥} dx x^{-1} sin(Kx) = 2p we have
ò_{-¥}^{¥} dx cos(xk) = 2pd(k)
and hence
F((i(x.b)(k) =
(2p)^{½N} d(b-k)
where d(x) = d(x_{1})d(x_{2})...d(x_{N})
is an N-D Dirac delta function.
Of course
F^{-1}(d(b-k))(y) =
(2p)^{-½N}(i(y¿b))^{}
and so
F^{-1}( F((i(x.b)(k) )^{}) ) (y)
= F^{-1}( (2p)^{½N} d(b-k) ) (y)
= (i(y¿b))^{} , and more generally the (2p)^{-½N} factors
in the definitions of F and F^{-1} serve to ensure that
F^{-1}F = 1 .
An obvious geometric generalisation is
F(a_{x})(k) º ò
(2p)^{-½N}
ò d^{N}x i^{-1} (xÙk)^{} a_{x}
with pseudovector k=k^{*}=ki^{-1} providing the conventional 1-vector Fourier transform
when i=i since then
xÙ(k^{*}) = (x.k)^{*} = -i(x.k) .
However for general grade k we must speak of left and right Fourier transforms
since (xÙb)^{} may not commute with a_{x}.
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 Ñ_{p} = Ñ_{b} (b¿Ñ) = Ñ_{b} Ð^{p}_{¯(b)} .
But (b¿Ñ) = Ð_{¯(b)} = 0 if ¯b=0 so we have
Ñ_{p} = Ñ_{b} Ð^{p}_{¯(b)} = Ñ_{b} Ð^{p}_{¯(b)}
.
The map coordinate based definition of the undirected tangential derivative
Ñ_{[CM] p}F(p) of a multivector-valued field
F(p) defined over a neighbourhood of point q within a M-curve C_{M} is
Ñ º
Ñ_{[CM] p} º
Ñ_{[Ip] p} º
å_{k=1}^{M} h^{k}_{p }Ð^{p}_{ hkp }
= å_{k=1}^{M} h^{k}_{p }Ð^{x}_{eðk}
where h^{i}_{p } are a reciprocal embedded frame for C_{M} ;
Ð_{ hkp } operaties in U_{N} ; and
Ð^{x}_{eðk} = (d/dx^{k}) operates in the mapspace M_{ap}
as
Ð_{eðk}F(p) = Ð^{x}_{eðk}F(¦(x))
= (Ñ_{p}¿¦^{Ñ}_{x}(e_{ðk}))F(p)
= F^{Ñp}(¦^{Ñ}_{x}(e_{ðk}))
= F^{Ñp}( h_{kp }) .
When the particular M-curve C_{M} under discussion is unambiguos, we will abbreviate
Ñ_{[CM] p} to Ñ_{p} and thence to Ñ.
We say F(p) is monogenic (aka. analytic) on C_{M} if Ñ F(p) = 0 over C_{M}.
Moving out of the mapspace, suppose instead that e_{1},e_{2},...e_{M} are an orthonormal basis for I_{p} (at a given p only) which we extend by e_{M+1},..e_{N} to
a fortuous universal basis for i at every p, though it coincides with I_{p}
only at the particular p of interest.
Then we can write
Ñ = ¯_{Ip}(Ñ_{p}) = å_{i=1}^{N.} ¯_{Ip}(e^{i})Ð_{ei}
= å_{i=1}^{M.} e^{i}Ð_{ei}
= å_{i=1}^{N.} e^{i}Ð_{¯Ip(ei)} .
For a more general basis for U_{N}, it is natural to define the tangential derivative by
Ñ_{p} º å_{i=1}^{N} e^{i}Ð_{¯Ip(ei)} .
We can also define a orthotangential derivative
Ñ_{^}
º Ñ_{^p}
º Ñ_{p} - Ñ_{p}
= å_{i=1}^{N} e^{i}Ð_{^Ip(ei)}
.
For 1-curve C= { p(t) } we have
Ñ_{[C] p} = h^{1}_{p }(dF(p(t))/dt)
= p'(t)^{-1}(dF(p(t))/dt)
, effectively the derivative with respect to arc length.
We can think of
Ñ_{[CM] p} as being
Ñ_{p} "restricted" to act within M-curve C_{M} ( The symbol Ñ is
visually suggestive of a "portion" of Ñ ).
It is the directed 1-derivative "splayed out" only over directions lieing within I_{p} .
We clearly have Ñ_{[UN] p} = Ñ_{p}
and Ñ_{[1]} = Ñ_{<0>}
.
The use of
h^{k}_{p } in the defintion of Ñ_{[CM]p} "counter scales the expansions"
of ¦^{Ñ} to ensure Ñ_{p} p º Ñ_{[CM]p} p = M .
Though ¯(Ñ) º ¯_{Ip}(Ñ_{p}) is a natural notation for the undirected tangential derivative, we favour Ñ º Ñ_{p} here to minimise confusion with the composition operator (¯_{Ip}Ñ)(a_{p}) º ¯_{Ip}(Ñ_{p}(a_{p})). Nonetheless, it is important to recognise that in essence Ñ = ¯(Ñ) = ¯(Ñ) abbreviating (¯(Ñ_{p}))(a_{p}_{Ñ}) = Ñ_{p}(a_{p}) " a_{p} .
The projection ¯ is equivalent to the tangential 1-differential of the "scalar" identity
multitensor 1_{p}(a)=1(a)=a
and so we have the alternate notation
¯ = 1^{Ñ} abbreviating
¯_{Ip}(a) = 1^{Ñ[Ip]}(a) " multivector a.
[ Proof :
(a¿Ñ_{ðp})p
=(a¿å_{i=1}^{M} h^{i}_{p }Ð^{p}_{ hip })p
=å_{i=1}^{M} (a¿ h^{i}_{p }) h_{ip }
=¯_{Ip}(a)
.]
Perhaps the best symbolic definition is Ñ_{p} º Ñ_{b} Ð^{p}_{¯(b)} .
The symmetry (self-adjointness) of projection gives
Ñ_{*}a_{p} = ¯(Ñ)_{*}a_{p} = Ñ_{*}(¯_{Ip}(a_{p}))
which with 1-vector operands understood we can write as
Ñ¿ = Ñ¿¯ . The tangential divergence (aka. contraction)
is the divergence of the projection.
.
[ Proof :
Ñ_{*}a_{p} = ¯_{Ip}(Ñ)_{*}a_{p}
= Ñ_{*}(¯_{Ip}(a_{p}))
.]
Recall that with regard to its action on functions satisfying the integrability condition Ð_{a}×Ð_{b} = 0 " a,b, we have ÑÙÑ=0 . This is not the case for Ñ, but we will later show that ¯(ÑÙÑ)=0 and ¯(ÑÙÑÙa_{p})=0 " a_{p}.
Normalisation condition I_{p}^{2}=±1 gives ¯_{Ip}(Ð_{a}I_{p})=0 which in turn yields Ñ I_{p} = Ñ Ù I_{p}
For p^{2} ³ 0, Ñ_{[bk]} |p|^{k} = Ñ_{[bk]}( (p^{2})^{½k} = k |p|^{k-2} ¯_{bk}(p) for nondegenerate k-blade b_{k}.
Suppose now that F_{p}=F(p) is a multivector valued field defined over C_{M} with
F_{p} not necessarily within I_{p}.
F_{p} induces a field in the mapspace F_{x} = F_{¦(x)} for which
Ñ_{x}F_{x} need not lie in I_{p} since
Ð_{eðk}F_{x} need not lie in I_{p} .
Even if we insist that F_{p} Î I_{p} so that ¯_{Ip}(F_{p})=F_{p}, then Ð_{eðk}F_{x} still need not lie in I_{p}.
Similarly Ð_{d}F_{p} need not lie in I_{p} even if dÎI_{p} and F_{p}ÎI_{p} " p.
Thus the tangential derivative "acts within" C_{M} but is not "confined to" C_{M} in that it can (at p) "return" multivectors
not contained in C_{M} at p. It differentiates along tangents but is not itself tangent.
For a derivative entirely "within" C_{M} we must look to
the coderivative.
Alternate Ñ Definition
The coordinate-independant definition of the
tangential derivative at q
discussed at length in Hestenes NFMP
is
Ñ_{[CM] p} º
I_{p}^{-1}
Lim_{|O| ® 0}
[ |O|^{-1} ò_{dO } d^{m-1}p F(p) ]
this being the geometric product of the inverted M-orientation I_{p}^{-1} at q and
the limit (finite or otherwise) of the multivector-valued directed integral
ò_{dO } d^{m-1}p F(p)
taken over the boundary dO of a small
M-curve O Ì C_{M}
enclosing q , divided by O content |O| as |O| ® 0 .
For M=N=3 and 1-field F(p) this is equivalent to the conventional integration-based definitions of
divergence and (the dual of) curl.
Indeed, for general N,
Ñ_{p} º Ñ_{[UN] p} provides a coordinate-independant definition of
the del-operator, suggesting a more fundamental consideration of differentiation as the
limit of the quotient of integrals.
One can continue to think in "coordinate" terms even when the symbolism is coordinate
independant and we will here retain the view of Ñ_{p} as a "splaying" of multivector-directed derivatives.
Our fundamental view of Ñ is essentially "that which satisfies" a_{*}Ñ = Ð_{a}
Fundamental Theorem of Calculus
Having defined integration over and differentaiation within an M-curve we can formulate 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.
ò_{dCM} d^{M-1}p F(p) = ò_{CM} d^{M}p (Ñ_{p}F(p)) º ò_{CM} d^{M}p Ñ_{[CM]}F(p) . where scaled M-blade d^{M}p = I_{p}d^{M}p = I_{p}|d^{M}p| is an elemental M-simplex for C_{M} at p.
The proof [ommitted here, see Hestenes] is an almost trivial consequence of the coordinate-independant definition Ñ_{[CM]}F . It has the following consequences.
A more general form of interest is
ò_{dCM} G_{p} d^{M-1}p F_{p}
= (-1)^{M-1}ò_{CM} (G_{p}Ñ_{p}) d^{M}p F_{p}
+ ò_{CM} G_{p} d^{M}p (Ñ_{p} F_{p}) _{[ HS 7-3.10 ]}
[ Proof :
ò_{dCM} G_{p} d^{M-1}p F_{p}
=
ò_{CM} G_{p}_{Ñ} d^{M}pÑ_{p} F_{p}
+ ò_{CM} G_{p} d^{M}pÑ_{p} F_{p}_{Ñ}
and recall that d^{M}p is a scaled M-blade pseudoscalar for the tangent space containing Ñ
and so G_{p}_{Ñ} d^{M}pÑ_{p} =
G_{p}_{Ñ}(d^{M}p.Ñ_{p}) =
(-1)^{M-1}G_{p}_{Ñ}(Ñ_{p}.d^{M}p) =
(-1)^{M-1}G_{p}_{Ñ}(Ñ_{p}d^{M}p) =
(-1)^{M-1}(G_{p}Ñ_{p})d^{M}p
.]
Greens Functions
A Greens function is a function used to express a solution to a differential equation with particular boundary conditions
as a definite integral. For example,
d/dx^{2} y(x) = ¦(x, y(x)) subject to y(a)=y_{0} and y'(a)=y_{0}' has solution
y(x) = y_{0} + y_{0}'(x-a) + ò_{a}^{x} dx"
ò_{a}^{x"} dx' ¦(x',y(x')) which can
be alternatively evaluated as
y(x) = y_{0} + y_{0}'(x-a) + ò_{a}^{x} dx' (x-x')¦(x',y(x'))
=
y_{0} + y_{0}'(x-a) + ò_{a}^{b} dx' G(x,x') ¦(x',y(x'))
where b³x and G(x,x') = (x-x') Hvsd(x-x') is the Greens function for
(d/dx)^{2} for the boundary conditions y(a)=y_{0} ; y'(a)=y_{0}' .
