Geometric Algebra of Manifolds

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

    Thus far we have considered fields over 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 [t₀,t₁] and is either closed (a "loop") , infinite, or "bounded" by two points. We will call a 1-curve with a specific parameterisation a path.
    A 2-curve is a surface and is either closed (eg. a 2-sphere), infinite, or "bounded" by a 1-curve.
    A 3-curve is a "solid" is either closed or bounded by a surface (2-curve). In general, M-curve 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}(x¹,x²,..,x^m) sending V^M ® U^N defined over a fully bounded finite subvolume (interior of a unimapped (M-1)-curve) M_apÌV^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_{k_p} º (d/dx^k)¦_{_q}(x) = Ð^x_{e_k}¦_{_q}(x) = (e_k¿Ñ_x)¦_{_q}(x) = ¦_{_q}^Ñ_x(e_k)
where Ð^x_{e_k} 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_{N_p} = ( h_{1_p}Ù h_{2_p}Ù... h_{N-1_p})^-1i .

Spherical Surface Example
We can formulate coventional spherical polar coordinates as ¦:V^M=Â³ ® U^N=Â³
¦(x)=¦(qe_q + fe_f + re_r) = r sinq cosfe₁ + r sinq sinfe₂ + r cosqe₃ = rR_q,f_§(e₃)
where e_q,e_f,e_r are an orthonormal basis for V^M and e₁,e₂,e₃ 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=Â³ with e_{_ð}ⁱ=eⁱ and regard ¦ as a transformation then
    ¦^Ñ_x(dx)) = (Ñ_x¿dx)¦(x)
    = (dqr cosq cosf - dfr sinq sinf + dr sinq cosf)e₁ + (dqrcos(q) sinf + dfr sinq cosf + dr sinq sinf) e₂ + (-dqr sinq + dr cosq) e₃ .

    The embedded frame in U^N is orthogonal.
    h_{q_p} = rR_q,f_§(e₁) ;
    h_{f_p} = e₃×¦(q,f,r) = (e₃Ù¦(q,f,r))^* = r sinqR_q,f_§(e₂);
    h_{r_p} = ¦(q,f,r)^~ = R_q,f_§(e₃)
    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=S₂ , 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)e₁ + (dq cosq sinf + df sinq cosf) e₂ - dq sinq e₃ .

    From a purist point of view, mapspace coordinates have the form qe_q+fe_f whereas tangent vectors are expressed in terms of e₁,e₂,e₃ but we will here consider the M_ap as existing in U_N and set e_q=e₁ , e_f=e₂.

Inverse Embedded Frame
    We can construct an inverse embedded M-frame hⁱ_{_p} of 1-vectors within the tangent space satisfying hⁱ_{_p}¿ h_{j_p} = d_{i j} via
    h^k_{_p} º (-1)^k-1( h_{1_p}Ù.. h_{k-1_p}Ù h_{k+1_p}Ù.. h_{M_p})( h_{1_p}Ù.. h_{M_p})^-1 .
    Note that the hⁱ_{_p}ÎI_p and are a frame for precisely the same subspace as are the embedded frame.

    We can invert ¦_{_q} to express the xⁱ(p) = eⁱ¿¦_{_q}^-1(p) as M scalar fields defined over a local (M-curve) neighbourhood of q.
    When M=N so that ¦_{_q} : U^N ® U^N we have
    hⁱ_{_p} º Ñ_p(xⁱ(p)) = Ñ_{¦_{_q}(x)}xⁱ = ¦_x^-D(eⁱ) .
[ Proof : h_{i_p}¿ h^j_{_p} = ¦_{_qx}^Ñ(eⁱ) ¿ ¦_{_qx}^-D(e_j) = eⁱ¿ ¦_{_qx}^D(¦_{_qx}^-D(e_j)) = eⁱ¿e_j = d_ij .]
    hⁱ_{_p} is the normal to the coordinate isosurface xⁱ(p)=xⁱ(q) at q. In a nonorthogonal embedded frame, we can have hⁱ_{_p}Ù h_{i_p} 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 hⁱ_{_p}^~ = R_q,f_§(e_i).
    ¦^D_p(dp) º Ñ_x(dp¿¦(x)) = Ñ_x(dp¹r sinq cosf + dp²r sinq sinf + dp³r cosq)
    = r(dp¹ cosq cosf + dp² cosq sinf - dp³ sinq)e_q + r(-dp¹ sinq sinf + dp² sinq cosf)e_f + (dp¹ sinq cosf + dp² sinq sinf + dp³ cosq)e_r     defined for dpÎÂ³ . This reduces to
    ¦^D_p(dp) = (dp¹ cosq cosf + dp² cosq sinf)e_q + (-dp¹ sinq sinf + dp² sinq cosf)e_f     when pÎS₂ and dp is in the tangent space at p.

    We have M scalar fields defined over C_M by xⁱ(p) = e_{_ð}ⁱ¿¦^-1(p) = e_{_ð}ⁱ¿x .
    For spherical mapping these are
    q(p¹e₁ + p²e₂ + p³e₃) = cos^-1(p³(p¹²+p²²+p³²)^-½)     [ p¹² = (p¹)² ]
    f(p¹e₁ + p²e₂ + p³e₃) = tan^-1(p²/p¹)
    r(p¹e₁ + p²e₂ + p³e₃) = (p¹²+p²²+p³²)^½

    We have Ñ_p q(p) = -(1-(p³²(p¹²+p²²+p³²)^-1)^-½ ( e¹(-p³x¹(p¹²+p²²+p³²)^-3/2 + e²(-p³x²(p¹²+p²²+p³²)^-3/2 + e³((p¹²+p²²)(p¹²+p²²+p³²)^-3/2
    = -((p¹²+x²²)(p¹²+p²²+p³²)^-1)^-½ (p¹²+p²²+p³²)^-_3/2( e¹(-p³p¹) + e²(-p³p²) + e³(p¹²+p²²) )
    =?= r^-2 h_{q_p} = r^-1 h_{q_p}^~
    Ñ_p f(p) = (r sinq)^-2 h_{f_p} = (r sinq)^-1 h_{f_p}^~
    Ñ_p r(p) = p^~ = h_{r_p} = h_{r_p}^~ .

Local Orientation
    The h_{k_p} define a C_M-point-dependant nondegenerate U_N M-blade J_p º h_{1_p}Ù..Ù h_{M_p} that spans the tangent space at p=¦_{_q}(x). C_M has tangent M-plane (1+q)ÙJ_q at q (or (e₀+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-1_p} then it is conventional to specify the orientation ("handedness") of I_p by defining I_p = I_{M-1_p} _np where _np = I_{M-1_p}^-1 I_p is the spur, the unit outward normal to the boundary at p.
    The normalisation condition I_p² = ±1 gives (Ð_aI_p)I_p + I_p(Ð_aI_p)=0 and taking the scalar part yields (Ð_aI_p)¿I_p=0 Þ ¯_{I_p}(Ð_aI_p)=0.
    Hence ¯_{I_p}(Ñ_pI_p) = 0 .
[ Proof :   Choosing an orthonormal frame with e₁Ùe₂Ù...e_M =I_p we obtain Ñ_pI_p = å_i=1^N eⁱ Ð_{e_i}I_p = å_i=M+1^N eⁱ Ð_{e_i}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 S₂ mapping, J_p = h_{q_p}Ù h_{f_p} = h_{q_p} h_{f_p} = sinqR_q,f_§(e₁)R_q,f_§(e₂) = sinqR_q,f_§(e₁₂) .
    Although J_p vansihes at the poles, it is natural to define I_p=R_q,f_§(e₁₂)     " p=R_q,f_§(e₃) ÎS₂ .

The Metric
                       Let a,b be two 1-vectors in an M-curve's mapspace M_apÌV_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¿_xb 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) = r²dqe_q + r² sin²qdfe_f + e_r     with associated line element length dx¿g_x(dx) = r²dq² + r² sin²qdf² + dr²
[ Proof : ¦^Ñ_x(dx)) = (Ñ_x¿dx)¦(x)
    = (dqr cosq cosf-dfr sinq sinf+dr sinq cosf)e₁ + (dqrcos(q) sinf+dfr sinq cosf+dr sinq sinf)e₂ + (-dqr sinq+dr cosq)e₃ .
    ¦^D_p(dp) º Ñ_x(dp¿¦(x)) = Ñ_x(dp¹r sinq cosf + dp²r sinq sinf + dp³r cosq)
    = r(dp¹ cosq cosf + dp² cosq sinf - dp³ sinq)e_q + r(-dp¹ sinx siny + dp² sinq cosf)e_f + (dp¹ sinq cosf + dp² sinq sinf + dp³ cosq)e_r
    Thus g_x(dx) = ¦^D_p(¦^Ñ_x(dx))
    = ¦^D_p( (dqr cosq cosf -dfr sinq sinf +dr sinq cosf)e₁ + (dqrcos(q) sinf +dfr sinq cosf +dr sinq sinf) e₂ + (-dqr sinq +dr cosq) e₃ )
    = ...[Tedious manipulations]... = r²dqe_q + r² sin²qdfe_f + e_r .]
    Restricting r=1 for S₂ gives g_x(dx) = g_x(dqe_q+dfe_f) = dqe_q + sin²qdfe_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 p₀ÎU^N is known to lie in C_M then another U^N point p₁ will lie in C_M iff there exists a path (1-curve) { p(t) Î U^N : t Î [t₀,t₁] } with p(t₀)=p₀ ; p(t₁)=p₁ ; and p'(t)ÎI_p(t) " tÎ[t₀,t₁]. Establishing whether this is, in fact, the case for a given p₀ and p₁ may be far from easy. If all we can say about ¦ with regard to p=¦(x) is that ¦^Ñ_x(i)=I_p then things can get tricky. ( If ¦^-1(p₁) is available, of course, then p₁ÎC_M Û ¦^-1(p₁)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-1_p}^* = i_{N-1_p}i^-1 = (i)² i_{N-1_p}¿i . n_p = (-1)^N-1e_N in the fortutious basis and n_p² = (-1)^N-1 i_{N-1_p}² i² = ±1 where M=N-1 . .

