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 Fp by
    FÑ[CM]pp(a) º FpÑ(a) º (a*Ñ[CM]p)F(p)   =   (a*¯(Ñp))F(p)   =   (¯(a)*Ñp)F(p)   =   FpÑ(¯(a)) º Fpѯ(a) which gives us
    Ñ   =   ѯ     abbreviating FpÑ(a) = Fpѯ(a) º FpÑ(¯(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_ei(Ð^p_eiF_p)(a,b,...) = å_i=1^M (Ð^p_eiF_p)(eⁱ²e_i,b,...) = å_i=1^M (Ð^p_eiF_p)(eⁱ,b,...)
    = å_i=1^M F_pÑ(Ð^p_eieⁱ,b,...) = F_pÑ(Ñ,b,...)  .]

Hypercurve
    For M=N-1 we have Ñ = ¯(Ñ) = Ñ - n_p(n_p¿Ñ) = Ñ - n_pÐ_np .
    Since n_p¿Ð_an_p = 0 " a , Ñ_pn_p lies within i_N-1p and ^[Ñ_pn_p] = n_p^-1ÙÐ_npn_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 ¯_Ip(a)-directed dirivative
    Ð = ¯ Ð_¯() º ¯_° Ð_¯()     abbreviating Ð_aF_p º ¯_IpÐ_¯IpaF_p

    The symbol Ð can thus be thought of as an abbreviation
    Ð   =   ¯ Ð_¯()   =   Ð_¯()¯ - ¯^Ñ     abbreviating Ð_b(a_p) = ¯( Ð_¯(b)a_p) = Ð_¯(b)(¯(a_p)) - ¯^Ñ(a_p,b)     _{[ HS 5-1.1 ]} . The underscore serves to remind us of a p and I_p (ie. C_M) dependance.  
[ Proof :   Ð_¯(d)(¯(a_p)) = ( Ð_¯(d)¯)(a_p) + ¯( Ð_¯(ap)) Þ Ð_d(a_p) º ¯( Ð_¯(d)(a_p)) = Ð_¯(d)(¯(a_p)) - ( Ð_¯(d)¯)(a_p) = Ð_¯(d)(¯(a_p)) - ¯^Ñ(a_p)  .]

    The directed coderivative of an extended field is defined by
    (Ð_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_pa_1p + Ð_pa_2p = Ð_p(a_1p+a_2p) ; and so forth ) and then intone "as defined so mote it be" three times at midnight. Another is based on projections [see General Relativity ].    Many appeal to a notion of "parallel transport" which is defined (or not) in a variety of ways.
    We define the undirected coderivative 1-vector operator as the projected tangential derivative
    ÑF_p º Ñ_[Ip]pF_p º ¯_Ip(Ñ_[Ip]pF_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_pÑ))     by the projected product rule .
    This is particularly clear when expressed in coordinate terms with a fortuitous basis as Ñ_p(¯(a_pÑ)) = å_i=1^N eⁱ Ð_¯(ei)(¯(a_pÑ)) = å_i=1^M eⁱ(¯( Ð_¯(ei)a_p)) = å_i=1^M ¯(eⁱ)(¯( Ð_¯(ei)a_p)) = å_i=1^M ¯(eⁱ( Ð_¯(ei)a_p)) = ¯(eⁱ(å_i=1^M  Ð_¯(ei)a_p)) = ¯(Ñ_p(Ap)) º Ñ_p(Ap) .

    For an extended field F_p(a_1p,a_2p,...a_kp) with a_ipÎU_N we have a-directed primary coderivative
    Ð_↓aF_p(a_1p,a_2p,...a_kp) º (Ð_aF_p)(a_1p,a_2p,...a_kp)
     = Ð_a(F_p(a_1p,a_2p,...a_kp)) - F_p(Ð_a(¯_Ip(a_1p)),a_2p,...a_kp) - F_p(a_ip,Ð_a(¯_Ip(a_2p)),,...a_kp) - F_p(a_ip,a_2p,...,Ð_a(¯_Ip(a_kp)))
and an i^th exterior  coderivative
    Ñ^i→¿_p F_p(a_1p,a_2p,...a_kp) º ¯_Ip(Ñ^i→¿_p F_p(a_1p,a_2p,...a_kp)) , 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 Ѳ , while not a scalar operator, does not increase grade .

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

Projection Differential (1.2)-tensor     ¯^Ñ

    ¯^Ñ(a_p,d) º ¯_Ip^Ñ_p(a_p,d) º Ð_d(¯_Ip(a_p)) - ¯_Ip(Ð_da_p)     (informally) the rate of change of ¯_Ip(a) in direction d , is of of less interest than ¯^Ñ,  the primary tangential 1-differential of ¯_Ip (ie. the second tangential 1-differential of the identity function) ,
    ¯^Ñ_p(a_p,d)   =   ¯^Ñ_p(a_p,¯d)   =   1^Ñ2(a_p,d)   =   ( Ð_¯(d)¯_Ip)(a_p)     =   Ð_¯(d)(¯(a_p)) -  ¯( Ð_¯(d)a_p)   =   2( Ð_¯(d)ׯ)(a_p)  .
    The symmetry ¯^Ñ(a,b) =¯^Ñ(b,a) for a,b Î I_p follows from the symmetry of 1^Ñ2 in U_N.
    Tangentially differentiating the outtermorphism result ¯_Ip(aÙb)=¯_Ip(a) Ù ¯_Ip(b) yields
    ¯^Ñ(aÙb,d) =  ¯^Ñ(a,d) Ù ¯(b) +  ¯(a) Ù ¯^Ñ(b,d)     _{[ HS 4-2.6 ]} and hence
    ¯^Ñ(a₁Ùa₂Ù....a_k,d) = å_i=1^k (-1)ⁱ⁺¹ ¯^Ñ(a_i,d)Ù¯_Ip(a₁)Ù... ¯_Ip(a_i-1)Ù ¯_Ip(a_i+1)Ù... ¯_Ip(a_k) . _{[ HS 4-2.31 ]}