[ Hvsd(x) is the Heaviside step function zero for x£0 and 1 for x>0,
exploited to allow us to replace the indefinite ò_{a}^{x} dx' with the definite
ò_{a}^{b} dx'
]
If G_{p,q} is a 1-vector-valued Green's function having two primary C_{M} point-valued arguments with
Ñ_{p} G_{p,q}
= G_{p,q} Ñ_{q} = 0 " p¹q
and Ñ_{p}G_{p,q} = -G_{p,q}Ñ_{q} = 1 at p=q, the Fundamental Theorem of Calculus provides
ò_{dCM} G_{p,q} d^{M-1}p F_{p}
= (-1)^{M-1}I_{Mq} F_{q}
+ ò_{CM} G_{p,q} d^{M}p (Ñ_{p} F_{p})
which we can write as
F_{q} = (-1)^{M}I_{Mq}^{-1} (
ò_{dCM} G_{p,q} d^{M-1}p F_{p}
- ò_{CM} G_{p,q} d^{M}p (Ñ_{p} F_{p}) )
_{[ HS 7-4.7 ]}
expressing an interior value F_{q} of F_{p} in terms of boundary values of F_{p} and interior values of Ñ_{p}F_{p} .
In essence, this provides a Ñ ^{-1} in that we can reconstruct F_{p} from ÑF_{p} provided we also have
"boundary contraints" specifying F_{p} over an enclosing surface.
If G_{p,q} Î I_{Mq} " p Î dC_{M}
we can commute I_{Mq} across G_{p,q} , incurring a sign change if M is even, to obtain
F_{q} =
- ò_{dCM} G_{p,q} I_{Mq}^{-1} d^{M-1}p F_{p}
+ ò_{CM} G_{p,q} I_{Mq}^{-1} d^{M}p (Ñ_{p} F_{p})
If F_{p} is monogenic so that
ò_{CM} G_{p,q} d^{M}p (Ñ_{p} F_{p}) vanishes,
we have a Geometric generalisation of Cauchy's Theorem
F_{q} = (-1)^{M}I_{Mq}^{-1} ò_{dCM} G_{p,q} d^{M-1}p F_{p}
= - ò_{dCM} G_{p,q} I_{Mq}^{-1} d^{M-1}p F_{p}
_{[ HS 7-4.10 ]} .
With M=N and I_{p}=i in a Euclidean space we have monogenic G_{p,q} = o_{N}^{-1} (p-q) |p-q|^{-N}
as the Greens function for Ñ_{p}.
General Form
Included for completeness. Casual reader should skip.
A differentiable M-form on an M-curve C_{M} is a point-dependant multivector-valued
linear function of the directed measure
L(p,d^{M}p)
= dp^{1}dp^{2}...dp^{M}L(x, h_{1p }Ù h_{2p }Ù.. h_{Mp }) .
The exterior differential of an (M-1)-form is the M-form defined by
dL(p,d^{M-1}p) º
Lim_{O ® 0} ò_{O} L(p,d^{M}p) / |O|
where O is a small volume enclosing x.
The general form of the fundamental theorem is
ò_{CM} dL(p,d^{M}p)
= ò_{dCM} L(p,d^{M-1}p)
but we will not pursue this here.
Poles and Residues
Cauchy's Theorem
Relates the value of a monogenic function at a point to the surface integral over the boundary
of any M-curve including the point.
F(a)
= (I_{Ma}o_{N})^{-1}
ò_{dCM} (p-a)|p-a|^{-N} _d2p F(p)
where C_{M} is a closed M-curve including a in its interior and I_{Ma}
is the orientation for C_{M} at a.
Thus the surface integral over a closed surface of a monogenic function is independant of (and so extremal for)
the actual surface integrated over, insofar as we can vary the surface without effecting the result
provided we don't move the boundary across any poles of F.
The surface integral is zero unless the enclosed volume contains one or more poles of F, in which case
the integral is the sum of the "residues" at the poles , each independant of the surface used. Closed
surface integrals of monogenic functions are thus discrete-valued and tend to vary discontinuously.
They provide a "regional quantisation" .
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 f_{1}(a) is an operator taking one parameter then f_{1}^{×}(a,b) º ½(f_{1}(a)×f_{1}(b))
If f_{2}(a,b) is an operator taking two parameters then f_{2}^{×}(a,b) º ½(f_{2}(b,a)-f_{2}(a,b))
f_{1}_{<k>}(a) º f_{1}(a_{<k>})
We define f_{1}^{¨} by f_{1}^{¨} (a) ¨ a = f_{1}(a) where ¨ is a multivector product.
Thus, for example, if tensor f_{p}(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 Ñ_{p} 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
f_{1}^{Ñ}g_{1} º (f_{1}^{Ñ})g_{1} ? The "differentialiser" ^{Ñ} converts
f_{1} 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
f_{1}^{Ñ}(a,b)
= (Ð_{b}f_{1})(a)
= Ð_{b}(f_{1}(a)) - f_{1}(Ð_{b}a) .
We will interpret the g_{1} in the "composite product" f_{1}^{Ñ}g_{1} as applying to the
"newest" rightmost nonprimary parameter so that
f_{1}^{Ñ}g_{1}(a,b)
º
f_{1}^{Ñ}(a,g_{1}(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
f_{1}^{Ñ}g_{1} 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
f_{1}^{Ñ}g_{1}(a,b) º f_{1}^{Ñ}(a,g_{1}(b)) as the LIFO convention.
Tangential 1-differential ^{Ñ}
We can define the tangential 1-differential of field F_{p}
by
F^{Ñ[CM]p}_{p}(a)
º
F_{p}^{Ñ}(a)
º (a_{*}Ñ_{[CM]p})F(p)
= (a_{*}¯(Ñ_{p}))F(p)
= (¯(a)_{*}Ñ_{p})F(p)
= F_{p}^{Ñ}(¯(a))
º F_{p}^{Ñ}¯(a)
which gives us
^{Ñ} = ^{Ñ}¯
abbreviating
F_{p}^{Ñ}(a) =
F_{p}^{Ñ}¯(a)
º
F_{p}^{Ñ}(¯(a)) .
; the a-directed tangential derivative
is equivalent to the ¯(a)-directed derivative.
Consider now (ÑF_{p})^{Ñ}(a) = ¯^{Ñ}(Ñ_{b},a)F_{p}^{Ñ}(b) + Ñ_{b}F_{p}^{Ñ2}(b,a) _{[ HS 4-1.18 ]}
More generally suppose F_{p}(a_{1},a_{2},...a_{k}) is an extensor tensor of k multivector variables
themselves p-dependant. We can take a 1-differential of the secondary differntial
Ñ_{a1}F_{p}(a_{1},a_{2},...a_{k}) to give
(Ñ_{a1}F_{p})^{Ñ}(a_{1},..,a_{k},b) =
¯^{Ñ}(Ñ_{a1},b)F_{p}(a_{1},..,a_{k})
+ Ñ_{a1}F_{p}^{Ñ}(a_{1},..,a_{k},b) .
For 1-vector a_{1} = ¯_{a1} we can write this as
(b_{*}Ñ)Ñ_{Þ} = ¯^{Ñ}(Ñ_{Þ}, b) + Ñ_{Þ}(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 (Ñ_{ß}¿Ñ_{Þ}) is equivalent to replacing the first nonprimary argument
of a tensor with Ñ_{Ü} .
[ Proof : (Ñ_{a}¿Ñ_{ß})F_{p}(a,b,..)
= å_{i=1}^{M} e^{i}^{2} Ð^{a}_{ei}(Ð^{p}_{ei}F_{p})(a,b,...)
= å_{i=1}^{M} (Ð^{p}_{ei}F_{p})(e^{i}^{2}e_{i},b,...)
= å_{i=1}^{M} (Ð^{p}_{ei}F_{p})(e^{i},b,...)
= å_{i=1}^{M} F_{p}_{Ñ}(Ð^{p}_{ei}e^{i},b,...)
= F_{p}_{Ñ}(Ñ,b,...)
.]
Hypercurve
For M=N-1 we have Ñ = ¯(Ñ) = Ñ - n_{p}(n_{p}¿Ñ) = Ñ - n_{p}Ð_{np} .
Since n_{p}¿Ð_{a}n_{p} = 0 " a , Ñ_{p}n_{p} lies within i_{N-1p} and
^[Ñ_{p}n_{p}] = n_{p}^{-1}ÙÐ_{np}n_{p} .
More generally, let n_{p} be a unit 1-field.
Ñ_{p} =
n_{p}^{2}Ñ_{p} =
n_{p}(n_{p}¿Ñ_{p})
+ n_{p}(n_{p}ÙÑ_{p})
corresponding to tangential and orthogonal components of
Ñ_{p} .
Directed Coderivative Ð_{()}
With regard to a multivector field F_{p} defined over C_{M} (but not necessarily confined within I_{p}) we can most simply define the
a-directed coderivative for aÎU_{N} as the projection of the a-directed tangential derivative, ie. the projection
of the ¯_{Ip}(a)-directed dirivative
Ð_{} = ¯ Ð_{¯()} º ¯_{°} Ð_{¯()}
abbreviating Ð_{a}F_{p} º ¯_{Ip}Ð_{¯Ipa}F_{p}
The symbol Ð_{} can thus be thought of as an abbreviation
Ð_{} = ¯ Ð_{¯()} = Ð_{¯()}¯ - ¯^{Ñ}
abbreviating
Ð_{b}(a_{p}) = ¯( Ð_{¯(b)}a_{p}) = Ð_{¯(b)}(¯(a_{p})) - ¯^{Ñ}(a_{p},b) _{[ HS 5-1.1 ]}
. The underscore serves to remind us of a p and I_{p} (ie. C_{M}) dependance.
[ Proof : Ð_{¯(d)}(¯(a_{p})) = ( Ð_{¯(d)}¯)(a_{p}) + ¯( Ð_{¯(ap)}) Þ
Ð_{d}(a_{p}) º ¯( Ð_{¯(d)}(a_{p})) = Ð_{¯(d)}(¯(a_{p})) - ( Ð_{¯(d)}¯)(a_{p})
= Ð_{¯(d)}(¯(a_{p})) - ¯^{Ñ}(a_{p})
.]
The directed coderivative of an extended field is defined by
(Ð_{d}F_{p})(a_{1},a_{2},...)
º Ð_{d}(F_{p}(a_{1},a_{2},..)) - F_{p}(Ð_{d}a_{1},a_{2},...)- F_{p}(a_{1},Ð_{d}a_{2},...)
= ¯( Ð_{¯(d)}(F_{p}(a_{1},a_{2},..)) ) - F_{p}(¯( Ð_{¯(d)}a_{1}),a_{2},...)- F_{p}(a_{1},¯( Ð_{¯(d)}a_{2}),...)
Note that this differs from _{[ HS 4-3.3 ]} which inserts subtracts
- F_{p}(Ð_{d}¯(a_{1}),a_{2},...)- F_{p}(a_{1},¯( Ð_{¯(d)}a_{2}),...)
Coderivative Ñ
The covariant derivative Ð_{a} 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 Ñ_{p}F_{p} Î I_{p} ; Ð_{p}a_{1}_{p} + Ð_{p}a_{2}_{p} = Ð_{p}(a_{1}_{p}+a_{2}_{p}) ; 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
ÑF_{p} º Ñ_{[Ip]p}F_{p} º ¯_{Ip}(Ñ_{[Ip]p}F_{p})
which we can express operationally as Ñ º ¯Ñ
noting carefully that this denotes operator composition
¯_{°}Ñ º ¯(Ñ(a_{p})) rather than
¯(Ñ)(a_{p})
= ¯(¯(Ñ))(a_{p})
= ¯(Ñ)(a_{p})
= Ñ(a_{p})
. 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 Ð_{a} = (a¿Ñ_{p}) .
Since
ÑF_{p} Î I_{p} for a scalar field F_{p}
the coderivative operator Ñ_{p} is equivalent to the tangential
derivative Ñ_{p} when acting on scalar fields, and for aÎI_{p} the a-directed coderivative
Ð_{a} is equivalent to the a-directed derivative Ð_{a} when acting on scalar fields.
Ñ_{p}(a_{p}) º ¯(Ñ_{p}(a_{p})) = Ñ_{p}(¯(a_{p}_{Ñ}))
by the projected product rule .