Projector 1-multitensor     ¯
    The projector of M-curve C_M is the multitensor
    ¯(a) º ¯_{I_p}(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 ¯_{i_{N-1_p}}(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^t₁F(t) dt is defined for a multivector-valued function F of a scalar t as the limit of a sum of elemental contributors.
    We can immediately adapt this for the line integral of a multivector valued field F(p) along a 1-curve p(t) t Î [t₀,t₁] by defining
    ò_C F(p) dp º ò_t₀^t₁F(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_{M_p[p]} and magnitude the conventional scalar measure of area (simplex content); these triangles tesselating to approximate the surface.
If the 2-curve is parameterised as p(x¹,x²) then the directed area element has form d²p = ( h_{1_p}Ù h_{2_p})dx¹dx² .

    For integrating over an M-curve C_M=¦(M_ap) we have contributary (M-1)-simplex elements
    d^Mp   =   ( h_{1_p}Ù h_{2_p}.. h_{M_p})dx¹dx²..dx^M   =   |¦^Ñ|I_pdx¹dx²..dx^M and we can think of M succesive scalar integrals of multivector F(p)I_p via
    ò_{C_M}F(p) d^mp = ò_{C_M}F(p)( h_{1_p}Ù h_{2_p}..Ù h_{M_p}) dx¹dx²...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_{C_M} I_p^-1dp . 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^Nx i^-1 (-i(x¿k)) a_x = (2p)^-½N ò |d^Nx| (-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 i²=-1.
    The inverse transform is
    F^-1(b_k)(x) º (2p)^-½N ò d^Nk 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^Nx 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 ò₀^¥ dx x^-1 sin(Kx) = 2p we have ò_-¥^¥ dx cos(xk) = 2pd(k) and hence F((i(x.b)(k) =    (2p)^½N d(b-k) where d(x) = d(x₁)d(x₂)...d(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^-1F = 1 .

    An obvious geometric generalisation is F(a_x)(k) º ò (2p)^-½N ò d^Nx 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 ¯_{I_p}(a)-directed derivative.
    Ð_¯(a) º Ð_{¯_{I_p}(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_{[C_M] 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_{[C_M] p} º Ñ_{[I_p] p} º å_k=1^M h^k_{_p}Ð^p_{h_{k_p}} = å_k=1^M h^k_{_p}Ð^x_{e_{_ðk}}
where hⁱ_{_p} are a reciprocal embedded frame for C_M ; Ð_{h_{k_p}} operaties in U_N ; and Ð^x_{e_{_ðk}} = (d/dx^k) operates in the mapspace M_ap as
    Ð_{e_{_ðk}}F(p) = Ð^x_{e_{_ðk}}F(¦(x)) = (Ñ_p¿¦^Ñ_x(e_{_ðk}))F(p) = F^Ñ_p(¦^Ñ_x(e_{_ðk})) = F^Ñ_p( h_{k_p}) .
    When the particular M-curve C_M under discussion is unambiguos, we will abbreviate Ñ_{[C_M] 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 e₁,e₂,...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 = ¯_{I_p}(Ñ_p) = å_i=1^N. ¯_{I_p}(eⁱ)Ð_{e_i} = å_i=1^M. eⁱÐ_{e_i} = å_i=1^N. eⁱÐ_{¯_{I_p}(e_i)} .
    For a more general basis for U_N, it is natural to define the tangential derivative by
    Ñ_p º å_i=1^N eⁱÐ_{¯_{I_p}(e_i)} .
    We can also define a orthotangential derivative Ñ_^ º Ñ_^p º Ñ_p - Ñ_p   =   å_i=1^N eⁱÐ_{^_{I_p}(e_i)} .

For 1-curve C= { p(t) } we have Ñ_{[C] p} = h¹_{_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_{[C_M] 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_{[U^N] p} = Ñ_p and Ñ_[1] = Ñ_<0> . The use of h^k_{_p} in the defintion of Ñ_{[C_M]p} "counter scales the expansions" of ¦^Ñ to ensure Ñ_p p º Ñ_{[C_M]p} p = M .

Though ¯(Ñ) º ¯_{I_p}(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 (¯_{I_p}Ñ)(a_p) º ¯_{I_p}(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 ¯_{I_p}(a) = 1^Ñ_{[I_p]}(a) " multivector a.
[ Proof : (a¿Ñ_ðp)p =(a¿å_i=1^M hⁱ_{_p}Ð^p_{h_{i_p}})p =å_i=1^M (a¿ hⁱ_{_p}) h_{i_p} =¯_{I_p}(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_*(¯_{I_p}(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 = ¯_{I_p}(Ñ)_*a_p = Ñ_*(¯_{I_p}(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_p²=±1 gives ¯_{I_p}(Ð_aI_p)=0 which in turn yields Ñ I_p = Ñ Ù I_p

For p² ³ 0, Ñ_{[b_k]} |p|^k = Ñ_{[b_k]}( (p²)^½k = k |p|^k-2 ¯_{b_k}(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_xF_x need not lie in I_p since Ð_{e_{_ðk}}F_x need not lie in I_p . Even if we insist that F_p Î I_p so that ¯_{I_p}(F_p)=F_p, then Ð_{e_{_ðk}}F_x still need not lie in I_p.
Similarly Ð_dF_p need not lie in I_p even if dÎI_p and F_pÎI_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_{[C_M] p} º I_p^-1 Lim_{|O| ® 0} [ |O|^-1 ò_dO d^m-1p 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-1p 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_{[U^N] 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_{dC_M} d^M-1p F(p) = ò_{C_M} d^Mp (Ñ_pF(p)) º ò_{C_M} d^Mp Ñ_{[C_M]}F(p) . where scaled M-blade d^Mp = I_pd^Mp = I_p|d^Mp| 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_{[C_M]}F . It has the following consequences.

if dC_M = f, ie. if C_M is closed, then ò_{C_M} d^Mp Ñ_{[C_M]}F(p) = 0 for any F(p) .
Since d²C_M = 0 for any C_M it follows that ò_{C_M} d^Mp p Ñ_{[C_M]}Ñ_{[dC_M]} F(p) = 0 for any F_p and C_M
A "volume" integral ò_{C_M} d^Mp F(p) can be replaced by a boundary integral provided we can find a "potential" H(p) with Ñ_{[C_M]p}H(p) = F_p, which we often can.
In the case M=N we have Ñ_{[C_M]} = Ñ and hence
ò_{C_M} i d^Np Ñ¦(p) = ò_{dC_M} in^-1d^N-1p ¦(p) Þ ò_{C_M} d^Np (Ñ¿¦(p) + ÑÙ¦(p) ) = ò_{dC_M} d^N-1p (n¿¦(p) + nÙ¦(p))
(where n is the (assumed postive signature) normal to hypercurve dC_M). This provides (scalar part) the divergence theorem (aka. Gauss's theorem) and (bivector part when N=3) the (dual of) a curl integral theorem
In the case M=1 we have
ò_C₁ dp Ñ_[C₁]F(p) = F(t₁)-F(t₀) providing F(t) from by the path directed integration of the gradient of F given F(t₀).

    A more general form of interest is
    ò_{dC_M} G_p d^M-1p F_p   =   (-1)^M-1ò_{C_M} (G_pÑ_p) d^Mp F_p     +     ò_{C_M} G_p d^Mp (Ñ_p F_p) _{[ HS 7-3.10 ]}
[ Proof : ò_{dC_M} G_p d^M-1p F_p   =   ò_{C_M} G_p_{_Ñ} d^MpÑ_p F_p     +     ò_{C_M} G_p d^MpÑ_p F_p_{_Ñ}     and recall that d^Mp is a scaled M-blade pseudoscalar for the tangent space containing Ñ and so G_p_{_Ñ} d^MpÑ_p = G_p_{_Ñ}(d^Mp.Ñ_p) = (-1)^M-1G_p_{_Ñ}(Ñ_p.d^Mp) = (-1)^M-1G_p_{_Ñ}(Ñ_pd^Mp) = (-1)^M-1(G_pÑ_p)d^Mp .]

Greens Functions

A Greens function is a function used to express a solution to a differential equation with particular boundary conditions as a definite integral. For example, d/dx² y(x) = ¦(x, y(x)) subject to y(a)=y₀ and y'(a)=y₀' has solution y(x) = y₀ + y₀'(x-a) + ò_a^x dx" ò_a^x" dx' ¦(x',y(x')) which can be alternatively evaluated as
y(x) = y₀ + y₀'(x-a) + ò_a^x dx' (x-x')¦(x',y(x')) = y₀ + y₀'(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)=y₀ ; y'(a)=y₀' . [ Hvsd(x) is the Heaviside step function zero for x£0 and 1 for x>0, exploited to allow us to replace the indefinite 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_pG_p,q = -G_p,qÑ_q = 1 at p=q, the Fundamental Theorem of Calculus provides
    ò_{dC_M} G_p,q d^M-1p F_p   =   (-1)^M-1I_{M_q} F_q + ò_{C_M} G_p,q d^Mp (Ñ_p F_p)     which we can write as
    F_q   =   (-1)^MI_{M_q}^-1 ( ò_{dC_M} G_p,q d^M-1p F_p     -     ò_{C_M} G_p,q d^Mp (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_pF_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_{M_q} " p Î dC_M we can commute I_{M_q} across G_p,q , incurring a sign change if M is even, to obtain
    F_q   =   - ò_{dC_M} G_p,q I_{M_q}^-1 d^M-1p F_p     +     ò_{C_M} G_p,q I_{M_q}^-1 d^Mp (Ñ_p F_p)
    If F_p is monogenic so that ò_{C_M} G_p,q d^Mp (Ñ_p F_p) vanishes, we have a Geometric generalisation of Cauchy's Theorem
    F_q   =   (-1)^MI_{M_q}^-1 ò_{dC_M} G_p,q d^M-1p F_p   =   - ò_{dC_M} G_p,q I_{M_q}^-1 d^M-1p 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^Mp) = dp¹dp²...dp^ML(x, h_{1_p}Ù h_{2_p}Ù.. h_{M_p}) .
    The exterior differential of an (M-1)-form is the M-form defined by
    dL(p,d^M-1p) º Lim_{O ® 0} ò_O L(p,d^Mp) / |O|     where O is a small volume enclosing x.
    The general form of the fundamental theorem is
    ò_{C_M} dL(p,d^Mp) = ò_{dC_M} L(p,d^M-1p)
but we will not pursue this here.

Poles and Residues

Cauchy's Theorem
Relates the value of a monogenic function at a point to the surface integral over the boundary of any M-curve including the point.
    F(a)   =   (I_{_Ma}o_N)^-1 ò_{dC_M} (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 f₁(a) is an operator taking one parameter then f₁^×(a,b) º ½(f₁(a)×f₁(b))
    If f₂(a,b) is an operator taking two parameters then f₂^×(a,b) º ½(f₂(b,a)-f₂(a,b))
    f₁_<k>(a) º f₁(a_<k>)

We define f₁^¨ by f₁^¨ (a) ¨ a = f₁(a) where ¨ is a multivector product.
Thus, for example, if tensor 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 f₁^Ñg₁ º (f₁^Ñ)g₁ ? The "differentialiser" ^Ñ converts f₁ from an operator or tensor taking one nonprimary argument to one taking two, it introduces a second non-primary 1-vector parameter. In nonabbreviated expressions we will add this parameter at the rightmost end of the parameter list with
    f₁^Ñ(a,b) = (Ð_bf₁)(a) = Ð_b(f₁(a)) - f₁(Ð_ba) . We will interpret the g₁ in the "composite product" f₁^Ñg₁ as applying to the "newest" rightmost nonprimary parameter so that f₁^Ñg₁(a,b) º f₁^Ñ(a,g₁(b)) .
    From a programmers' perspective, we can think of "parsing" our composite operator expressions from left to right pushing introduced parameters onto a stack. When a "compositional irregularity" such as f₁^Ñg₁ 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 f₁^Ñg₁(a,b) º f₁^Ñ(a,g₁(b)) as the LIFO convention.

   Tangential 1-differential ^Ñ

    We can define the tangential 1-differential of field F_p by
    F^Ñ_{[C_M]p}_p(a) º F_p^Ñ(a) º (a_*Ñ_{[C_M]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_bF_p^Ñ²(b,a) _{[ HS 4-1.18 ]}

    More generally suppose F_p(a₁,a₂,...a_k) is an extensor tensor of k multivector variables themselves p-dependant. We can take a 1-differential of the secondary differntial Ñ_a₁F_p(a₁,a₂,...a_k) to give
    (Ñ_a₁F_p)^Ñ(a₁,..,a_k,b) = ¯^Ñ(Ñ_a₁,b)F_p(a₁,..,a_k) + Ñ_a₁F_p^Ñ(a₁,..,a_k,b) .
    For 1-vector a₁ = ¯_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 eⁱ² Ð^a_{e_i}(Ð^p_{e_i}F_p)(a,b,...) = å_i=1^M (Ð^p_{e_i}F_p)(eⁱ²e_i,b,...) = å_i=1^M (Ð^p_{e_i}F_p)(eⁱ,b,...)
= å_i=1^M F_p_{_Ñ}(Ð^p_{e_i}eⁱ,b,...) = F_p_{_Ñ}(Ñ,b,...) .]

Hypercurve
    For M=N-1 we have Ñ = ¯(Ñ) = Ñ - n_p(n_p¿Ñ) = Ñ - n_pÐ_{n_p} .
    Since n_p¿Ð_an_p = 0 " a , Ñ_pn_p lies within i_{N-1_p} and ^[Ñ_pn_p] = n_p^-1ÙÐ_{n_p}n_p .
    More generally, let n_p be a unit 1-field. Ñ_p = n_p²Ñ_p = n_p(n_p¿Ñ_p) + n_p(n_pÙÑ_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 ¯_{I_p}(a)-directed dirivative
Ð = ¯ Ð_¯() º ¯_° Ð_¯() abbreviating Ð_aF_p º ¯_{I_p}Ð_{¯_{I_p}a}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_{¯(a_p)}) 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_dF_p)(a₁,a₂,...) º Ð_d(F_p(a₁,a₂,..)) - F_p(Ð_da₁,a₂,...)- F_p(a₁,Ð_da₂,...) = ¯( Ð_¯(d)(F_p(a₁,a₂,..)) ) - F_p(¯( Ð_¯(d)a₁),a₂,...)- F_p(a₁,¯( Ð_¯(d)a₂),...)
    Note that this differs from _{[ HS 4-3.3 ]} which inserts subtracts - F_p(Ð_d¯(a₁),a₂,...)- F_p(a₁,¯( Ð_¯(d)a₂),...)

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_pF_p Î I_p ; Ð_pa₁_p + Ð_pa₂_p = Ð_p(a₁_p+a₂_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_{[I_p]p}F_p º ¯_{I_p}(Ñ_{[I_p]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 eⁱ Ð_{¯(e_i)}(¯(a_p_{_Ñ})) = å_i=1^M eⁱ(¯( Ð_{¯(e_i)}a_p)) = å_i=1^M ¯(eⁱ)(¯( Ð_{¯(e_i)}a_p)) = å_i=1^M ¯(eⁱ( Ð_{¯(e_i)}a_p)) = ¯(eⁱ(å_i=1^M Ð_{¯(e_i)}a_p)) = ¯(Ñ_p(Ap)) º Ñ_p(Ap) .

    For an extended field F_p(a₁_p,a₂_p,...a_k_p) with a_i_pÎU_N we have a-directed primary coderivative
    Ð_¯_aF_p(a₁_p,a₂_p,...a_k_p) º (Ð_aF_p)(a₁_p,a₂_p,...a_k_p)
    = Ð_a(F_p(a₁_p,a₂_p,...a_k_p)) - F_p(Ð_a(¯_{I_p}(a₁_p)),a₂_p,...a_k_p) - F_p(a_i_p,Ð_a(¯_{I_p}(a₂_p)),,...a_k_p) - F_p(a_i_p,a₂_p,...,Ð_a(¯_{I_p}(a_k_p)))
and an i^th exterior coderivative
    Ñ^i®¿_p F_p(a₁_p,a₂_p,...a_k_p) º ¯_{I_p}(Ñ^i®¿_p F_p(a₁_p,a₂_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 Ñ² , 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 ¯_{I_p}^Ñ_p(a_p,d) º Ð_d(¯_{I_p}(a_p)) - ¯_{I_p}(Ð_da_p)     (informally) the rate of change of ¯_{I_p}(a) in direction d , is of of less interest than ¯^Ñ, the primary tangential 1-differential of ¯_{I_p} (ie. the second tangential 1-differential of the identity function) ,
    ¯^Ñ_p(a_p,d)   =   ¯^Ñ_p(a_p,¯d)   =   1^Ñ²(a_p,d)   =   ( Ð_¯(d)¯_{I_p})(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 ¯_{I_p}(aÙb)=¯_{I_p}(a) Ù ¯_{I_p}(b) yields
    ¯^Ñ(aÙb,d) = ¯^Ñ(a,d) Ù ¯(b) + ¯(a) Ù ¯^Ñ(b,d)     _{[ HS 4-2.6 ]} and hence
    ¯^Ñ(a₁Ùa₂Ù....a_k,d) = å_i=1^k (-1)ⁱ⁺¹ ¯^Ñ(a_i,d)Ù¯_{I_p}(a₁)Ù... ¯_{I_p}(a_i-1)Ù ¯_{I_p}(a_i+1)Ù... ¯_{I_p}(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))¿¯^Ñb) = Ñ_Þ¿(¯¯^Ñb)) = Ñ_Þ¿0 . .]

Ñ_ÞÙ(¯^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_aF_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_p²((a_p¿n_p^Ñ(b))n_p + (a_p¿n_p)n_p^Ñ(b)) .
    Thus ¯^Ñ(¯(a_p),b) = -n_p² (¯(a_p)¿n_p^Ñ(b))n_p = -n_p² (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_aÐ_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_{[C_M]p}¯_{I_p}(a_p) º Ñ_ß ¯_{I_p}(a_p) = (Ñ_b Ð^p_¯(b) ¯_{I_p})(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¯](^_{I_p}(a_p)) + [Ñ¯](¯_{I_p}(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¯](¯_{I_p}(a_<k>))    ;     [Ñ¯](a)_<k-1> = ¯_{I_p}([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¯](^_{I_p}(a)) = (Ñ_p¿¯_{I_p})(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 ¯_{I_p}(b)¿[Ñ¯](a) = ¯_{I_p}(b)¿[ÑÙ¯](a) is normal to C_M , ie. ¯_{I_p}( ¯_{I_p}(b)¿[Ñ¯](a)) = 0 .

[Ñ¯](a₁Ùa₂Ù...a_k) = å_i=1^k. (-1)ⁱ⁺¹ [ [Ñ¯](a_i) Ù ¯_{I_p}(a₁Ù..Ùa_i-1Ùa_i+1..Ùa_k) + ¯_{I_p}([Ñ¯](a_i) × (a₁Ù..Ùa_i-1Ùa_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¯^Ñb) 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_Þ²Ù¯^Ñ     _{[ HS 4-2.16 ]}.
[ Proof : [Ñ¯](a) º (Ñ_p¯))(a) = Ñ_b¯^Ñ_p(a,b) so .... ??? .]