This is particularly clear when expressed in coordinate terms with a fortuitous basis as
Ñ_{p}(¯(a_{p}_{Ñ}))
= å_{i=1}^{N} e^{i} Ð_{¯(ei)}(¯(a_{p}_{Ñ}))
= å_{i=1}^{M} e^{i}(¯( Ð_{¯(ei)}a_{p}))
= å_{i=1}^{M} ¯(e^{i})(¯( Ð_{¯(ei)}a_{p}))
= å_{i=1}^{M} ¯(e^{i}( Ð_{¯(ei)}a_{p}))
= ¯(e^{i}(å_{i=1}^{M} Ð_{¯(ei)}a_{p}))
= ¯(Ñ_{p}(Ap))
º Ñ_{p}(Ap)
.
For an extended field F_{p}(a_{1}_{p},a_{2}_{p},...a_{k}_{p}) with a_{i}_{p}ÎU_{N} we have
a-directed primary coderivative
Ð_{¯}_{a}F_{p}(a_{1}_{p},a_{2}_{p},...a_{k}_{p}) º
(Ð_{a}F_{p})(a_{1}_{p},a_{2}_{p},...a_{k}_{p})
= Ð_{a}(F_{p}(a_{1}_{p},a_{2}_{p},...a_{k}_{p}))
- F_{p}(Ð_{a}(¯_{Ip}(a_{1}_{p})),a_{2}_{p},...a_{k}_{p})
- F_{p}(a_{i}_{p},Ð_{a}(¯_{Ip}(a_{2}_{p})),,...a_{k}_{p})
- F_{p}(a_{i}_{p},a_{2}_{p},...,Ð_{a}(¯_{Ip}(a_{k}_{p})))
and an i^{th} exterior coderivative
Ñ^{i®¿}_{p} F_{p}(a_{1}_{p},a_{2}_{p},...a_{k}_{p})
º
¯_{Ip}(Ñ^{i®¿}_{p} F_{p}(a_{1}_{p},a_{2}_{p},...a_{k}_{p}))
, the projection of the i^{th} conveyed divergence.
The cogradifying substitution rule that applying the operator (Ñ_{ß}¿Ñ_{Þ}) = (Ñ_{ß}¿_csgrad) º ¯_{°}(Ñ_{ß}¿Ñ_{Þ}) is equivalent to replacing the first nonprimary parameter with Ñ_{Ü} defined by follows immediately as the projection of tne tangential gradifying substitution rule.
We will see that ÑÙÑÙa_{p} = 0 " path-independant a_{p} , so that Ñ^{2} , while not a scalar operator, does not increase grade .
Hypercurve
For M=N-1 we have Ñn_{p} = Ñn_{p} .
Projection Differential (1.2)-tensor ¯^{Ñ}
¯^{Ñ}(a_{p},d) º ¯_{Ip}^{Ñp}(a_{p},d) º
Ð_{d}(¯_{Ip}(a_{p})) - ¯_{Ip}(Ð_{d}a_{p})
(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}(a_{p},d) =
¯^{Ñp}(a_{p},¯d) =
1^{Ñ2}(a_{p},d)
= ( Ð_{¯(d)}¯_{Ip})(a_{p})
= Ð_{¯(d)}(¯(a_{p})) - ¯( Ð_{¯(d)}a_{p})
= 2( Ð_{¯(d)}×¯)(a_{p}) .
The symmetry
¯^{Ñ}(a,b)
=¯^{Ñ}(b,a)
for a,b Î I_{p} follows from the symmetry of 1^{Ñ2} in U_{N}.
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}(a_{1}Ùa_{2}Ù....a_{k},d)
= å_{i=1}^{k} (-1)^{i+1}
¯^{Ñ}(a_{i},d)Ù¯_{Ip}(a_{1})Ù...
¯_{Ip}(a_{i-1})Ù
¯_{Ip}(a_{i+1})Ù...
¯_{Ip}(a_{k}) .
_{[ 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}(I_{p},d) = 0 and hence ¯^{Ñ}(I_{p},d)¿I_{p} = 0
" d .
Ñ_{Þ}¿(¯^{Ñ}¯) = 0 abbreviating
Ñ_{b}¿¯^{Ñ}(¯a,b) = 0 " a,b _{[ HS 4-2.17a ]} .
[ Proof : Ñ_{Þ}¿¯^{Ñ}(¯a,b) (¯(Ñ_{Þ}))¿¯^{Ñ}b)
= Ñ_{Þ}¿(¯¯^{Ñ}b)) = Ñ_{Þ}¿0 .
.]
Ñ_{Þ}Ù(¯^{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
(ÑF_{p})^{Ñ}(a) = ¯^{Ñ}(Ñ_{a},b)F_{p}^{Ñ}(a))
+ Ñ_{a}F_{p}^{Ñ2}(a,b)
provided aÎI_{p} .
[ Proof :
aÎI_{p} Þ F_{p}^{Ñ}(a)
= (a¿Ñ)F_{p}
= (¯(a)¿Ñ)F_{p}
= (¯(a)¿Ñ)F_{p}
= F_{p}^{Ñ}(¯(a))
.]
Hypercurve
For M=N-1 we have ¯^{Ñ}(a_{p},b) = (Ð_{b}¯)(a_{p})
= -n_{p}^{2}((a_{p}¿n_{p}^{Ñ}(b))n_{p} + (a_{p}¿n_{p})n_{p}^{Ñ}(b)) .
Thus ¯^{Ñ}(¯(a_{p}),b)
= -n_{p}^{2} (¯(a_{p})¿n_{p}^{Ñ}(b))n_{p}
= -n_{p}^{2} (a_{p}¿n_{p}^{Ñ}(b))n_{p} ;
Thus ¯^{Ñ}(n_{p},b) = -n_{p}^{Ñ}(b) .
, the normalisation condition on n_{p} providing n_{p}¿n_{p}^{Ñ}(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}(¯F_{p},b,a) - (¯^{Ñ})^{2}(¯F_{p},a,b) =
Ð_{a}(Ð_{b}(¯F_{p}))-Ð_{b}(Ð_{a}(¯F_{p}))
[ Proof : Recalling ¯_{=}¯^{Ñ}=0 we have:
Ð_{a}Ð_{b}(¯F_{p})
= Ð_{a}( Ð_{¯(b)}(¯F_{p}) - ¯^{Ñ}(¯F_{p},b) )
= ¯( Ð_{¯(a)}( Ð_{¯(b)}(¯F_{p}) - ¯^{Ñ}(¯F_{p},b) ))
= ¯_{=}( Ð_{¯(a)} Ð_{¯(b)})(F_{p}) - ¯( Ð_{¯(a)}(¯^{Ñ}(¯F_{p},b)) )
= ¯_{=}( Ð_{¯(a)} Ð_{¯(b)})(F_{p})
- ¯[ ( Ð_{¯(a)}¯^{Ñ})(¯F_{p},b)
+ ¯^{Ñ}( Ð_{¯(a)}(¯F_{p}),b)
+ ¯^{Ñ}(¯F_{p}, Ð_{¯(a)}b) ]
= ¯_{=}( Ð_{¯(a)} Ð_{¯(b)})(F_{p})
- ¯¯^{Ñ2}(¯F_{p},b,a)
- ¯¯^{Ñ}( Ð_{¯(a)}(¯F_{p}),b)
= ¯_{=}( Ð_{¯(a)} Ð_{¯(b)})(F_{p})
- ¯_{=}(¯^{Ñ2})(F_{p},b,a)
- ¯¯^{Ñ}( ( Ð_{¯(a)}¯)F_{p})+¯( Ð_{¯(a)}F_{p}),b)
= ¯_{=}( Ð_{¯(a)} Ð_{¯(b)})(F_{p})
- ¯_{=}(¯^{Ñ2})(F_{p},b,a)
- ¯(¯^{Ñ})^{2}(F_{p},a,b)
Þ (Ð_{a}×Ð_{b})F_{p}
=
(¯^{Ñ})^{2}^{×}(F_{p},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 C_{M} to be the undirected tangential 1-derivative
of the projector , an abbreviation of
[Ñ¯](a_{p}) º Ñ_{ß}_{[CM]p}¯_{Ip}(a_{p})
º Ñ_{ß} ¯_{Ip}(a_{p})
= (Ñ_{b} Ð^{p}_{¯(b)} ¯_{Ip})(a_{p})
= Ñ_{b} ¯^{Ñ}_{}(a_{p},b)
= Ñ_{b} ¯^{Ñ}_{}(a_{p},b)
= Ñ_{b} ¯^{Ñ}(a_{p},b)
.
[Ñ¯](a_{p}) = 0 so [Ñ¯] annihilates scalars.
[Ñ¯](a_{p}) decomposes as [Ñ¯](^_{Ip}(a_{p})) + [Ñ¯](¯_{Ip}(a_{p}))
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(¯(a_{p})) = Ñ(¯(a_{p})) + ^Ñ(¯(a_{p}))
= Ñ¯(a_{p}) + ^( [Ñ¯](a_{p}) + Ñ¯(a_{p}_{Ñ}) )
= Ñ¯(a_{p}) + ^([Ñ¯](a_{p}) + Ñ(a_{p}))
= Ñ¯(a_{p}) + ^[Ñ¯](a_{p})
= Ñ¯(a_{p}) + [Ñ¯]¯(a_{p})
= (Ñ+[Ñ¯])¯(a_{p}) .
.]
With tangential operands ¯(a_{p})=a_{p} 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)) = (Ñ_{p}¿¯_{Ip})(a)
is expressible
as (Ñ_{b}[ÑÙ¯](b))¿a where
1-vector
Ñ_{b}[ÑÙ¯](b) =
º Ñ_{Þ}[ÑÙ¯](b) =
Ñ_{Þ}¿[ÑÙ¯](b)
º Ñ_{Þ}¿[ÑÙ¯](b)
[¯(Ñ)]
= ¯_{Ñ}(Ñ) = Ñ_{Þ} [ÑÙ¯] = Ñ_{Þ}¿[ÑÙ¯]
= [Ñ¿¯]^{¿}
_{[ HS 4-2.20 ]}
is known as the spur 1-field of C_{M} and lies outside I_{p} .
[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) + (Ñ_{b}[ÑÙ¯](b)).a .
1-vector ¯_{Ip}(b)¿[Ñ¯](a) = ¯_{Ip}(b)¿[ÑÙ¯](a) is normal to C_{M} , ie. ¯_{Ip }( ¯_{Ip}(b)¿[Ñ¯](a)) = 0 .
[Ñ¯](a_{1}Ùa_{2}Ù...a_{k}) = å_{i=1}^{k.} (-1)^{i+1} [ [Ñ¯](a_{i}) Ù ¯_{Ip}(a_{1}Ù..Ùa_{i-1}Ùa_{i+1}..Ùa_{k}) + ¯_{Ip}([Ñ¯](a_{i}) × (a_{1}Ù..Ùa_{i-1}Ùa_{i+1}..Ùa_{k}) ] _{[ HS 2.41c ]} provides the extension of [Ñ¯] to general multivectors.
[Ñ¯] = Ñ_{Þ2} ¯^{Ñ}
abbreviating [Ñ¯](a_{p}) = Ñ_{b}¯^{Ñ}b)
expressing the shape as the secondary tangential derivative of ¯^{Ñ}
_{[ HS 4-2.14 ]}.
[ Proof : Ð^{b}_{d} ¯^{Ñ}(a_{p},b)
= Ð^{b}_{d}( Ð_{¯(b)}(¯(a_{p})) - ¯( Ð_{¯(b)}a_{p}))
= Lim_{e ® 0} e^{-1}[ Ð_{¯(b+ed)}(¯(a_{p})) - ¯( Ð_{¯(b+ed)}a_{p}))
- Ð_{¯(b)}(¯(a_{p})) - ¯( Ð_{¯(b)}a_{p}))]
= Ð_{¯(d)}(¯(a_{p})) - ¯( Ð_{¯(d)}a_{p}))
= ( Ð_{¯(d)}¯)(a_{p})
so we have both
Ñ_{ß} ¯(a_{p})
º (Ñ¯)(a_{p}) = Ñ_{b}¯^{Ñ}(a_{p},b) º Ñ_{Þ2}¯^{Ñ}(a_{p},b)
and
Ñ_{¯ß} ¯(a_{p})
º [Ñ¯](a_{p}) = Ñ_{b}¯^{Ñ}(a_{p},b) º Ñ_{Þ2}¯^{Ñ}(a_{p},b)
Since ¯^{Ñ}(a,b)=¯^{Ñ}(b,a) the result follows.