    Ñ² = [Ñ¯](Ñ) + ¯Ñ²
[ Proof : Ñ²(a_p) =Ñ(¯_Ñ(a_p)) = [Ñ¯](Ñ) + ¯_{Ñ(Ñ(a_p))} = [Ñ¯](Ñ)(a_p) + ¯_{Ñ(Ñ(a_p))} = [Ñ¯](Ñ)(a_p) + ¯_{ÑÑ(a_p)}
    = [Ñ¯](Ñ)(a_p) + ¯_{¯_ÑÑ(a_p)} = [Ñ¯](Ñ)(a_p) + ¯_{Ñ²(a_p)} = [Ñ¯](Ñ)(a_p) + ¯_{(Ñ¿Ñ)a_p} .]
    Hence ÑÙÑ = [Ñ¯](Ñ) = [ÑÙ¯](Ñ) with regard to path-independant functions. _{[ HS 4-3.10b ]} .

We have the following Shape Properties:

[ÑÙ¯]^ = 0 ie. [ÑÙ¯](^_{I_p}(a_p)) = 0 " a_p _{[ HS 4-2.37a ]}
[Ñ¿¯]¯ = 0 . ie. [Ñ¿¯](¯_{I_p}(a_p)) º (Ñ¿¯)(¯(a_p)) = 0 " a_p _{[ HS 4-2.37b ]}
¯[ÑÙ¯] = 0     when acting on 1-vectors ie. ¯[ÑÙ¯](a)=0 " a . _{[ HS 4-2.30 ]}
[ Proof : ¯(a)¿[ÑÙ¯](b) = ¯^Ñ(¯(a),b) _{[ HS 4-2.26 ]} Þ ¯( ¯(a)¿[ÑÙ¯](b) ) = ¯¯^Ñ(¯(a),b) = ¯₌¯^Ñ(a,b) = 0 .
    Þ ??? ¯( a¿¯([ÑÙ¯](b)) )=0 ???? .]
^[Ñ¿¯] = 0     follows immediately from ^¯₁=0 with regard to 1-vectors.
I[Ñ¯] = I×[Ñ¯] = I×[ÑÙ¯]     abbreviating I_p×[ÑÙ¯](a) = I_p×[Ñ¯](a) = -[Ñ¯](a)×I_p = -[ÑÙ¯](a)×I_p = -[ÑÙ¯](a)I_p    (4-4.5)
[ Proof : 3 Þ [ÑÙ¯](a)¿I_p = 0 and since [ÑÙ¯](a)ÙI_p=0 the result follows from bivector nature of [ÑÙ¯] .]
    Hence [ÑÙ¯](a) = ([Ñ¯](a)×I_p)I_p^-1 = -I_p_-1([ÑÙ¯](a))
    Also I_p.[ÑÙ¯] = I_pÙ[ÑÙ¯] = 0     for 1-vector arguments     _{[ HS 4-4.3 ]} which yields
    [ÑÙ¯]² = 0 .
[ Proof : ¯_{I_p}([ÑÙ¯](a)) º ¯ [ÑÙ¯](a) = 0 Þ [ÑÙ¯](a)=^([ÑÙ¯](a)) Þ [ÑÙ¯]([ÑÙ¯](a)) = 0 by 1. .]
¯₌[Ñ¯]     º     ¯[Ñ¯]¯     =     0        _{[ HS 4-2.45 ]}
[ Proof : ¯([Ñ¯](¯(a))) = [Ñ¯](^(¯(a))) = [Ñ¯](0) = 0 .]
^₌[Ñ¯]     º     ^[Ñ¯]^     =     0
[ Proof : ^[Ñ¯]^ = ^¯[Ñ¯] = 0 .]
¯^Ñ(a,b) = ¯(a)×[ÑÙ¯](b) - ¯( a×[ÑÙ¯](b) ) = ¯(a)×[Ñ¯](b) - ¯( a×[Ñ¯](b) )
is crucially important because it expresses ¯^Ñ geometrically in terms of [ÑÙ¯] and ¯ .
[ Proof : See _{[ HS 4-2.33 ]}. .]
In particular, it gives ¯^Ñ(¯(a),b) = ¯(a)×[ÑÙ¯](b) = ¯(a)×[Ñ¯](b) .
[ Proof : The projected product rule gives ¯(¯(a)[ÑÙ¯](b)) = ¯(a)¯([ÑÙ¯](b)) = 0 and as ¯(¯(a)Ù[ÑÙ¯](b)) = ¯(a)Ù¯([ÑÙ¯](b)) = 0 we must have ¯( ¯(a)×[ÑÙ¯](b) ) = 0 .]
[ÑÙ¯](b) = I_p^-1¯^Ñ(I_p,b) = I_p(Ð_bI_p^-1 = -¯^Ñ(I_p,b)I_p^-1 expresses [ÑÙ¯] geometrically in terms of ¯^Ñ and I_p without further differention and provides perhaps the simplest mental picture of the curl tensor as the bivector which acting geoemtrically on I_p returns -Ð^p_aI_p , the negated rate of change of I_p in direction a. .
[ Proof : a=I_p in 6 gives ¯^Ñ(I_p,b) = I_p×[ÑÙ¯](b) = - [ÑÙ¯](b)I_p by 4. .]
¯^Ñ(a,b)=¯(a)¿[ÑÙ¯](b) - ¯(a¿[ÑÙ¯](b)) _{[ HS 4-2.27 ]}
[ Proof : a×b_<2> = a¿b_<2> combined with 5 .]
(¯^Ñ)²(F_p,a,b) =   ¯( [Ñ¯](b)×(¯(F_p)×[Ñ¯](a)) ) - ¯( F_p×[Ñ¯](a) )×[Ñ¯](b)
[ Proof : Applying Shape Property 6 twice while exploiting Shape Property 3
    ¯^Ñ(¯^Ñ(F_p,a),b) =   ¯(¯^Ñ(F_p,a))×[Ñ¯](b) - ¯( ¯^Ñ(F_p,a)×[Ñ¯](b) )
    = ¯(¯(F_p)×[Ñ¯](a) - ¯( F_p×[Ñ¯](a) ))×[Ñ¯](b)     -     ¯( (¯(F_p)×[Ñ¯](a) - ¯( F_p×[Ñ¯](a) ))×[Ñ¯](b) )
    = 0 - ¯²( F_p×[Ñ¯](a) ))×[Ñ¯](b) - ¯( (¯(F_p)×[Ñ¯](a) )×[Ñ¯](b) )   + 0     by the projected bivector commutation rule .] If ¯(F_p)=F_p this reduces to
    ¯^Ñ(¯^Ñ(F_p,a),b) =   ¯( [Ñ¯](b)×(F_p×[Ñ¯](a)) )     _{[ HS 5-1.3 ]}   which we can abbrviate to
    ¯^Ñ²¯ = - ¯[Ñ¯]×[Ñ¯]×¯ .
[Ñ¯](I_p) = Ñ_pÙI_p = Ñ_pI_p _{[ HS 4-2.44 ]}
Pad -----
¯_{I_p}(b)¿[ÑÙ¯](a) = ¯_{I_p}(a)¿[ÑÙ¯](b) _{[ HS 4-2.28 ]}
¯^Ñ_{I_p}(a,d) = ¯_{I_p}(b) ¿ [ÑÙ¯](d) - ¯_{I_p}(a¿[ÑÙ¯](d)) _{[ HS 4-2.27 ]}
¯^Ñ_{I_p}([ÑÙ¯](a),d) = - ¯(I_p)([ÑÙ¯](a)×[ÑÙ¯](d))
Pad --++-------
a_*(¯_{I_p}^Ñ_p(d,b)) = b_*(¯_{I_p}^Ñ_p(d,a))
¯_{I_p}^Ñ(d,aÙb) = ¯_{I_p}^Ñ(d,a) Ù ¯_{I_p}(b) + ¯_{I_p}(b) Ù ¯_{I_p}^Ñ_p(d,b)
¯_{I_p}^Ñ_p(b,¯_{I_p}(a)) = ¯_{I_p}^Ñ_p(a,¯_{I_p}(b))
--- end list

    [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_jÙb_k) = [ÑÙ¯](a_jÙb_k) = Ñ_cÙ( ¯^Ñ(¯(a_j),c)Ù¯(b_k) + (¯(a_j)Ù¯^Ñ(¯(b_k),c) )
    = Ñ_cÙ( ¯^Ñ(¯(a_j),c)Ù¯(b_k) + (-1)^jk¯^Ñ(¯(b_k),c)Ù¯(a_j) ) = (Ñ_cÙ¯^Ñ(¯(a_j),c))Ù¯(b_k) + (-1)^_jk(Ñ_cÙ¯^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_cÙ¯^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_p² n_pn_p^Ñ(a) = n_p² n_pÙn_p^Ñ(a)
[ Proof : Ð_bi_{N-1_p} = Ð_b(n_pi) = (Ð_bn_p)i = n_p^Ñ(b)i so Shape Property 7 gives
    [ÑÙ¯](a) = i_{N-1_p}^-1( Ð_¯(a)i_{N-1_p}) = i^-1n_p^-1n_p^Ñ(a)i = n_p² _psiinvd(n_pn_p^Ñ(a)) = n_p² _psiinvd(n_pÙn_p^Ñ(a)) = n_p² n_pÙn_p^Ñ(a)     since i commutes with all bivectors. .]
    [¯(Ñ)] = n_p² n_p(Ñ_p¿n_p) = n_p² n_p(Ñ_p¿n_p)
[ Proof : Ñ_a[ÑÙ¯](a) = -n_p² Ñ_a[n_p^Ñ(a)n_p] = -n_p² (Ñ_an_p^Ñ(a))n_p = -n_p² (Ñ_aÐ_an_p)n_p = -n_p² (Ñ_pn_p)n_p = -n_p² n_p(Ñ_p¿n_p) .]
    Thus [Ñ¿¯](a) = [¯(Ñ)].a = -n_p² (Ñ¿n_p)(a_p¿n_p)

Squape 1-multitensor     [Ñ¯]²

    The shape tensor [Ñ¯] raises the grade of ¯_{I_p}(a) while lowering the grade of ^_{I_p}(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_aÐ_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_aÙ¯[Ñ¯]^×(¯(aÙb)) = 0 _{[ HS 5-1.11 ]} .

We also have
Ñ_ßÙ¯[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.

¯[Ñ¯]^×   =   -¯^Ñ[ÑÙ¯]     abbreviating ¯( [ÑÙ¯]^×(a,b) ) = -¯^Ñ([ÑÙ¯](a),b) = ¯^Ñ([ÑÙ¯](b),a)     _{[ HS 5-1.5b ]}
[ Proof : Substituting a_p=[ÑÙ¯](a) into Shape Property 6   and exploiting Shape Property 3 :
    ¯^Ñ([ÑÙ¯](a),b) = ¯([ÑÙ¯](a))×[ÑÙ¯](b) - ¯( [ÑÙ¯](a)×[ÑÙ¯](b) ) = 0 -¯( [Ñ¯](a)×[Ñ¯](b) ) .]
¯[Ñ¯]^× = [ÑÙ¯]^Ñ = (Ð[ÑÙ¯]) abbreviating ¯[Ñ¯]^×(a,b) = (Ð_b[ÑÙ¯])(a) º Ð_b([ÑÙ¯](a))-[ÑÙ¯](Ð_ba) _{[ HS 5-1.5c ]}
[ Proof : (Ð_b[ÑÙ¯])(a) = Ð_¯(b)(¯([ÑÙ¯](a))) - ¯^Ñ([ÑÙ¯](a),b) = Ð_¯(b)0 - ¯^Ñ([ÑÙ¯](a),b) and result follows from 1. .]
¯[Ñ¯]^××¯    =   -2(¯^Ñ)²^×¯     abbreviating ¯([Ñ¯]^×(a,b))×¯(F_p) = 2(¯^Ñ(¯^Ñ(¯F_p,a),b)-¯^Ñ(¯^Ñ(¯F_p,b,a))     _{[ HS 5-1.4 ]}
[ Proof : Shape Property 9 gives
    ¯^Ñ(¯^Ñ(¯F_p,a),b) - ¯^Ñ(¯^Ñ(¯F_p,b),a) =   ¯( [Ñ¯](b)×(¯(F_p)×[Ñ¯](a)) ) -   ¯( [Ñ¯](a)×(¯(F_p)×[Ñ¯](b)) )
    =   ¯( ¯(F_p)×([Ñ¯](b)×[Ñ¯](a)))     by Jacobi identity
    =    ¯(F_p)×¯([Ñ¯](b)×[Ñ¯](a)))     by projected bivector commutation rule .]
¯[Ñ¯]^××¯   =   -2Ð^×¯     abbreviating ¯([Ñ¯]^×(a,b))×¯(F_p) = - 2(Ð_a×Ð_b)¯(F_p)     _{[ HS 5-1.4 ]}
[ Proof : Follows immediately from (¯^Ñ)²^×¯ = Ð^×¯ and the previous identity. .]
    c_{_pa}b)×F_p º -(Ð_a×Ð_b)F_p with F_pÎI_p understood is the traditional approach in much of the literature. For functions satisfying the integrability condition, directed derivatives commute but directed coderivatives may not. In restricting Ñ_p to act "within and withon" I_p, we "break" the commutivity of directed derivatives, but the commutativity "survives" both in the form of the antisymmetry of the Riemann 2-curvature so defined and also in the Ricci Identity
    ÑÙÑÙ b₂_p = 0 for any bivector field b₂_p
    This enables us to also regard the Riemann curvature as a pointer-dependant bivector-specific 1-tensor
    c_{_paÙb}c) º c_{_paÙb}×c which we shall see later has the geometric interpretation of the 1-vector change in c if it is "paralell transported" around (the projection into C_M of) a tiny planar loop through p in aÙb , divided by the content of that loop.
    It is easy to fail to appreciate its true significance of this. In consequence of the integrability condition of 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.
¯[Ñ¯]² = Ñ_Þ ¯[Ñ¯]^× = Ñ_Þ ¿ ¯[Ñ¯]^× = Ñ_Þ ¿ ¯[Ñ¯]^× provides the more traditional definition of the Rici 1-tensor as the contraction of the Riemann curvature 2-tensor . _{[ HS 5-1.17 ]}
[ Proof : ¯[Ñ¯]([Ñ¯](b)) = [Ñ¯]([Ñ¯](¯b) ) = [Ñ¯]([ÑÙ¯](b)) = (Ñ_aÐ^p_a¯)([ÑÙ¯](b)) = Ñ_a¯^Ñ([ÑÙ¯](b),a) = Ñ_a¿¯^Ñ([ÑÙ¯](b),a) combined with Curvature Identity 1 . Finally, Ñ_Þ¿¦_p(a_p) = ¯(Ñ_Þ¿¦_p(a_p)) = Ñ_Þ¿¦_p(a_p) for any 1-vector-valued ¦_p(a_p) . .]
¯₌[Ñ¯]^×(aÙb) = (b¿Ñ_p) [ÑÙ¯](a) = Ð_b[ÑÙ¯](a) _{[ HS 5-1.5c ]}
expresses ¯₌[Ñ¯]^×(aÙb) as the b-directed primary coderivative of [ÑÙ¯](a), the geometric operator that sends I_p to Ð^p_aI_p .
[ Proof : ??? .]
¯₌[Ñ¯]^×(aÙb) = ¯^Ñ_{I_p}([ÑÙ¯](b),a) = Ñ_uÙ¯^Ñ_aÑ_v ¯^Ñ(¯^Ñ(v,b),a) _{[ HS 5-1.5b ]} .
[ Proof : ??? .]
¯₌[Ñ¯]^×(aÙb) = (Ñ_uÙÑ_v)(¯^Ñ_a(u)¿¯^Ñ_b(v)) _{[ HS 5-1.5d ]} .
[ Proof : ??? .]
Ñ_ßÙ¯₌[Ñ¯]^× = 0 abbreviating (ÑÙ¯₌[Ñ¯]^×)(aÙb)=0 _{[ HS 5-1.13a ]} .
[ Proof : ??? .]
¯₌[Ñ¯]^×(Ñ_ÜÙb)   =   ÑÙ₍[Ñ¯]²¯₎(b)     _{[ HS 5-1.22 ]}.
[ Proof : ¯₌[Ñ¯]^×_{_Ñ}(ÑÙb) º ¯(¯₌[Ñ¯]^×_{_Ñ}(ÑÙb)) = ¯((Ñ_Þ¿Ñ)¯₌[Ñ¯]^×(aÙb))     by the tangential gradifying substition rule
    = ¯( Ñ_Þ¿(ÑÙ¯₌[Ñ¯]^×(aÙb) + ÑÙ(Ñ_Þ.¯₌[Ñ¯]^×(aÙb)) )     via a¿(bÙc)=(a.b)c-bÙ(a.c)
    = ¯( Ñ¿0 + ÑÙ₍[Ñ¯]²¯₎(b)) ) º ÑÙ₍[Ñ¯]²¯₎(b)     by the Bianchi identity. .]

We thus have Ñ_ßÙ[Ñ¯]^{^×}(aÙb) º (ÑÙ[Ñ¯]^{^×}¯)(aÙb) = Ñ_aÙ¯^Ñ_{I_p}([ÑÙ¯](b),a) = 0 which we can write as the Bianchi identity
Ñ_ßÙ¯₌[Ñ¯]^× = 0 _{[ HS 5-1.13a ]}.
[ Proof : (Ð_c¯₌[Ñ¯]^×)(aÙb) = Ð_c( Ð_¯(v)Ù The gradifying substitutuin rule gives _..... ????? .]

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_p² (n_pn_p^Ñ(a))×(n_pn_p^Ñ(b)) = -n_p⁴ (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_Þ² ¯[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_pÙÑ_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 ^NC_M scalar fields i_{ij..m_p} = 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 N² sclar fields although with regard to an extended tangential frame at p it can be regarded as an identity matrix having the last N-M terms in the lead diagonal zeroed.

The Riemann curvature tensor c_{_paÙb}, as a bivector valued function of bivectors, constarined within and upon I_p requires ^MC₂² = 1/4M²(M-1)² scalar fields in theory although the Bianchi symmetries reduce the generality considerably. At a fixed given p Î C_M, mapping bivectors in I_p to bivectors in I_p induces the obvious C_x(aÙb) = ¦^-Ñ_p(c_{_p¦^Ñ_p(aÙb)}) 2-tensor acting in the mapspace which we can represent with 1/4M²(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=1⁴ eⁱ¿c_{_pe_iÙb} = å_i=1⁴ eⁱ¿ (( Ð_¯(b)w_{e_i}) - ( Ð_{¯(e_i)}w_b) + (w_b×w_{e_i}))    , 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 M² 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=1⁴ Cⁱ_i .
    The mapspace Einstein tensor is given by M² 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 p₀ÎC_M there is a unique "tangent matcher" path in U^N for v_p :
    C_{i_p} = { p(t) : t Î [t₀,t₁] } [ where t₀<0<t₁ ] with p(0)=p₀ ; p(t) Î C_M and (¶/¶t)p(t) = v_p(t) " t Î [t₀,t₁] . ( p(0)=p₀ ensures that the parameterisation is unique up to choice of domain range [t₀,t₁])
    Such paths are called integral curves or streamlines for 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_p² 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
    u₁(p)Ùu₂(p)Ù..u_M(p) = a_pI_p     " p Î C_M where scalar field a_p > 0 " p .
    An obvious example is u_{i p}=¦^Ñ_x(e_i)= h_{i_p} 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_{l_i}(F(p)) º ¶F(p_i(l_i))/¶l_i . We will sometime ommit the brackets and write d_{l_i}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_{l_i}_p . We will accordingly use   the symbol Ð_{¯(u_{i p})} interchangeably with d_{l_i} .