.]
The grade-tangency associations of [Ñ¯] immediately provide
[Ñ¯]^ =
[Ñ¿¯] = Ñ_{Þ2}¿¯^{Ñ}
;
[Ñ¯]¯ =
[ÑÙ¯] = Ñ_{Þ2}Ù¯^{Ñ}
_{[ HS 4-2.16 ]}.
[ Proof : [Ñ¯](a) º (Ñ_{p}¯))(a) = Ñ_{b}¯^{Ñp}(a,b) so .... ???
.]
Ñ^{2} = [Ñ¯](Ñ) + ¯Ñ^{2}
[ Proof : Ñ^{2}(a_{p}) =Ñ(¯_{Ñ}(a_{p})) = [Ñ¯](Ñ) + ¯_{Ñ(Ñ(ap))}
= [Ñ¯](Ñ)(a_{p}) + ¯_{Ñ(Ñ(ap))}
= [Ñ¯](Ñ)(a_{p}) + ¯_{ÑÑ(ap)}
= [Ñ¯](Ñ)(a_{p}) + ¯_{¯ÑÑ(ap)}
= [Ñ¯](Ñ)(a_{p}) + ¯_{Ñ2(ap)}
= [Ñ¯](Ñ)(a_{p}) + ¯_{(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:
[Ñ¯] = Ñ_{Þ} ¯^{Ñ} enables us to grade extend [Ñ¯] from the grade extension of_prl0g()
as
[Ñ¯](aÙb) = Ñ_{c}¯^{Ñ}(aÙb,c)
= Ñ_{c}(¯^{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¯](¯(a_{j})Ù¯(b))
= ([Ñ¯](¯(a_{j}))Ù¯(b) + (-1)^{j}¯(a_{j})Ù([Ñ¯]¯(b))
_{[ HS 4-2.40a ]}
[ Proof :
[Ñ¯](¯(a_{j})Ù¯(b_{k}))
= [Ñ¯]¯(a_{j}Ùb_{k})
= [ÑÙ¯](a_{j}Ùb_{k})
= Ñ_{c}Ù( ¯^{Ñ}(¯(a_{j}),c)Ù¯(b_{k}) + (¯(a_{j})Ù¯^{Ñ}(¯(b_{k}),c) )
= Ñ_{c}Ù( ¯^{Ñ}(¯(a_{j}),c)Ù¯(b_{k}) + (-1)^{jk}¯^{Ñ}(¯(b_{k}),c)Ù¯(a_{j}) )
= (Ñ_{c}Ù¯^{Ñ}(¯(a_{j}),c))Ù¯(b_{k})
+ (-1)^{_jk}(Ñ_{c}Ù¯^{Ñ}(¯(b),c))Ù¯(a_{j})
= ([ÑÙ¯](¯(a_{j}))Ù¯(b_{k}) + (-1)^{jk}([ÑÙ¯]¯(b))Ù¯(a_{j})
= ([ÑÙ¯](¯(a_{j}))Ù¯(b_{k}) + (-1)^{jk+j(k+1)}¯(a_{j})Ù[ÑÙ¯]¯(b_{k})
= ([Ñ¯](¯(a_{j}))Ù¯(b_{k}) + (-1)^{j}¯(a_{j})Ù([Ñ¯]¯(b_{k}))
.]
In particular, [Ñ¯]¯(aÙb) = [Ñ¯]¯(a)Ù¯(b)-[Ñ¯]¯(b)Ù¯(a) .
[Ñ¯](^(a_{j})Ùb) =
[Ñ¯](^(a_{j})Ù¯(b)) =
[Ñ¯](^(a_{j}))Ù¯(b)
+ (-1)^{j}¯_{Ñ}(^(a_{j}))Ù(Ñ¿b) _{[ HS 4-2.40b ]}
[ Proof :
[Ñ¯](^(a_{j})Ù¯(b_{k}))
= Ñ_{c}¯^{Ñ}(^(a_{j})Ù¯(b_{k}) ,c)
= Ñ_{c}(¯^{Ñ}(^(a_{j}),c)Ù¯(b_{k}))
= Ñ_{c}¿(¯^{Ñ}(^(a_{j}),c)Ù¯(b_{k}))
+ (Ñ_{c}Ù¯^{Ñ}(^(a_{j}),c))Ù¯(b_{k})
= (Ñ_{c}.¯^{Ñ}(^(a_{j}),c))Ù¯(b_{k}) + (-1)^{j} ¯^{Ñ}(^(a_{j}),c)(Ñ_{c}.¯(b_{k}))
+ [ÑÙ¯](^(a_{j}))Ù¯(b_{k}) by the expanded inner product rule
= ([Ñ¯]^(a_{j}) + (-1)^{j} ¯^{Ñ}(^(a_{j}),c)(Ñ_{c}.¯(b_{k}))
+ 0
.]
====
The primary 1-differential of the extended curl (2;1)-tensor is
[ÑÙ¯]^{Ñ}(a,b) º Ð_{¯(b)} [ÑÙ¯](a_{p}) - [ÑÙ¯]( Ð_{¯(b)}(a_{p}))
.
Hypercurve
For M=N-1 we have [ÑÙ¯](a) = n_{p}^{2} n_{p}n_{p}^{Ñ}(a) = n_{p}^{2} n_{p}Ùn_{p}^{Ñ}(a)
[ Proof : Ð_{b}i_{N-1p}
= Ð_{b}(n_{p}i)
= (Ð_{b}n_{p})i
= n_{p}^{Ñ}(b)i so Shape Property 7 gives
[ÑÙ¯](a) =
i_{N-1p}^{-1}( Ð_{¯(a)}i_{N-1p})
= i^{-1}n_{p}^{-1}n_{p}^{Ñ}(a)i
= n_{p}^{2} _psiinvd(n_{p}n_{p}^{Ñ}(a))
= n_{p}^{2} _psiinvd(n_{p}Ùn_{p}^{Ñ}(a))
= n_{p}^{2} n_{p}Ùn_{p}^{Ñ}(a) since i commutes with all bivectors.
.]
[¯(Ñ)] = n_{p}^{2} n_{p}(Ñ_{p}¿n_{p})
= n_{p}^{2} n_{p}(Ñ_{p}¿n_{p})
[ Proof :
Ñ_{a}[ÑÙ¯](a) = -n_{p}^{2} Ñ_{a}[n_{p}^{Ñ}(a)n_{p}]
= -n_{p}^{2} (Ñ_{a}n_{p}^{Ñ}(a))n_{p}
= -n_{p}^{2} (Ñ_{a}Ð_{a}n_{p})n_{p}
= -n_{p}^{2} (Ñ_{p}n_{p})n_{p}
= -n_{p}^{2} n_{p}(Ñ_{p}¿n_{p})
.]
Thus [Ñ¿¯](a) = [¯(Ñ)].a = -n_{p}^{2} (Ñ¿n_{p})(a_{p}¿n_{p})
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 I_{p}) 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 I_{p} .
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}(¯(a_{p}))
= (ÑÙÑ)(¯(a_{p}))
= (ÑÙÑ)×(¯(a_{p})) " a_{p} .
Hence ÑÙÑ is grade-preserving.
[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) = -Ñ_{p}[ÑÙ¯](b) = -Ñ_{p}¿[ÑÙ¯](b) .
_{[ HS 5-1.19 ]}
[ Proof :
Ñ_{a} ¯[Ñ¯]^{×}(a,b) =
-Ñ_{a}Ð_{a}[ÑÙ¯](b)
= -Ñ_{p}[ÑÙ¯](b) by Curvature Identity 2 below
.]
¯[Ñ¯]^{2} is symmetric (self-adjoint).
[ Proof :
???
.]
Curvature 2-tensor [Ñ¯]^{×}
Recall that the commutator product of two bivectors is itself a bivector.
If I_{p} satisfies the integrability condition then
the antisymmetric full curvature 2-tensor
c_{paÙb} 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}(a_{p},b) º Ð_{¯(b)}([ÑÙ¯](a_{p})) - [ÑÙ¯]( Ð_{¯(b)}a_{p}) .
[ 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 Ð_{ßc}([Ñ¯]^{×}(a,b))
º (Ð_{c}([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} Ð_{ßc}([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. 1^{Ñ2} [ÑÙ¯] = - 1^{Ñ} [ÑÙ¯]^{×} = - 1^{Ñ} [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 I_{p} 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 I_{p} 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 : Ð_{ßc} ¯[Ñ¯]^{×}(aÙb)
º (Ð_{c} ¯[Ñ¯]^{×})(aÙb)
= ¯^{Ñ} , c)) + ¯( Ð_{ßc}[Ñ¯]^{×}(aÙb))
Þ S_{¬abc} (Ð_{c} ¯[Ñ¯]^{×})(aÙb)
= S_{¬abc} ¯([Ñ¯]^{×}(aÙb))×[Ñ¯](c) .
Hence
S_{¬abc} (Ð_{c} ¯[Ñ¯]^{×})(aÙb)
= S_{¬abc} ¯^{Ñ} , c) + ¯( S_{¬abc} Ð_{ßc}[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} (Ð_{c} ¯[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 I_{p}
_{[ HS 4-2.48 ]}.
Since [Ñ¯]^{2} preserves grade we must have
Ñ_{a}¿[Ñ¯]^{2}(aÙb) = -2Ñ_{a}¿(¯[Ñ¯]^{×}(a,b))
= -2¯(Ñ_{a})¿([Ñ¯]^{×}(a,b))
= -2Ñ_{a}¿([Ñ¯]^{×}(a,b))
and similarly
Ñ_{a}¿[Ñ¯]^{2}(aÙb) = -2Ñ_{a}¿(¯[Ñ¯]^{×}(a,b))
= [Ñ¯]^{2}(b) .
The Riemann curvature is protractionless
Ñ_{a}Ù¯[Ñ¯]^{×}(¯(aÙb)) = 0
_{[ HS 5-1.11 ]} .
We also have
Ñ_{ß}Ù¯[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.
It is easy to fail to appreciate its true significance of this. In consequence of the integrability condition of I_{p}, it is possible to evaluate (Ð_{a}×Ð_{b})F_{p} without having to differentiate F_{p} by applying a particular linear function independant of F_{p} to F_{p}. This is why ÑÙÑ is essentially geometric, with no differentiating component.
Hypercurve
For M=N-1 we have [Ñ¯]^{×}(a,b) = ¯[Ñ¯]^{×}(a,b) = - n_{p}^{Ñ}(aÙb) .
[ Proof : [Ñ¯](a)×[Ñ¯](b) = [ÑÙ¯](a)×[ÑÙ¯](b) =
n_{p}^{2} (n_{p}n_{p}^{Ñ}(a))×(n_{p}n_{p}^{Ñ}(b))
= -n_{p}^{4} (n_{p}^{Ñ}(a)×n_{p}^{Ñ}(b))
= - n_{p}^{Ñ}(a)Ùn_{p}^{Ñ}(b)
º - n_{p}^{Ñ}(aÙb)
.]
Hence [Ñ¯]^{2}¯(b) = Ñ_{a}¿¯[Ñ¯]^{×}(a,b)
= -Ñ_{a}¿(n_{p}^{Ñ}(a)Ùn_{p}^{Ñ}(b))
= ???
Scalar Curvature (0;1)-tensor R = Ñ_{Þ} [Ñ¯]^{2}¯
The scalar curvature (aka. total curvature)
is traditionally presented as the second contraction of the Rieman curvature
R º
(Ñ_{Þ}¿)^{2} ¯[ÑÙ¯]^{×}
= Ñ_{Þ}^{2} ¯[ÑÙ¯]^{×}
= Ñ_{Þ} [Ñ¯]^{2}¯ ;
but is also, perhaps more fundamentally,
the tangential point divergence of the spur Ñ¿[¯(Ñ)] _{[ HS 5-1.21 ]} .