    We introduce the notation _{Dl_i}¬ⁱp 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.
    _{Dl_i}¬ⁱp º ò_{C_{i_p}_{[0,Dl_i]}} 1 dp = ò₀^Dl_i p_i'(l_i) dl_i
    We can extend _{Dl_i}¬ⁱ to act on general multivector fields by _{Dl_i}¬ⁱF(p) º F(_{Dl_i}¬ⁱp) .
    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¬ⁱp) = e^ed_{l_i}F(p) and in particular [ Set F(p)=p ] we have _e¬ⁱp = e^ed_{l_i}(p) which we can write as
    _e¬ⁱ = e^ed_{l_i} .

    We might be tempted to try to parameterise C_M using y:M_ap®C_M defined by
    y(Dl₁, Dl₂, ..Dl_m, q₀) º _{Dl_m}¬^M_{Dl_m-1}¬^M-1..._Dl₂¬²_Dl₁¬¹q₀ =_( )= _{Dl_m}¬^M(_{Dl_m-1}¬^M-1(..._Dl₂¬²(_Dl₁¬¹q₀))..) .
    [ That is, starting from a fixed reference point q₀ÎC_M we follow the v₁(p) stream line for arclength Dl₁ to a new point _Dl₁¬¹q₀ from which we follow the v₂(p) streamline for arc length Dl₂ to _Dl₂¬²_Dl₁¬¹p₀ and so forth. After M such steps we arrive a final point q_MÎC_M which we can associate with parameters (Dl₁,..,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_{l_i} (and hence the _{Dl_i}¬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^ed_{l_i} × e^ed_{l_j})(p) = ... = e²(d_{l_i}×d_{l_j})(p)   + _O(e³) .
    Taking e®0 we can visualise (d_{l_i}×d_{l_j})(p) º [ u_{i p},u_{j_p} ](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_{j_p}](v) º d_{l_i}(d_{l_j}[v]) - d_{l_j}(d_{l_i}[v]) = 2(d_{l_i}×d_{l_j})v
which we can write as [u_{i p},u_{j_p}] = d_{l_i}u_{j_p} - d_{l_j}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_{l_i} . Consider two points p₀ and p₀ in C_M with p₀=_{Dl_i}¬ⁱp₀ for small Dl_i>0. Clearly p₀=_{-Dl_i}¬ⁱp₀ .
    For j¹i, the specific u_{j_p} streamline through p₀ (i¹1) at p with associated directed derivative d_{l_j} generates a distinct path p_j^*(l_j) = _{-Dl_i}¬ⁱp_j(l_j) passing through point p₀ = _{-Dl_i}¬ⁱp₀ known as the Lie drag of u_{j_p} . [ Here ^* indicates "new" or "modifed" rather than a multivector dual ] . This path need not be the streamline at p₀ 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 p₀ a tangent vector written (ⁱÝ_{-Dl_i}(u_{j_p}))(p₀) fully within I_p₀ and an associated streamlined differential operator d_{l_j^*} = (ⁱÝ_{-Dl_i}(d_{l_j}))(p₀) .
    Though d_{l_j^*} is not one of the d_{l_k}, it is not general. Because of its construction it commutes with d_{l_i} (ie. d_{l_j^*}(d_{l_i}(F))) = d_{l_i}(d_{l_j^*}(F))) . [ Proof : .... .]
    We refer to     (ⁱÝ_{-Dl_i}(u_{j_p}))(p₀)     ( or (ⁱÝ_{-Dl_i}(d_{l_j}))(p₀) ) as the Lie drag of u_{j_p} ( or d_{l_j} ) from p₀ to p₀= _{-Dl_i}¬ⁱp₀ .
    We define the Lie drag of a scalar field f(p) defined over C_M by (ⁱÝ_{-Dl_i}(f))(p) º f(_{-Dl_i}¬ⁱp) so that (ⁱÝ_e(f))(p) = e^ed_{l_i}f .                                              7

Lie Derivative
    The 1-vector Lie derivative of a scalar field is simply £_{d_{l_i}}f(p) = d_{l_i}f(p) .

    Since the Lie drag of u_{j_p} (or d_{l_j}) from p₀ to p₀ can be meaningfully subtracted from u_{i p} (or d_{l_i}) at p₀ . The Lie dervivative at p₀ of u_{j_p} with regard to 1-field u_{i p} is an operator defined (by its action on a general 1-field F) as
    ( £_{d_{l_i}}(d_{l_j}))F(p) º Lim_{Dl_i ® 0} [ ( (ⁱÝ_{-Dl_i}(d_{l_j}))(p₀)(F(p)) - d_{l_j}(F(p)) ) / Dl_i ] ï_p₀ = Lim_{Dl_i ® 0} [2(d_{l_i}×d_{l_j})F ï_p₀ ]
[ Proof : d_{l_j^*}F ï_p₀ = d_{l_j^*}F ï_p₀ - Dl_i(d_{l_i}(d_{l_j^*}(F))))ï_p₀ + _O(Dl_i²) = d_{l_j^*}F ï_p₀ + Dl_i(d_{l_i}(d_{l_j}(F))ï_p₀ + _O(Dl_i²) - Dl_i(d_{l_j^*}(d_{l_i}(F))ï_p₀ + _O(Dl_i²)
    Hence Lim_{Dl_i ® 0} [ ( (ⁱÝ_{-Dl_i}(d_{l_j}))(p₀) - d_{l_j} )F / Dl_i ] ï_p₀ = Lim_{Dl_i ® 0} [((d_{l_j^*} - d_{l_j} )F ï_p₀ ) / Dl_i ] = Lim_{Dl_i ® 0} [(d_{l_i}(d_{l_j}(F)) - d_{l_j^*}(d_{l_i}(F)) ) ï_p₀ ] which provides the result assuming d_{l_j^*} can be safely replaced in the limit by d_{l_j} .]
    Omitting the brackets , we thus have the 1-vector operator identity
    £_{u_{i p}}u_{j_p} = £_{d_{l_i}}d_{l_j} = 2 d_{l_i}×d_{l_j} = [ u_{i p},u_{j_p} ] .

Covariant Frame
    Let us now identify the spaces U_N and V_N and adopt fixed frames {e_i} and {eⁱ} in both. With regard to ¿ within V_N , {eⁱ} is not a reciprocal frame for {e_i}. Rather, we need the point-dependant (nonorthormal) covariant frame {eⁱ º g_x^-1(eⁱ)} which itself has ¿ reciprocal frame {e_i º g_x(e_i)}
[ Proof : eⁱ¿e_j = g_x(g_x^-1(eⁱ))e_j = eⁱ¿e_j     Also eⁱ¿e_j = g_x^-1(eⁱ)¿g_x(e_j) = g_x(g_x^-1(eⁱ))¿e_j = eⁱ¿e_j .]
    For brevity we ommit an x subscript from e_i , eⁱ , 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_av_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 vⁱe_i as the 1-vector v_p^£ º å_i=1^N vⁱe_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_ivⁱe_i where vⁱ º eⁱ¿v_p .
    (0;1)-tensor v_p has "covariant" coordinate representation å_iv_ieⁱ where v_i º v_p(e_i) = e_i¿v_p and eⁱ is the 1-covector of e_i which we associate with the 1-vector eⁱ.

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^Dl_i_p₀(a) denote the parallel transport of a from p₀ for arclength Dl_i along the u_{i p} streamline.
We obviously want ⁱ^Dl_i_p₀(a)ÙI_{_{Dl_i}¬ⁱp₀} = 0 and we might therefore think of defining ⁱ^Dl_i_p(a) = ¯_{I_{_{Dl_i}¬ⁱp₀}}(u_{j_p})~ whenever ¯_{I_{_{Dl_i}¬ⁱp₀}}(u_{j_p}) is nonnull but instead we simply postulate a 1-vector ⁱ^Dl_i_p(a) within I_{_{Dl_i}¬ⁱp₀} that satifies ⁱ^Dl_i_p(a)² = a² . We will henceforth assume a to be a unit vector.

The normalisation condition for ⁱ^Dl_i_p(a) means that the right-connection b_a,p(Dl_i) º a^-1ⁱ^Dl_i_p(a) for ⁱ^Dl_i at p satisfies b_a,pb_a,p^§ = 1 and is thus a 1-rotor expressable as a 2-spinor e^{w_{u_{i p} a _p}(Dl_i)} , where w_{u_{i p} a _p}(Dl_i) = cos^-1(a^-1¿ⁱ^Dl_i_p(a) (a^-1Ùⁱ^Dl_i_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_{u_{i p} a _p}(e) = _O(e²) .
    ⁱ^e_p(a) º ae^{w_{u_{i p} a _p}(e)} = a + a¿w_{u_{i p} a _p}(e) + _O(e²)
where for given scalar e, w_{u_{i 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_{j_p},d_{l_i},e) º u_{j_p} - ⁱ^e_p(u_{j_p}) = -u_{j_p}¿w_{u_{i p} u_{j_p} _p}(e) + _O(e²)
      is a 1-vector lying in general neither in I_p nor I_{_e¬ⁱp}. 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¬ⁱu_{j_p} - ⁱ^-e_{_e¬ⁱp}(_e¬ⁱu_{j_p})] = -g_p(u_{j_p},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_{j_p},u_{i p}) º Lim_{e ® 0}[e^-1g_{j p}(u_{i p},e) ] we obtain the connection, the 1-vector component of d_{l_i} u_{j_p} ï_p that exists outside I_p.
    g_p(u_{j_p},u_{i p}) = -u_{j_p}¿w_{u_{i p} u_{j_p}} = w_{u_{i p} u_{j_p}}×u_{j_p} .
    g_p(u_{j_p},u_{i p}) is orthogonal to both u_{i p} and u_{j_p} at p .