Ñ_{p}¿[Ñ¯]^{2}¯(a) = Ñ_{p}¿[Ñ¯]^{2}¯(a) = Ñ_{p}¿[Ñ¯]^{2}¯(a)
2 [Ñ¯]^{2}_{Ñ}(Ñ) = ÑR
_{[ HS 5-1.23 ]}
[ Proof :
Contracting the contracted Bianchi identity yields
Ñ_{b}¿((ÑÙ_{(}[Ñ¯]^{2}¯_{)})(b)) =
Ñ_{b}¿(¯_{=}[Ñ¯]^{×}_{Ñ}(ÑÙb))
Þ
(Ñ_{b}¿Ñ)_{(}[Ñ¯]^{2}¯_{)}_{Ñ}(b) -
Ñ(Ñ_{b}¿_{(}[Ñ¯]^{2}¯_{)}_{Ñ}(b)) = -
Ñ_{b}¿(¯_{=}[Ñ¯]^{×}_{Ñ}(bÙÑ))
Þ
_{(}[Ñ¯]^{2}¯_{)}_{Ñ}(Ñ)) - Ñ(R_{Ñ}) = -_{(}[Ñ¯]^{2}¯_{)}_{Ñ}(Ñ))
Þ
2_{(}[Ñ¯]^{2}¯_{)}_{Ñ}(Ñ)) = Ñ(R_{Ñ}) .
.]
Einstein 1-tensor (1-½Ñ_{Þ})[Ñ¯]^{2}¯
2_{(}[Ñ¯]^{2}¯_{)}_{Ñ}(Ñ)) = Ñ(R_{Ñ}) means that the symmetric (self-adjoint)
Einstein 1-tensor
(1-½Ñ_{Þ})[Ñ¯]^{2}¯(a) = _{(}[Ñ¯]^{2}¯_{)}(a) - ½Ra
= [Ñ¯]^{2}¯(a) - (Ñ¿[¯(Ñ)])a
has (1-½Ñ_{Þ})[Ñ¯]^{2}¯_{Ñ}(Ñ) = ¯(1-½Ñ_{Þ})[Ñ¯]^{2}¯_{Ñ}(Ñ) = 0 and so is zero (point) codivergent, ie.
[Ñ¿(1-½Ñ_{Þ})[Ñ¯]^{2}¯](a) = [Ñ¿(1-½Ñ_{Þ})[Ñ¯]^{2}¯](a) = 0 " a Î I_{p}
.
_{[ HS 5-1.23 ]}
It's (directional) cocontraction is Ñ_{a}¿((1-½Ñ_{Þ})[Ñ¯]^{2}¯(a)) = Ñ_{a}¿((1-½Ñ_{Þ})[Ñ¯]^{2}¯(a))
= ½(2-M)R .
(1-½Ñ_{a})(Ñ_{p}ÙÑ_{p}) = 0
The Coordinate based approach
By defining an M-curve by means of I_{p} 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 I_{p} can be expressed in U_{N} via ^{N}C_{M} scalar fields
i_{ij..mp} = e^{ij...m}¿I_{p} though the normalisation
condition renders one such field redundnat apart from sign. In the mapspace ¦^{Ñ}^{-1}(I_{p}) is everywhere the map pseudoscalar.
In U_{N} the projector 1-tensor ¯ can be expressed with regard to a universal basis as N^{2} 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 c_{paÙb}, as a bivector valued function of bivectors, constarined within and upon I_{p} requires ^{M}C_{2}^{2} = 1/4M^{2}(M-1)^{2} scalar fields in theory although the Bianchi symmetries reduce the generality considerably. At a fixed given p Î C_{M}, mapping bivectors in I_{p} to bivectors in I_{p} induces the obvious C_{x}(aÙb) = ¦^{-Ñ}_{p}(c_{p¦Ñp(aÙb)}) 2-tensor acting in the mapspace which we can represent with 1/4M^{2}(M-1)^{2} scalar fields C^{kl}_{ij} º e^{kl}¿C_{x}(e_{ij})
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
C^{k}_{ijl} º e^{k}¿(C_{x}(e_{ij})×e_{l})
c_{paÙb} and C_{x}(aÙb) are alternate representations of the same "thing",
although c_{p} defined over subspace I_{p} of U_{N} can more readily be
"extended" out of C_{M}.
We can contract the curvature tensor to obtain
the Ricci 1-tensor)
c_{pb} º Ñ_{a}¿c_{paÙb}
= å_{i=1}^{4} e^{i}¿c_{peiÙb}
= å_{i=1}^{4} e^{i}¿
(( Ð_{¯(b)}w_{ei}) - ( Ð_{¯(ei)}w_{b}) + (w_{b}×w_{ei}))
, 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
C^{k}_{j} º e^{k}¿ C_{x}(e_{i})
= å_{j=1}^{M} C^{kj}_{ij}
or with C_{ij} º
å_{k=1}^{M} C^{k}_{ijk}
which itself contracts to the scalar curvature
R = å_{i=1}^{4} C^{i}_{i} .
The mapspace Einstein tensor is given by M^{2} scalar fields
G^{k}_{j} º
C^{k}_{j} - 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 v_{p} defined over M-curve C_{M} such that
v_{p} Î I_{p} .
It follows from a mathematical result concerning the unique solvability of ordinary differential equations
that, provided v_{p} is sufficiently smooth and is nowhere 0-valued, for every p_{0}ÎC_{M}
there is a unique "tangent matcher" path in U^{N} for v_{p} :
C_{i p} =
{ p(t) : t Î [t_{0},t_{1}] } [ where t_{0}<0<t_{1} ]
with p(0)=p_{0} ;
p(t) Î C_{M} and (¶/¶t)p(t) = v_{p(t)} " t Î [t_{0},t_{1}] .
( p(0)=p_{0} ensures that the parameterisation is unique up to choice of domain range [t_{0},t_{1}])
Such paths are called integral curves or streamlines for v_{p}.
We will use both terms
interchangeably here. Streamlines cannot intercept eachother
(other than at points with v_{p}=0 which we have disallowed) .
Since every pÎC_{M} has such a curve, the set of all
integral curves for each v_{p} (known as a congruence for v_{p}) "fills" and effectively defines C_{M}.
It is important to note that the 1-curve streamlines of field v_{p} are independant of the magnitude of v_{p},
(given v_{p}^{2} 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 u_{i p} defined over M-curve C_{M} satisfying
u_{1}(p)Ùu_{2}(p)Ù..u_{M}(p) = a_{p}I_{p}
" p Î C_{M} where scalar field a_{p} > 0 " p .
An obvious example is u_{i p}=¦^{Ñ}_{x}(e_{i})= h_{ip } where C_{M}=¦(M_{ap}) is a point embedding specification for C_{M}.
Given a general multivector-valued function F(p) defined over C_{M} ,
u_{i p} generates
at each pÎC_{M} a specific "streamlined" derivative of F which is the tangential derivative
within the 1-curve u_{i p} streamline. We will write this as
d_{li}(F(p)) º ¶F(p_{i}(l_{i}))/¶l_{i} . We will sometime ommit the
brackets and write d_{li}F(p).
If we so differentiate the identity map F(p)=1(p)=p we recover u_{i p}
and, indeed, one can think of 1-field u_{i p} as being the
streamlined derivative operator field d_{li}_{p} .
We will accordingly use the symbol
Ð_{¯(ui p)} interchangeably with d_{li}
.
We introduce the notation _{ Dli}¬^{i }p to indicate the point p_{i}(Dl_{i}) Î C_{M} reached
by moving from p by arclength Dl_{i} along the l_{i} (ie. the u_{i p}) streamline through p.
_{ Dli}¬^{i }p º
ò_{Ci p [0,Dli]} 1 dp = ò_{0}^{Dli} p_{i}'(l_{i}) dl_{i}
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 Dl_{i} within the neighbourhood of ¦ applicability.
For small Dl_{i}=e we can apply Taylor's Theorem to the streamline l_{i}-parameterisation and obtain
F(_{ e}¬^{i }p) = e^{edli}F(p)
and in particular
[ Set F(p)=p ]
we have
_{ e}¬^{i }p = e^{edli}(p) which we can write as
_{ e}¬^{i } = e^{edli} .
We might be tempted to try to parameterise C_{M} using y:M_{ap}®C_{M} defined by
y(Dl_{1}, Dl_{2}, ..Dl_{m}, q_{0}) º
_{ Dlm}¬^{M }_{ Dlm-1}¬^{M-1 }..._{ Dl2}¬^{2 }_{ Dl1}¬^{1 }q_{0} =_{( )}=
_{ Dlm}¬^{M }(_{ Dlm-1}¬^{M-1 }(..._{ Dl2}¬^{2 }(_{ Dl1}¬^{1 }q_{0}))..) .
[
That is, starting from a fixed reference point q_{0}ÎC_{M}
we follow the v_{1}(p) stream line for arclength Dl_{1} to a new point
_{ Dl1}¬^{1 }q_{0} from which we follow the v_{2}(p) streamline for arc length Dl_{2}
to _{ Dl2}¬^{2 }_{ Dl1}¬^{1 }p_{0} and so forth.
After M such steps we arrive a final point
q_{M}ÎC_{M} which we can associate with parameters
(Dl_{1},..,Dl_{m}).
]
Any M scalars (sufficiently small that we stay within the neighbourhood) interpreted in this way define a point in C_{M},
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 C_{M}.
Only if the d_{li} (and hence the _{ Dli}¬^{i }) everywhere commute with eachother do
the l_{i} provide a true coordinate system.
[ Proof :
...
.]
(_{ e}¬^{i } × _{ e}¬^{j }) is a 1-field defined over C_{M} returning
a U_{N} 1-vector
(_{ e}¬^{i } × _{ e}¬^{j })(p)
= (e^{edli} × e^{edlj})(p)
= ...
= e^{2}(d_{li}×d_{lj})(p) + _{O}(e^{3}) .
Taking e®0 we can visualise (d_{li}×d_{lj})(p) º [ u_{i p},u_{jp} ](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
[u_{i p},u_{jp}](v) º d_{li}(d_{lj}[v])
- d_{lj}(d_{li}[v])
= 2(d_{li}×d_{lj})v
which we can write as
[u_{i p},u_{jp}] = d_{li}u_{jp} - d_{lj}u_{i 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 u_{i p} generates a congruence for C_{M} so let us fix on u_{i p} as a "preferred" congruence
with associated dervivative field d_{li} .
Consider two points p_{0} and p_{0} in C_{M} with p_{0}=_{ Dli}¬^{i }p_{0} for small Dl_{i}>0.
Clearly p_{0}=_{ -Dli}¬^{i }p_{0} .
For j¹i, the specific u_{jp} streamline through p_{0} (i¹1) at p with
associated directed derivative d_{lj} generates a distinct path
p_{j}^{*}(l_{j}) = _{ -Dli}¬^{i }p_{j}(l_{j}) passing through point p_{0} = _{ -Dli}¬^{i }p_{0}
known as the Lie drag
of u_{jp} .
[ Here ^{*} indicates "new" or "modifed" rather than a multivector dual
]
. This path need not be the streamline at p_{0}
of any of our u_{k p} 1-fields and, in particular, it need not be the
l_{j} streamline there .
It is however, a streamline having at
p_{0} a tangent vector written (^{i}Ý_{-Dli}(u_{jp}))(p_{0}) fully within I_{p0} and an associated
streamlined differential operator
d_{lj*} = (^{i}Ý_{-Dli}(d_{lj}))(p_{0}) .
Though d_{lj*} is not one of the d_{lk}, it is not general. Because of its construction it commutes with
d_{li} (ie. d_{lj*}(d_{li}(F))) = d_{li}(d_{lj*}(F))) .
[ Proof :
....
.]
We refer to
(^{i}Ý_{-Dli}(u_{jp}))(p_{0}) ( or (^{i}Ý_{-Dli}(d_{lj}))(p_{0}) ) as
the Lie drag of u_{jp} ( or d_{lj} ) from p_{0} to p_{0}= _{ -Dli}¬^{i }p_{0} .
We define the Lie drag of a scalar field f(p) defined over C_{M} by
(^{i}Ý_{-Dli}(f))(p) º f(_{ -Dli}¬^{i }p)
so that
(^{i}Ý_{e}(f))(p) = e^{edli}f
. 7
Lie Derivative
The 1-vector Lie derivative of a scalar field is simply £_{dli}f(p) = d_{li}f(p) .
Since the Lie drag of u_{jp} (or d_{lj}) from p_{0} to p_{0} can
be meaningfully subtracted from u_{i p} (or d_{li}) at p_{0} .
The Lie dervivative at p_{0} of u_{jp} with regard to
1-field u_{i p} is an operator defined (by its action on
a general 1-field F) as
( £_{dli}(d_{lj}))F(p)
º Lim_{Dli ® 0} [ ( (^{i}Ý_{-Dli}(d_{lj}))(p_{0})(F(p)) - d_{lj}(F(p)) ) / Dl_{i} ] ï_{p0}
= Lim_{Dli ® 0} [2(d_{li}×d_{lj})F ï_{p0} ]
[ Proof :
d_{lj*}F ï_{p0}
= d_{lj*}F ï_{p0} - Dl_{i}(d_{li}(d_{lj*}(F))))ï_{p0} + _{O}(Dl_{i}^{2})
= d_{lj*}F ï_{p0}
+ Dl_{i}(d_{li}(d_{lj}(F))ï_{p0} + _{O}(Dl_{i}^{2})
- Dl_{i}(d_{lj*}(d_{li}(F))ï_{p0} + _{O}(Dl_{i}^{2})
Hence
Lim_{Dli ® 0} [ ( (^{i}Ý_{-Dli}(d_{lj}))(p_{0}) - d_{lj} )F / Dl_{i} ] ï_{p0}
= Lim_{Dli ® 0} [((d_{lj*} - d_{lj} )F ï_{p0} ) / Dl_{i} ]
= Lim_{Dli ® 0} [(d_{li}(d_{lj}(F)) - d_{lj*}(d_{li}(F)) ) ï_{p0} ]
which provides the result assuming d_{lj*} can be safely replaced in the limit by d_{lj}
.]
Omitting the brackets , we thus have the 1-vector operator identity
£_{ui p}u_{jp} = £_{dli}d_{lj} = 2 d_{li}×d_{lj} = [ u_{i p},u_{jp} ] .
Covariant Frame
Let us now identify the spaces U_{N} and V_{N} and adopt fixed frames
{e_{i}} and {e^{i}} in both.
With regard to ¿ within V_{N} , {e^{i}} is not a reciprocal frame for {e_{i}}. Rather, we need
the point-dependant (nonorthormal) covariant frame
{e^{ i} º g_{x}^{-1}(e^{i})} which itself has ¿ reciprocal frame
{e_{i} º g_{x}(e_{i})}
[ Proof :
e^{ i}¿e_{j} = g_{x}(g_{x}^{-1}(e^{i}))e_{j} = e^{i}¿e_{j}
Also e^{ i}¿e_{j} = g_{x}^{-1}(e^{i})¿g_{x}(e_{j}) = g_{x}(g_{x}^{-1}(e^{i}))¿e_{j} = e^{i}¿e_{j}
.]
For brevity we ommit an x subscript from e_{i} , 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 V^{M}) . For any 1-field ( (1;0)-tensor) v_{p} , ¿ induces a scalar-valued directional function
( a (0;1)-tensor)
v_{p}(a) º a¿v_{p} known as a covector field.
We can recover v_{p} at a given p from its covector as v_{p} = Ñ_{a}v_{p}(a) . A vector
and its covector are essentially alternate representations of the same "thing". If we "feed"
v_{p} "to" its covector we get "squared magnitude" 0-tensor v_{p}(v_{p}) = v_{p}¿v_{p}
For ¿ in Â_{N }, if 1-vector v_{p} is regarded as a 1×N matrix
"column vector"
, then
its covector is the N×1 matrix "row vector" (or transpose) v_{p}^{T} .
For ¿ in Â_{p,q} , we take the transpose
of the conjugate v_{p}^{†} .
With regard to ¿, we will here define the 1-covector of v_{p} = å_{i=1}^{N} v^{i}e_{i} as the 1-vector
v_{p}^{£} º å_{i=1}^{N} v^{i}e_{i} so that v_{p}^{£} ¿_{+} a = v_{p}¿a .
In a Euclidean space, v_{p}^{£} = v_{p} .
(1;0)-tensor v_{p} has "contravariant" coordinate representation å_{i}v^{i}e_{i} where v^{i} º e^{i}¿v_{p} .
(0;1)-tensor v_{p} has "covariant" coordinate representation å_{i}v_{i}e^{ i} where v_{i} º v_{p}(e_{i}) = e_{i}¿v_{p}
and e^{ i} is the 1-covector of e_{i} 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 qÎC_{M}
a function
_{q}^{p C}(a) Î I_{p} giving the tangent vector at pÎC_{M} "parallel" to aÎI_{q} at q
ÎC_{M} ; as obtained by
"parallel transportation" of a "along" a given path C from q to p (eg. along a minimal length geodesic).
Let ^{i}^{Dli}_{p0}(a) denote the parallel transport of a from p_{0} for arclength Dl_{i} along the u_{i p} streamline.
We obviously want
^{i}^{Dli}_{p0}(a)ÙI_{ Dli¬i p0} = 0 and we might
therefore think of defining
^{i}^{Dli}_{p}(a) = ¯_{I Dli¬i p0}(u_{jp})~ whenever ¯_{I Dli¬i p0}(u_{jp}) is nonnull
but instead we simply postulate a 1-vector
^{i}^{Dli}_{p}(a) within I_{ Dli¬i p0} that satifies
^{i}^{Dli}_{p}(a)^{2} = a^{2} . We will henceforth assume a to be a unit vector.
The normalisation condition for ^{i}^{Dli}_{p}(a) means that the right-connection b_{a,p}(Dl_{i}) º a^{-1}^{i}^{Dli}_{p}(a) for ^{i}^{Dli} at p satisfies b_{a,p}b_{a,p}^{§} = 1 and is thus a 1-rotor expressable as a 2-spinor e^{wui p a p(Dli)} , where w_{ui p a p}(Dl_{i}) = cos^{-1}(a^{-1}¿^{i}^{Dli}_{p}(a) (a^{-1}Ù^{i}^{Dli}_{p}(a)^{~} is a U_{N} 2-blade. [ we assume Dl_{i} is sufficiently small and C_{M} 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.
^{i}^{e}_{p}(a) ¿ ^{i}^{e}_{p}(b) = a¿b
and we can acheive this by insisting that u_{i p}¿w_{ui p a p}(e) = _{O}(e^{2}) .
^{i}^{e}_{p}(a) º
ae^{wui p a p(e)}
= a + a¿w_{ui p a p}(e) + _{O}(e^{2})
where for given scalar e, w_{ui p a p}(e) is a pure U_{N}-bivector 2-field defined at every
pÎC_{M}
for all aÎI_{p} .
Parallel displacement along the i^{th} streamline thus preserves orthonormality and corresponds to a Lorentz rotation
having U_{N} rotor R_{p 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 C_{M}.
Linear Connection
g_{p}(u_{jp},d_{li},e)
º
u_{jp} - ^{i}^{e}_{p}(u_{jp})
= -u_{jp}¿w_{ui p ujp p}(e) + _{O}(e^{2})
is a 1-vector lying in general neither
in I_{p} 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
Lim_{e ® 0}[_{ e}¬^{i }u_{jp} - ^{i}^{-e}_{ e¬i p}(_{ e}¬^{i }u_{jp})] = -g_{p}(u_{jp},u_{i 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 Â_{1,3}) fiducial frame is unfortunate.
]
Taking g_{p}(u_{jp},u_{i p}) º Lim_{e ® 0}[e^{-1}g_{j p}(u_{i p},e) ] we obtain
the connection, the 1-vector component of d_{li} u_{jp} ï_{p} that exists outside I_{p}.
g_{p}(u_{jp},u_{i p}) = -u_{jp}¿w_{ui p ujp } = w_{ui p ujp }×u_{jp}
.
g_{p}(u_{jp},u_{i p}) is orthogonal to both u_{i p} and u_{jp} at p .
A linear connection (aka. affine connection) derives from an assumed
bilinearity of g_{p} which can then
can be fully defined by M^{2} 1-fields
g_{p}( h_{jp }, h_{ip })
which then effectively define _{q}^{p C}(a).
Indeed, some authors define ^{i}^{Dli} as the consequence of a given linear connection.
We say the connection is symmetric if
g_{p}(u_{i p},u_{jp})
= g_{p}(u_{jp},u_{i p}) .
Suppose now that the u_{i p} are everywhere mutually orthogonal so that {u_{i p}} provides
an orthonormal frame for I_{p} with inverse frame {u^{i}_{p}}. Because w_{ui p ujp } is a 2-blade perpendicular to I_{p}, we have
w_{ui p ujp } ×u_{i p} = 0 i¹j . Thus we can define M pure bivectors
w_{ui p p} º å_{j=1}^{M} w_{ui p ujp } which satisfy
g_{p}(u_{jp},u_{i p}) = w_{ui p p}×u_{jp}
The connection is symmetric if u_{jp}¿w_{ui p p} = u_{i p}¿w_{ujp p} " 1£i,j£M .
A linear connection arises if w_{ui p a } is linear in a.
[ Proof :
Since the u_{i p} are orthogonal
g_{p}((å_{j=1}^{M} b^{j}u_{jp}),(å_{i=1}^{M} a^{i}u_{i p}))
= (å_{j=1}^{M} b^{j}u_{jp})¿w_{ui p åi 1M=}^{} a^{i}u_{i p})}
= (å_{j=1}^{M} b^{j}u_{jp})¿(å_{i=1}^{M} a^{i}w_{ui p ui p })
= å_{i=1}^{M} å_{j=1}^{M} a^{i}b^{j}(u_{jp}¿w_{ui p ui p })
= å_{i=1}^{M} å_{j=1}^{M} a^{i}b^{j}g_{p}(u_{jp},u_{i p})
.]
Within the mapspace
Parallel transport in the mapspace
^{i}^{e}_{x}(b) º ¦^{Ñ}_{x+eei}^{-1}( ^{i}^{e}_{¦(x)}(¦^{Ñ}_{p}(b)) )
preserves ¿ instead of ¿.
[ Proof :
^{i}^{e}_{p}(a) ¿_{x+eei} ^{i}^{e}_{p}(b)
= ^{i}^{e}_{p}(¦^{Ñ}_{x}(a))¿^{i}^{e}_{p}(¦^{Ñ}_{x}(b))
= ¦^{Ñ}_{p}(a)¿¦^{Ñ}_{x}(b)
= a¿_{x}b
.]
Let b be a map direction at xÎM_{ap} with corresponding C_{M} direction a=¦^{Ñ}_{x}(b)
at p=¦(x).
It may not make any geometrical "sense" to subtract a map 1-vector "at" x+ee_{i} from one "at" x
but we can still do it!
G_{x}(b,e_{i},e)
º
b - ^{i}^{e}_{x}(b)
= b - ¦^{Ñ}_{x+eei}^{-1}(^{i}^{e}_{p}(¦^{Ñ}_{x}(b)))
= b - ¦^{Ñ}_{x+eei}^{-1}(¦^{Ñ}_{p}(b) - g_{p}(¦^{Ñ}_{p}(b),d_{li},e))
is, for given e_{i} and scalar e, a map-1-vector-valued function of map-1-vector b, linear in b if
the C_{M} connection g_{p} is linear. We
then have G_{x}(e_{i},e_{j},e) = å_{k=1}^{M} G^{k}_{ij}_{x}(e)e_{k} .
Letting
G_{x}(b,e_{i}) = Lim_{e ® 0}(e^{-1}G_{x}(b,e_{i},e))
we achieve an affine connection for the mapspace, symmetric if g is.
G_{x}(a,b) = ¦^{-Ñ}_{x}(¦^{Ñ2}_{x}(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 u_{i 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 I_{p}.
Ð_{ui p}(b) = d_{li}b + g_{p}(b,d_{li})
= d_{li}b + w_{ui p ujp p}×b
[ Proof :
Ð_{ui p}(b(p)) º
Lim_{e ® 0}[
^{i}^{-e}_{ e¬i p}(b(_{ e}¬^{i }p)) - v_{j}(p) / e ]
= Lim_{e ® 0}[
( ^{i}^{-e}_{ e¬i p}(b(_{ e}¬^{i }p)) - b(_{ e}¬^{i }p)) + ( b(_{ e}¬^{i }p) - b(p)) / e ]
= g_{p}(b,d_{li}) +d_{li}u_{jp}
.]