A linear connection (aka. affine connection) derives from an assumed bilinearity of g_p which can then can be fully defined by M² 1-fields g_p( h_{j_p}, h_{i_p}) which then effectively define _q^p C(a). Indeed, some authors define ⁱ^Dl_i as the consequence of a given linear connection.
We say the connection is symmetric if g_p(u_{i p},u_{j_p}) = g_p(u_{j_p},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 {uⁱ_p}. Because w_{u_{i p} u_{j_p}} is a 2-blade perpendicular to I_p, we have w_{u_{i p} u_{j_p}} ×u_{i p} = 0     i¹j . Thus we can define M pure bivectors w_{u_{i p} _p} º å_j=1^M w_{u_{i p} u_{j_p}} which satisfy g_p(u_{j_p},u_{i p}) = w_{u_{i p} _p}×u_{j_p}
    The connection is symmetric if u_{j_p}¿w_{u_{i p} _p} = u_{i p}¿w_{u_{j_p} _p}     " 1£i,j£M .
    A linear connection arises if w_{u_{i p} a} is linear in a.
[ Proof : Since the u_{i p} are orthogonal g_p((å_j=1^M b^ju_{j_p}),(å_i=1^M aⁱu_{i p})) = (å_j=1^M b^ju_{j_p})¿w_{u_{i p} å_{i ₁}M=} aⁱu_{i p})}
= (å_j=1^M b^ju_{j_p})¿(å_i=1^M aⁱw_{u_{i p} u_{i p}}) = å_i=1^M å_j=1^M aⁱb^j(u_{j_p}¿w_{u_{i p} u_{i p}}) = å_i=1^M å_j=1^M aⁱb^jg_p(u_{j_p},u_{i p}) .]

Within the mapspace
Parallel transport in the mapspace ⁱ^e_x(b) º ¦^Ñ_{x+ee_i}^-1( ⁱ^e_¦(x)(¦^Ñ_p(b)) ) preserves ¿ instead of ¿.
[ Proof : ⁱ^e_p(a) ¿_{x+ee_i} ⁱ^e_p(b) = ⁱ^e_p(¦^Ñ_x(a))¿ⁱ^e_p(¦^Ñ_x(b)) = ¦^Ñ_p(a)¿¦^Ñ_x(b) = a¿_xb .]

    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+ee_i}^-1(ⁱ^e_p(¦^Ñ_x(b))) = b - ¦^Ñ_{x+ee_i}^-1(¦^Ñ_p(b) - g_p(¦^Ñ_p(b),d_{l_i},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^-1G_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¬ⁱp) back from _e¬ⁱp 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_{u_{i p}}(b) = d_{l_i}b + g_p(b,d_{l_i}) = d_{l_i}b + w_{u_{i p} u_{j_p} _p}×b
[ Proof : Ð_{u_{i p}}(b(p)) º Lim_{e ® 0}[ ⁱ^-e_{_e¬ⁱp}(b(_e¬ⁱp)) - v_j(p) / e ]
    = Lim_{e ® 0}[ ( ⁱ^-e_{_e¬ⁱp}(b(_e¬ⁱp)) - b(_e¬ⁱp)) + ( b(_e¬ⁱp) - b(p)) / e ] = g_p(b,d_{l_i}) +d_{l_i}u_{j_p} .]
    In the limit, since l_i is the natural (proper) parameterisation, we can consider d_{l_i} to be equivalent to the N-D directional derivative Ð_{u_{i 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^Dl_i_p(u_{j_p}) = e^Dl_iÐ_{l_i}_e¬ⁱu_{j_p} = e^Dl_iÐ_{l_i} e^Dl_id_{l_i} u_{j_p} .
[ Proof : ⁱ^Dl_i_p(u_{j_p}) = v_j(_{Dl_i}¬ⁱp) + Dl_iÐ_{l_i}v_j(_{Dl_i}¬ⁱp) + Dl_i²Ð(u_{i p})²v_j(_{Dl_i}¬ⁱp) + ... .]

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_pÐ_{u_{k p}} = Ñ_{[C_M]} + å_k=1^M u^k_p(w_{u_{k p}}×) .

Within the mapspace
    Ð_a induces a differential operator within M-D parameter space M_apÌV_M
    Ð_a^ð(B(x)) = ¦^-Ñ_p(Ð_{¦^Ñ_p(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_{e_i}^ðe_j = å_k=1^M G^k_jie_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^Dl_i_p(u_{i p}) = u_{i p}(_e¬ⁱp)     " 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_{u_{i 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_{u_{i 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 h₁,h₂ and normal n=h₁×h₂ forming the SYMBOL">t) is the curvature of the 1-curve confined in trihedron) at p. Taking p as the origin, the induced coordinates (x¹,x²,x³) satisfy
    x³ = ½ (¶²x³/¶(x¹)²) x¹² + (¶²x³/¶x¹¶x²) x¹x² + ½ (¶²x³/¶x²¶) x² |²
    By rotation about e₃ we can diagnonalise this form as
    x³ = ½ (k₁(x¹)² + k₂(x²)²)     where k₁,k₂ are known as principle curvatures.
    If the surface is parameterised as p(l₁,l₂) with h_{1_p} = ¶p/¶l₁ , h_{2_p} = ¶p/¶l₂ then
    ¶²p/¶l_i¶l_j º ¶ h_{i_p}/¶l_j = å_k=1² G^k_ij h_{k_p} + b_ij h_{3_p}     i,j,k Î {1,2}
    ¶ h_{3_p}/¶l_i = - å_{k l=1}² g^klb_li h_{k_p}     i Î {1,2}

2-Curvature
    The second directional coderivative operator Ð_{l_i}Ð_{l_j} is not symmetric in i,j.
    We can take the skewsymmetrol of it to obtain the Rieman Curvature operator
    c_{_pu_{i p}Ùu_{j_p}}(u)   =   c_{_pu_{i p}Ùu_{j_p}}×u   =   2(Ð_{l_i}×Ð_{l_j})u = ( Ð_{¯(l_i)}w_{u_{j_p}} - Ð_{¯(l_j)}w_{u_{i p}} + w_{u_{i p}}×w_{u_{j_p}}) . u
[ Proof :

Ð_bÐ_au = ( Ð_¯(b) + w_b×)( Ð_¯(a) + w_a×)u
= ( Ð_¯(b) + w_b×)( Ð_¯(a)u + ½(w_au-uw_a)) = ( Ð_¯(b) Ð_¯(a)u + ½( Ð_¯(b)w_au- Ð_¯(b)uw_a) +w_b×( Ð_¯(a)u +   ½(w_au-uw_a))
= Ð_¯(b) Ð_¯(a)u + ½( Ð_¯(b)w_au - Ð_¯(b)uw_a + w_b Ð_¯(a)u - Ð_¯(a)uw_b ) + ½w_b×(w_au-uw_a)
= Ð_¯(b) Ð_¯(a)u + ½( Ð_¯(b)(w_au) + w_b Ð_¯(a)u - Ð_¯(a)(uw_b) - Ð_¯(b)(uw_a) ) + 1/4( w_bw_au - w_buw_a - w_auw_b + uw_aw_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_au)- Ð_¯(a)(w_bu) + 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 ¦(p₀) + ¦^Ñ_p(dp) for ¦ near p₀.

Geometric interpretation
    If [u_{i p},u_{j_p}]=0 so that we have a loop _{-Dl_i}¬^j_{-Dl_i}¬ⁱ_{Dl_j}¬^j_{Dl_i}¬ⁱ p = p then
    c_{_pu_{i p}Ùu_{j_p}}(u_{k p}) = (Dl_iDl_j)^-1 ^j^-Dl_j( ⁱ^-Dl_i( ^j^Dl_j( ⁱ^Dl_i(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_{_pu_{i p}}u_{j_p})(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_{j_p} , divided by the "area" of that loop. This must lie in I_p so we have
    c_{_pu_{i p}Ùu_{j_p}}(u_{k p}) = å_l=1⁴ c^l_ijku_{l_p}     where scalar c^l_ijk º u^l_{_p}¿c_{_pu_{i p}Ùu_{j_p}}(u_{k p}) .
[ Proof : ^j^-Dl_j ⁱ^-Dl_i ^j^Dl_j ⁱ^Dl_i(u_{k p}) = e^{-Dl_jÐ_{l_j}} e^{-Dl_iÐ_{l_i}} e^Dl_jÐ_{l_j} e^Dl_iÐ_{l_i}u_{k p} = ...[tedious expansion]... = u_{k p} + 2Dl_iDl_j(Ð_{l_i}×Ð_{l_j})u_{k p} + _O(Dl³) .]

Symmetries and Bianci Relations
    It is possible to express C_{_xe_ij} in terms of the g_ij and their derivatives and doing so reveals some more symmetries:
    C^l_kij º e^l¿(C_{_xe_ij}(e_k)) = eⁱ¿(C_{_xe_lk}(e_j)) = -e^k¿(C_{_xe_ij}(e_l)) .
    The last of these reflects the reversability of parallel transport.
    We also have (from the G expression for Cⁱ_jkl) the first Bianci identity
   e_k¿C_{_xe_ij} + e_j¿C_{_xe_ki} + e_i¿C_{_xe_jk} = 0 .
    which can also be expressed as     C_{_xe_ij} ¿ e_kl = C_{_xe_kl} ¿ 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 M⁴ element matrix representation of C_x has only M²(M²-1)/12 independant elements (20 out of 256 for N=4).

Ð_{e_i}^ðC_{_xe_jk} + Ð_{e_j}^ðC_{_xe_ki} + Ð_{e_k}^ðC_{_xe_ij} = 0₂ is known as the second Bianci identity.