In the limit, since l_{i} is the natural (proper) parameterisation, we can
consider d_{li} to be equivalent to the N-D directional derivative Ð_{ui p}
Accordingly we can define a directed coderivative operator
Ð_{a} º Ð_{a} + w_{a}×
for a given direction aÎI_{p} which when applied at p to a 1-field over and within C_{M}
returns a 1-vector in I_{p}.
We then have
^{i}^{Dli}_{p}(u_{jp}) = e^{DliÐli}_{ e}¬^{i }u_{jp}
= e^{DliÐli} e^{Dlidli} u_{jp} .
[ Proof :
^{i}^{Dli}_{p}(u_{jp}) =
v_{j}(_{ Dli}¬^{i }p)
+ Dl_{i}Ð_{li}v_{j}(_{ Dli}¬^{i }p)
+ Dl_{i}^{2}Ð_{ }(u_{i p})^{2}v_{j}(_{ Dli}¬^{i }p) + ...
.]
Ð_{a} can be regarded as a differential over and "within" C_{M}, although its component parts Ð_{a} and w_{a}× cannot.
Ð_{a} generates a 1-vector codel-operator
Ñ_{p} º å_{k=1}^{M} u^{k}_{p}Ð_{uk p}
= Ñ_{[CM]} + å_{k=1}^{M} u^{k}_{p}(w_{uk p}×)
.
Within the mapspace
Ð_{a} induces a differential operator within M-D parameter space M_{ap}ÌV_{M}
Ð_{a}^{ð}(B(x)) = ¦^{-Ñ}_{p}(Ð_{¦Ñp(a)}(b(p)))
where b(p) º B(¦^{-1}(p)) .
Ð_{a}^{ð}(v_{p}) = Ð_{a}(v_{p}) + G(a,v_{p})
where G is linear as a consequence of the linearity of ¦^{-Ñ}_{p} .
In particular,
G_{p}(e_{j},e_{i})
º Ð_{ei}^{ð}e_{j} =
å_{k=1}^{M} G^{k}_{ji}e_{k}
where point-dependant scalar
G^{k}_{ji}
is known as a Christoffel symbol.
Geodesics
We say a 1-field u_{i p} is geodesic if
^{i}^{Dli}_{p}(u_{i p}) = u_{i p}(_{ e}¬^{i }p) " pÎC_{M}, Dl_{i}
where Dl_{i} is assumed small enough to remain in the neighbourhood.
Geodesic fields are thus invarient under parallel transport and satisfy
Ð_{ui p}u_{i p} = 0.
The physical interpretation of geodesics are the potential trajectories of "free falling" particles
and (assuming a natural streamline parameterisation p_{i}'(l_{i})^{2} = 1)
we can then regard the
p_{i}"(l_{i}) + g(p_{i}'(l_{i}),p_{i}'(l_{i})) = 0
derived from Ð_{ui p}(u_{i 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 C_{M}. 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 Lim_{d ® 0}[ d^{-1} q_{Ð}(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 h_{1},h_{2} and normal n=h_{1}×h_{2} forming the SYMBOL">t) is the curvature of the 1-curve confined in
trihedron) at p. Taking p as the origin, the induced coordinates (x^{1},x^{2},x^{3}) satisfy
x^{3} = ½ (¶^{2}x^{3}/¶(x^{1})^{2}) x^{1}^{2} +
(¶^{2}x^{3}/¶x^{1}¶x^{2}) x^{1}x^{2} + ½ (¶^{2}x^{3}/¶x^{2}¶) x^{2} |^{2}
By rotation about e_{3} we can diagnonalise this form as
x^{3} = ½ (k_{1}(x^{1})^{2} + k_{2}(x^{2})^{2})
where k_{1},k_{2} are known as principle curvatures.
If the surface is parameterised as p(l_{1},l_{2})
with h_{1p } = ¶p/¶l_{1} , h_{2p } = ¶p/¶l_{2}
then
¶^{2}p/¶l_{i}¶l_{j} º ¶ h_{ip }/¶l_{j} =
å_{k=1}^{2} G^{k}_{ij} h_{kp } + b_{ij} h_{3p } i,j,k Î {1,2}
¶ h_{3p }/¶l_{i} = - å_{k l=1}^{2} g^{kl}b_{li} h_{kp } i Î {1,2}
2-Curvature
The second directional coderivative operator
Ð_{li}Ð_{lj} is not symmetric in i,j.
We can take the skewsymmetrol of it to obtain the
Rieman Curvature operator
c_{pui pÙujp}(u) = c_{pui pÙujp}×u =
2(Ð_{li}×Ð_{lj})u =
( Ð_{¯(li)}w_{ujp} - Ð_{¯(lj)}w_{ui p}
+ w_{ui p}×w_{ujp}) . u
[ Proof :
.]
Ð_{b}Ð_{a}u =
( Ð_{¯(b)} + w_{b}×)( Ð_{¯(a)} + w_{a}×)u
=
( Ð_{¯(b)} + w_{b}×)( Ð_{¯(a)}u + ½(w_{a}u-uw_{a}))
= ( Ð_{¯(b)} Ð_{¯(a)}u + ½( Ð_{¯(b)}w_{a}u- Ð_{¯(b)}uw_{a})
+w_{b}×( Ð_{¯(a)}u + ½(w_{a}u-uw_{a}))
=
Ð_{¯(b)} Ð_{¯(a)}u + ½( Ð_{¯(b)}w_{a}u - Ð_{¯(b)}uw_{a}
+ w_{b} Ð_{¯(a)}u - Ð_{¯(a)}uw_{b} )
+ ½w_{b}×(w_{a}u-uw_{a})
=
Ð_{¯(b)} Ð_{¯(a)}u +
½( Ð_{¯(b)}(w_{a}u) + w_{b} Ð_{¯(a)}u
- Ð_{¯(a)}(uw_{b})
- Ð_{¯(b)}(uw_{a}) )
+ 1/4(
w_{b}w_{a}u - w_{b}uw_{a}
- w_{a}uw_{b} + uw_{a}w_{b} )
Þ 2(¶_{b}×¶_{a})u =
2( Ð_{¯(b)}× Ð_{¯(a)} + Ð_{¯(b)}×(w_{a}×) + (w_{b}×)× Ð_{¯(a)} +
(w_{b}×)×(w_{a}×))u
=
Ð_{¯(b)}(w_{a}u)- Ð_{¯(a)}(w_{b}u) + w_{b} Ð_{¯(a)}u -w_{a} Ð_{¯(b)}u
+ (w_{b}×w_{a})×u
= ( Ð_{¯(b)}w_{a})u - ( Ð_{¯(a)}w_{b})u + (w_{b}×w_{a})×u
=
( Ð_{¯(b)}w_{a})×u - ( Ð_{¯(a)}w_{b})×u + (w_{b}×w_{a})×u
Given a linear connection, the second-derivative curvature operator c_{paÙb}(v) is thus a
geometric commutatator product with a particular bivector c_{paÙb} .
The more general definition of the curvature operator is 2(Ð_{a}×Ð_{b}) - Ð_{[a,b]} but we are most interested in the case when [a,b]=0 ( ie. Ð_{a}×Ð_{b} =0 ) .
c_{paÙb} º
Ð_{¯(b)}w_{a} - Ð_{¯(a)}w_{b} + w_{b}×w_{a}
=_{( )}= ( Ð_{¯(b)}w_{a}) - ( Ð_{¯(a)}w_{b}) + (w_{b}×w_{a})
is known as the curvature 2-tensor (aka. Riemann-Christoffel tensor).
It is a C_{M}-point-dependant I_{p}-bivector-valued function of I_{p}-bivectors acting as a "second order correction"
for the linear approximator ¦(p_{0}) + ¦^{Ñ}_{p}(dp) for ¦ near p_{0}.
Geometric interpretation
If [u_{i p},u_{jp}]=0 so that we have a loop
_{ -Dli}¬^{j }_{ -Dli}¬^{i }_{ Dlj}¬^{j }_{ Dli}¬^{i } p = p
then
c_{pui pÙujp}(u_{k p}) =
(Dl_{i}Dl_{j})^{-1} ^{j}^{-Dlj}( ^{i}^{-Dli}( ^{j}^{Dlj}( ^{i}^{Dli}(u_{k p}) ))))
which is clearly dependant solely on value of u_{k p} and the geometry of C_{M} . In particular, it is independant
of any differential of u_{k p}.
We thus have the geometric interpretation of c_{pui p}u_{jp})(u_{k p}) as the
1-vector change in u_{k p} resulting from parallel
transport of u_{k p} around (the projection into C_{M} of) a tiny planar loop through p in u_{i p}Ùu_{jp}
, divided by the "area" of that loop. This must lie in I_{p} so we have
c_{pui pÙujp}(u_{k p}) = å_{l=1}^{4} c^{l}_{ijk}u_{lp}
where scalar c^{l}_{ijk} º u^{l}_{p}¿c_{pui pÙujp}(u_{k p}) .
[ Proof :
^{j}^{-Dlj} ^{i}^{-Dli} ^{j}^{Dlj} ^{i}^{Dli}(u_{k p})
=
e^{-DljÐlj}
e^{-DliÐli}
e^{DljÐlj}
e^{DliÐli}u_{k p}
= ...[tedious expansion]...
= u_{k p} + 2Dl_{i}Dl_{j}(Ð_{li}×Ð_{lj})u_{k p} +
_{O}(Dl^{3})
.]
Symmetries and Bianci Relations
It is possible to express C_{xeij} in terms of the g_{ij} and their derivatives
and doing so reveals some more symmetries:
C^{l}_{kij} º
e^{l}¿(C_{xeij}(e_{k}))
= e^{i}¿(C_{xelk}(e_{j}))
= -e^{k}¿(C_{xeij}(e_{l}))
.
The last of these reflects the reversability of parallel transport.
We also have (from the G expression for C^{i}_{jkl})
the first Bianci identity
e_{k}¿C_{xeij} + e_{j}¿C_{xeki} + e_{i}¿C_{xejk} = 0 .
which can also be expressed as
C_{xeij} ¿ e_{kl}
= C_{xekl} ¿ e_{ij}
.
It follows that
a¿C_{x}(bÙc))
+ b¿C_{x}(cÙa)
+ c¿C_{x}(aÙb) = 0 " 1-vector a,b,c Î I_{p}.
As a result, the M^{ 4} element matrix representation of C_{x}
has only M^{ 2}(M^{ 2}-1)/12 independant elements (20 out of 256 for N=4).
Ð_{ei}^{ð}C_{xejk} + Ð_{ej}^{ð}C_{xeki} + Ð_{ek}^{ð}C_{xeij} = 0_{2} is known as the second Bianci identity.
The codivergence of the Ricci tensor
Ñ^{ð}_{x}¿ c_{pa}
º å_{i}e^{i}¿Ð_{ei} c_{pa}
= ½Ð_{a}C_{x}
[ Proof :
Ñ^{ð}_{x}¿ c_{pb}
=
Ñ^{ð}_{x}¿(Ñ_{a}¿c_{pa}b))
Þ ... ???
.]
Tortion
The differential operator Ð_{a} Ð_{¯(b)} - Ð_{b} 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 p}u_{jp} = Ð_{ui p} Ð_{¯(ujp)} - Ð_{ujp} Ð_{¯(ui p)}
.
Suppose now that u_{i p} is a geodesic field and 1-vector b_{p} is Lie dragged along geodesic u_{i p} congruence so
that £_{ui p}b_{p} = 0 . We must then have
Ð_{ui p}^{2} Ð_{¯(bp)} =
2(Ð_{ui p}×Ð_{bp}) Ð_{¯(ui p)} .
[ Proof : Ð_{ui p} Ð_{¯(bp)} = Ð_{bp} Ð_{¯(ui p)}
Þ Ð_{ui p}^{2} Ð_{¯(bp)} = Ð_{ui p}Ð_{bp} Ð_{¯(ui p)}
= 2(Ð_{ui p}×Ð_{bp}) Ð_{¯(ui p)} +
Ð_{bp}Ð_{ui p} Ð_{¯(ui p)}
= 2(Ð_{ui p}×Ð_{bp}) Ð_{¯(ui p)}
since u_{i p} geodesic Þ Ð_{ui 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
ds^{2} = dx^{2} + dy^{2} so that the length constraint is
a = ò_{0}^{1} ds = ò_{x0}^{x1} dx(1+y'^{2})^{½}
where y' º dy/dx.