    The codivergence of the Ricci tensor Ñ^ð_x¿ c_{_pa} º å_ieⁱ¿Ð_{e_i} c_{_pa} = ½Ð_aC_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
    £_{u_{i p}}u_{j_p} = Ð_{u_{i p}} Ð_{¯(u_{j_p})} - Ð_{u_{j_p}} Ð_{¯(u_{i 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 £_{u_{i p}}b_p = 0 . We must then have
    Ð_{u_{i p}}² Ð_{¯(b_p)} = 2(Ð_{u_{i p}}×Ð_{b_p}) Ð_{¯(u_{i p})} .
[ Proof : Ð_{u_{i p}} Ð_{¯(b_p)} = Ð_{b_p} Ð_{¯(u_{i p})} Þ Ð_{u_{i p}}² Ð_{¯(b_p)} = Ð_{u_{i p}}Ð_{b_p} Ð_{¯(u_{i p})} = 2(Ð_{u_{i p}}×Ð_{b_p}) Ð_{¯(u_{i p})} + Ð_{b_p}Ð_{u_{i p}} Ð_{¯(u_{i p})} = 2(Ð_{u_{i p}}×Ð_{b_p}) Ð_{¯(u_{i p})} since u_{i p} geodesic Þ Ð_{u_{i p}} Ð_{¯(u_{i p})} = 0 .]
    Thus we have a second geometric interpretation of the curvature tensor as the second derivative of a vector dragged along a geodesic confluence, informally: a measure of geodesic "splay".

[Under Construction]

Extremal M-Curves
    A standard problem in classical mechanics is to determine the form of the curve assumed by a rope of length a hanging motionless between two fixed points a distance less than a apart under a uniform gravitational field, which is an extremal 1-curve in N=2 dimensions.
    If we assume the rope to hold a curve shape p(s)=(x(s),y(x)) (the Y axis being gravitationally vertical) that does not loop "over" itself vertically, we have ds² = dx² + dy² so that the length constraint is a = ò₀¹ ds = ò_x₀^x₁ dx(1+y'²)^½ where y' º dy/dx.
    The mechanical constraint is that total gravitational potential energy ò₀¹ 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 z₀ and z₁ such that y(x₀,z₀,z₁)=y(x₀) ; y(x₁,z₀,z₁)=y(x₁) " z₀,z₁ (boundary condition) ; y(x,0,0)=y(x) ; and y(x,z₀,z₁) is twice continuosuly differentiable in all parameters.
    Forming K(z₀,z₁) = ò_+x0^x₁ dx h(x,y(x,z₀,z₁),y'(x,z₀,z₁)) the extremal condition requires ¶K/¶z₀ and ¶K/¶z₁ to vanish at z₀=z₁=0 and this leads (via integration by parts) to the Euler-Lagrange equation ¶h/¶y = d/dx(¶h/¶y') .
    In the case of the hanging rope, h(x,y,y') = (rgy-l)(1+y'²)^½ and setting l = -rgy₀ we have h(x,y,y') = rg(y-y₀)(1+y'²)^½ = rg(y-y₀)(1+(y-y₀)'²)^½ for some constant y₀ .
    Setting h = y-y₀ the Euler-Lagrange equation is rg(1+h'²)^½ = d/dx   rghh'(1+h'²)^-½ with first order simplification
    h' rghh'(1+(y-y₀)'²)^-½ - rgh(1+h'²)^½ = b Þ rg(1+h'²)^-½h = b having solution h = b cosh(x-c)/b) so the rope has shape y₀ + b cosh((x-c)/b) where c,y₀ and b are chosen to match the given endpoints.

We say an M-curve C_M is M-extremal for an action functional ¦_{C_M}:U^N ® U_N if some particular magnitude measure (content, scalar part, square, modulus or whatever) of ò_{C_M} d^M-1p ¦_{C_M}(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 ¦_{C_M}. 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 x₁(t),..,x_k(t) varies from time t₀ to t₀ so as to 1-extremise a (usually real scalar) integral measure
    S(t) º S(t₀) + ò_t₀^t₀ dt L(t,x₁(t),..,x_k(t),x'(t),...,x_k'(t))     with the action functional L(t,x₁(t),..,x₁'(t)) known as the Lagrangian of the system.
    L is usually assumed to be real-scalar valued and independant of p"(t) and higher derivatives and is traditionally assumed to seperate into "kinetic" and "potential" componenets independent of time and velocity respectively as L(t,p(t),p'(t)) = T(p(t),p'(t)) - V(t,p(t)) but more generally we might postulate a multivector-valued Lagrangian of k multivector-valued variables and their first temporal derivatives L(t,x₁(t),..,x_k(t),x₁'(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_{B_ase(t)} d^N-1p L(t,p,p') that spacially integrated function is known as a Lagrangian density. Note that such requires a "velocity" p' be associated with every point p in 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 z₀ and z₁ such that (s₀,z₀,z₁)=(s₀) ; (s₁,z₀,z₁)=(s₁) " z₀,z₁ (boundary condition) ; (s,0,0)=(s) ; and (s,z₀,z₁) is twice continuosuly differentiable in all parameters. Forming K(z₀,z₁) = ò_s₀^s₁ ds¦(s,(s,z₀,z₁),'(s,z₀,z₁)) the extremal condition requires ¶K/¶z₀ and ¶K/¶z₁ to vanish at z₀=z₁=0 and this leads (via integration by parts) to the Euler-Lagrange equation ¶¦/¶ = (d/ds)(¶¦/¶') .
    In particular cases where ¦(s,,') = ¦(,') so that ¶¦/¶s=0 and there is no explicit s dependence, we have d/ds('¶¦/¶' - ¦)= 0 giving a first order differential equation '¶¦/¶' - h = constant .
[ Proof :   y" ¶h/¶y' + y'd/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,x₁,x₂,..x_k,x₁',x₂',..x_k') subject to M scalar constraints g_i(s,x₁,x₂,...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,x₁,..,_xK,x₁',..,x_k') = ¦(s,x₁,..,_xK,x₁',..,x_k') + å_j=1^M l_j(s)g_j(s,x₁,..,x_k,x₁',..,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,x₁,x₂,..,x_k,x₁',x₂',..,+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_{x_i'})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
    ¶_xF(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 eⁱ(¶/¶xⁱ).

Generalised Momentum

    When F is a scalar-valued Lagrangian we have (x_i'_*¶_{x_i'})L = x_i'_*(¶_{x_i'}L) and multivector m = Ñ_{x_i'}L(x₁,..x_k,x₁',..,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^-1a_x¿' to L introduces qc^-1a_x to the momentum

If L(t,x₁(t),..,x_k(t),x₁'(t),..,x_k'(t)) = L(x₁(t),..,x_k(t),x₁'(t),..,x_k'(t)) so that there is no explicit t dependance then the Hamiltonian H(x₁(t),..,x_k(t),x₁'(t),..,x_k'(t)) º å_i=1^k (x_i'_*Ñ_{x_i'})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'_*¶_{x_i'})L = Ñ_{x_i}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_jx_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_jx_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_iy_i' - å_i=1^k ¶¦/¶y_i'y_i"   =   å_i=1^k y_i' d/dx ¶¦/¶y_i' - ¶¦/¶ - å_i=1^k ¶¦/¶y_iy_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 ò₀^t ds|d¹p¿g(d¹p)|^½ . Integrating a square root can be messy but fortunately such paths also extremise ò₀^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 ò₀^t ds ds L(p(s),p'(s),s) by solving the N Euler-Lagrange equations d/ds(¶L/¶pⁱ') = ¶L/¶pⁱ) for i=1,2,...N .
    L(p,p',s) will generally depend on p via _gp but if _gp has a symmetry such that ¶L/¶pⁱ=0 for a particular coordinate i then ¶L/¶pⁱ' is constant along any geodesic path.
    Geodesic flow can consequently be viewed as arising from a Lagrangian density L(p(t),p'(t)) = |p'(t)¿_gp(p'(t))|^½ . with particles following timelike trajectories that extremise (minimally) their proper time (arclength).
    If t is proper time parameterisation then L(p(t),p'(t)) = |p'(t)²|^½ = 1 along a geodesic.
    Electrodynamical forces due to a 1-vector four-potential 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_pÙa_p .
[ Proof : L = m(-p'²)^½ - q p'¿a_p Þ Ñ_pL = 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_pa_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_pÙa_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]}²y_p = 0 over C_M .
In electrodynamics we have L = - 2^-4p^-1 f_p² + c^-1j_p¿a_p where c is scalar lightspeed and f_p = Ñ_pÙa_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 ¯_{I_p}(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_{i_p} (potentially nonorthonormal "tangent frame").
    Their g_i is our e_i (orthonormal "fiducial (basis) frame").

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

Ð_bÐ_au	=	( Ð_¯(b) + w_b×)( Ð_¯(a) + w_a×)u
	=	( Ð_¯(b) + w_b×)( Ð_¯(a)u + ½(w_au-uw_a)) = ( Ð_¯(b) Ð_¯(a)u + ½( Ð_¯(b)w_au- Ð_¯(b)uw_a) +w_b×( Ð_¯(a)u + ½(w_au-uw_a))
	=	Ð_¯(b) Ð_¯(a)u + ½( Ð_¯(b)w_au - Ð_¯(b)uw_a + w_b Ð_¯(a)u - Ð_¯(a)uw_b ) + ½w_b×(w_au-uw_a)
	=	Ð_¯(b) Ð_¯(a)u + ½( Ð_¯(b)(w_au) + w_b Ð_¯(a)u - Ð_¯(a)(uw_b) - Ð_¯(b)(uw_a) ) + 1/4( w_bw_au - w_buw_a - w_auw_b + uw_aw_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_au)- Ð_¯(a)(w_bu) + 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