The mechanical constraint is that total gravitational potential energy
ò_{0}^{1} ds rgy(s) = rg ò_{x0}^{x1} 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
ò_{x0}^{x1} dx¦(x,y,y') subject to a constraint
ò_{x0}^{x1} 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
z_{0} and z_{1} such that
y(x_{0},z_{0},z_{1})=y(x_{0}) ; y(x_{1},z_{0},z_{1})=y(x_{1}) " z_{0},z_{1} (boundary condition)
; y(x,0,0)=y(x) ; and y(x,z_{0},z_{1}) is twice continuosuly differentiable in all parameters.
Forming K(z_{0},z_{1}) = ò_{+x0}^{x1} dx h(x,y(x,z_{0},z_{1}),y'(x,z_{0},z_{1})) the extremal condition requires
¶K/¶z_{0} and ¶K/¶z_{1} to vanish at z_{0}=z_{1}=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 = -rgy_{0} we have
h(x,y,y') = rg(y-y_{0})(1+y'^{2})^{½}
= rg(y-y_{0})(1+(y-y_{0})'^{2})^{½} for some constant y_{0} .
Setting h = y-y_{0} the Euler-Lagrange equation is
rg(1+h'^{2})^{½} =
d/dx rghh'(1+h'^{2})^{-½}
with first order simplification
h' rghh'(1+(y-y_{0})'^{2})^{-½} -
rgh(1+h'^{2})^{½} = b
Þ
rg(1+h'^{2})^{-½}h = b
having solution h = b cosh(x-c)/b) so the rope has shape
y_{0} + b cosh((x-c)/b) where c,y_{0} and b are chosen to
match the given endpoints.
We say an M-curve C_{M} is M-extremal for an action functional
¦_{CM}:U^{N} ® U_{N} if some particular magnitude measure (content, scalar part, square, modulus or whatever) of
ò_{CM} d^{M-1}p ¦_{CM}(p) is maximised (or minimised) by C_{M} in the sense that
integrating other any M-curve which deviates only slightly from C_{M} will produce
a result no higher (or lower) than the C_{M} integral. C_{M} is a "locally optimal"
M-curve for a particular integration ¦_{CM}. We specify the dependancy of ¦ on C_{M} to allow
action functionals dependant on the geometric properties of the "sampling curve" C_{M} , 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 F_{C1} = F(s,p(s),p'(s)).
Hamilton's principle provides that the state of a system characterised at time t by k multivector variables
x_{1}(t),..,x_{k}(t) varies from time t_{0} to t_{0} so as to 1-extremise
a (usually real scalar) integral measure
S(t) º S(t_{0}) + ò_{t0}^{t0} dt L(t,x_{1}(t),..,x_{k}(t),x'(t),...,x_{k}'(t))
with the action functional L(t,x_{1}(t),..,x_{1}'(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,x_{1}(t),..,x_{k}(t),x_{1}'(t),..,x_{k}'(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 B_{ase} space)
L = ò_{Base(t)} d^{N-1}p 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 B_{ase}(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
z_{0} and z_{1} such that
(s_{0},z_{0},z_{1})=(s_{0}) ; (s_{1},z_{0},z_{1})=(s_{1}) " z_{0},z_{1} (boundary condition)
; (s,0,0)=(s) ; and (s,z_{0},z_{1}) is twice continuosuly differentiable in all parameters.
Forming K(z_{0},z_{1}) = ò_{s0}^{s1} ds¦(s,(s,z_{0},z_{1}),'(s,z_{0},z_{1})) the extremal
condition requires
¶K/¶z_{0} and ¶K/¶z_{1} to vanish at z_{0}=z_{1}=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/dx¶h/¶y' - ¶h/¶-¶h/¶yy' - ¶h/¶y'y"
= y'(d/dx¶h/¶y' - ¶h/¶y) - ¶h/¶
= 0
.]
A nongeometric generalisation is to extremise ò_{s0}^{s1} ds¦(s,x_{1},x_{2},..x_{k},x_{1}',x_{2}',..x_{k}')
subject to M scalar constraints g_{i}(s,x_{1},x_{2},...x_{k})=0 i=1,2,..,M
(Note the absence of any g dependance on the x_{i}').
To do so we form
h(s,x_{1},..,_xK,x_{1}',..,x_{k}') = ¦(s,x_{1},..,_xK,x_{1}',..,x_{k}')
+ å_{j=1}^{M} l_{j}(s)g_{j}(s,x_{1},..,x_{k},x_{1}',..,x_{k}')
where l_{j}(s) are M arbitary functions of s
and obtain Euler-Lagrange formulae
¶h/¶x_{i} = (d/ds)(¶h/¶x_{i}') for i=1,2,..k .
Generalising geometrically, we have F(s,x_{1},x_{2},..,x_{k},x_{1}',x_{2}',..,+x_{k}')
with geometric Euler-Lagrange Equations
¶F/¶x_{i} = (d/ds)(¶F/¶x_{i}') " i=1,2,..k
and first order equation
å_{i=1}^{k} (x_{i}'_{*}Ñ_{xi'})F - F = constant (independant of s)
if ¶F/¶s = 0.
Considering the x_{i} 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
¶_{x}F(s,x,x') = (d/ds)(¶_{x'}F(s,x,x'))
where ¶_{x} = å_{ijk..} e^{ijk..}¶/¶x^{ijk..}
over all blades comprising x space so that, for example,
¶_{x} = å_{i=1}^{N} e^{i}(¶/¶x^{i}).
Generalised Momentum
When F is a scalar-valued Lagrangian we have (x_{i}'_{*}¶_{xi'})L =
x_{i}'_{*}(¶_{xi'}L) and multivector m = Ñ_{xi'}L(x_{1},..x_{k},x_{1}',..,x_{k})
is known as generalised spacial momenta or cononical momenta.
Typically x_{i}' is spacial 1-vector valued and so momenutum is a spacial 1-vector.
If L is independant of x_{i} so that ¶L/¶x_{i}=0 then the i^{th} Euiler-Lagrange equation provides
(d/dt) ¶L/¶x_{i}' = 0 so M_{i} = ¶L/¶x_{i}' 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^{-1}a_{x}¿' to L introduces
qc^{-1}a_{x} to the momentum
If
L(t,x_{1}(t),..,x_{k}(t),x_{1}'(t),..,x_{k}'(t))
= L(x_{1}(t),..,x_{k}(t),x_{1}'(t),..,x_{k}'(t))
so that there is no explicit t dependance then
the Hamiltonian H(x_{1}(t),..,x_{k}(t),x_{1}'(t),..,x_{k}'(t))
º å_{i=1}^{k} (x_{i}'_{*}Ñ_{xi'})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 = (x_{i}'_{*}¶_{xi'})L = Ñ_{xi}S(t,x_{i}) and in particular we have the Hamilton-Jacobi equation
H(x_{i},x_{i}',t) = -¶S/¶t
[ Proof : For any ¦(,')
d/dt ( å_{j=1}^{k} x_{i}' ¶¦/¶x_{j}' - ¦)
= å_{j=1}^{k} x_{j}" ¶¦/¶x_{j}'
+ å_{i=1}^{k} x_{j}' d/dt ¶¦/¶x_{j}'
- ¶¦/¶t
- å_{j=1}^{k} ¶¦/¶x_{j}x_{j}'
- å_{j=1}^{k} ¶¦/¶x_{j}'x_{j}"
=
å_{j=1}^{k} x_{j}' d/dt ¶¦/¶x_{j}'
- ¶¦/¶t
- å_{j=1}^{k} ¶¦/¶x_{j}x_{j}'
=
å_{j=1}^{k} x_{i}'( d/dt(¶¦/¶x_{j}') - ¶¦/¶x_{j} )
- ¶¦/¶t
=
å_{j=1}^{k} x_{j}'( d/dt(¶¦/¶x_{j}') - ¶¦/¶x_{j} )
if ¶¦/¶t = 0
= 0
if d/dt(¶¦/¶x_{j}') = ¶¦/¶x_{j} j=1,2,..,k
which are the Euler-Lagrange equations.
d/dx ( å_{i=1}^{k} y_{i}' ¶¦/¶y_{i}' - ¦)
= å_{i=1}^{k} y_{i}" ¶¦/¶y_{i}'
+ å_{i=1}^{k} y_{i}' d/dx ¶¦/¶y_{i}'
- ¶¦/¶
- å_{i=1}^{k} ¶¦/¶y_{i}y_{i}'
- å_{i=1}^{k} ¶¦/¶y_{i}'y_{i}"
=
å_{i=1}^{k} y_{i}' d/dx ¶¦/¶y_{i}'
- ¶¦/¶
- å_{i=1}^{k} ¶¦/¶y_{i}y_{i}'
=
å_{i=1}^{k} y_{i}'( d/dx(¶¦/¶y_{i}') - ¶¦/¶y_{i} )
- ¶¦/¶
=
å_{i=1}^{k} y_{i}'( d/dx(¶¦/¶y_{i}') - ¶¦/¶y_{i} )
if ¶¦/¶ = 0
= 0
if d/dx(¶¦/¶y_{i}') = ¶¦/¶y_{i} 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 ò_{0}^{t} ds|d^{1}p¿g(d^{1}p)|^{½} . Integrating a square root can be messy
but fortunately such paths also extremise
ò_{0}^{t} ds ds p'(s)¿g_{p(s))}(p'(s))
where p'(s) denotes the (d/ds)p(s)
. _Be Thus we set
L(p,p',s) =
p'¿g_{p}(p')|^{½} and
minimise scalar integral ò_{0}^{t} ds ds L(p(s),p'(s),s) by solving the N Euler-Lagrange equations
d/ds(¶L/¶p^{i}') =
¶L/¶p^{i}) 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/¶p^{i}=0 for a particular coordinate i then
¶L/¶p^{i}' 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 a_{p} (typically with Ñ_{p}¿a_{p}=0) are introduced by adding scalar
-q p'(t)¿ a_{p(t)}
to the Lagrangian density leading to the Lorentz force law
mp"(t) = qf_{p}.p'(t)
where f_{p} = Ñ_{p}Ùa_{p} .
[ Proof :
L = m(-p'^{2})^{½} - q p'¿a_{p}
Þ Ñ_{p}L =
Ñ_{p}(m(-p'^{2})^{½} - q p'¿ a_{p})
= -qÑ_{p}(p'¿a_{p}) ;
Ñ_{p'}L =
-m(-p'(t)^{2})^{-½} p'(t) - qa_{p}
so the Euler-Lagrange equation is
qp'¿Ñ_{p}a_{p} =
(d/dt)(m(-p'^{2})^{-½} p' + qa_{p} )
= mp" + q(p'¿Ñ_{p})a_{p}
Þ mp" = q(Ñ_{p}(a_{p}¿p') - (p'¿Ñ_{p})a_{p}))
= q(Ñ_{p}Ùa_{p})¿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]}y_{p})^{2} is extremised for integration over C_{M}
(subject to constraint of given boundary values over dC_{M}) if y_{p}
satisfies the spacial Laplace equation Ñ_{p [e123]}^{2}y_{p} = 0
over C_{M} .
In electrodynamics we have L = - 2^{-4}p^{-1} f_{p}^{2}
+ c^{-1}j_{p}¿a_{p}
where c is scalar lightspeed and f_{p} = Ñ_{p}Ùa_{p} extremised by solutions to
Maxwell equations.
Next : 4D Spacetime
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References/Source Material
David Hestenes "New Foundations For Mathematical Physics" Websource
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 P_{b}(a) is our ¯^{Ñ}(a,b) ("differential of projector").
Their S_{a} 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 d_{a} is our
Ð_{aß} º ¯Ð_{¯aß} ("extensor coderivative").
Their (a.¶) is our Ð_{a}=(a¿Ñ) ("directed derivative").
Their e_{i} is our h_{ip } (potentially nonorthonormal "tangent frame").
Their g_{i} is our e_{i} (orthonormal "fiducial (basis) frame").