    ¯¯^Ñ   =   ¯^Ñ^       ;     ^¯^ѯ   =   ¯^ѯ       _{[ 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¿(¯^ѯ) = 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)^2ׯ = Ð ^ׯ     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     [ѯ]
    We define the shape [ѯ] of C_M to be the undirected tangential 1-derivative of the projector , an abbreviation of
    [ѯ](a_p) º Ñ_ß[CM]p¯_Ip(a_p) º Ñ_ß ¯_Ip(a_p) = (Ñ_b Ð^p_↓(b) ¯_Ip)(a_p) = Ñ_b ¯^Ñ(a_p,b) = Ñ_b ¯^Ñ(a_p,b) = Ñ_b ¯^Ñ(a_p,b) .

    [ѯ](a_p) = 0 so [ѯ] annihilates scalars.

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

    For general operands we have   [ѯ](a)_<k+1> = [ѯ](¯_Ip(a_<k>))    ;     [ѯ](a)_<k-1> = ¯_Ip([ѯ](a_<k>))     and so
    ѯ = ѯ + [ѯ]¯ = ѯ + [ÑÙ¯]     _{[ HS 4-3.7a ]} with Ñ¿¯ = Ñ¿¯ . Hence
    ^ѯ = [ѯ]¯ = [ÑÙ¯]¯
[ Proof :  Ñ(¯(a_p)) = Ñ(¯(a_p)) + ^Ñ(¯(a_p)) = ѯ(a_p) + ^( [ѯ](a_p) + ѯ(a_pÑ) ) = ѯ(a_p) + ^([ѯ](a_p) + Ñ(a_p))
    = ѯ(a_p) + ^[ѯ](a_p) = ѯ(a_p) + [ѯ]¯(a_p) = (Ñ+[ѯ])¯(a_p) .  .]
    With tangential operands ¯(a_p)=a_p understood we can write this as
    Ñ = Ñ + [ѯ] = Ñ + [ÑÙ¯]       _{[ HS 4-3.6a ]} and   ^Ñ = [ѯ] = [ÑÙ¯] .

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

    Scalar ¯₁ [ѯ](a) =[ѯ](^_Ip(a)) = (Ñ_p¿¯_Ip)(a) is expressible as (Ñ_b[ÑÙ¯](b))¿a where 1-vector
    Ñ_b[ÑÙ¯](b) = º Ñ_Þ[ÑÙ¯](b) = Ñ_Þ¿[ÑÙ¯](b) º Ñ_Þ¿[ÑÙ¯](b)

    [¯(Ñ)]   =   ¯_Ñ(Ñ)   =   Ñ_Þ [ÑÙ¯]   =   Ñ_Þ¿[ÑÙ¯]   =   [Ñ¿¯]^¿     _{[ HS 4-2.20 ]}
is known as the spur 1-field of C_M and lies outside I_p .

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

    1-vector ¯_Ip(b)¿[ѯ](a) = ¯_Ip(b)¿[ÑÙ¯](a) is normal to C_M , ie. ¯_Ip ( ¯_Ip(b)¿[ѯ](a)) = 0 .

    [ѯ](a₁Ùa₂Ù...a_k) = å_i=1^k.  (-1)ⁱ⁺¹ [ [ѯ](a_i) Ù ¯_Ip(a₁Ù..Ùa_i-1Ùa_i+1..Ùa_k) + ¯_Ip([ѯ](a_i) × (a₁Ù..Ùa_i-1Ùa_i+1..Ùa_k) ]     _{[ HS 2.41c ]} provides the extension of [ѯ] to general multivectors.

    [ѯ]  = Ñ_Þ² ¯^Ñ     abbreviating [ѯ](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 (ѯ)(a_p) = Ñ_b¯^Ñ(a_p,b) º Ñ_Þ²¯^Ñ(a_p,b)     and Ñ_¯ß ¯(a_p) º [ѯ](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 [ѯ] immediately provide
    [ѯ]^ = [Ñ¿¯] = Ñ_Þ²¿¯^Ñ     ;     [ѯ]¯ =  [ÑÙ¯] = Ñ_Þ²Ù¯^Ñ       _{[ HS 4-2.16 ]}.
[ Proof :  [ѯ](a) º (Ñ_p¯))(a) = Ñ_b¯^Ñ_p(a,b) so .... ???  .]

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

    We have the following Shape Properties:

1.   [ÑÙ¯]^ = 0   ie. [ÑÙ¯](^_Ip(a_p)) = 0  " a_{p       [ HS 4-2.37a ]}
2.   [Ñ¿¯]¯ = 0 . ie. [Ñ¿¯](¯_Ip(a_p)) º (Ñ¿¯)(¯(a_p)) = 0   " a_{p     [ HS 4-2.37b ]}
3. ¯[ÑÙ¯] = 0     when acting on 1-vectors ie. ¯[ÑÙ¯](a)=0  " a . _{[ HS 4-2.30 ]}
   [ Proof : ¯(a)¿[ÑÙ¯](b) = ¯^Ñ(¯(a),b) _{[ HS 4-2.26 ]} Þ ¯( ¯(a)¿[ÑÙ¯](b) ) = ¯¯^Ñ(¯(a),b)   = ¯₌¯^Ñ(a,b) = 0 .
       Þ ??? ¯( a¿¯([ÑÙ¯](b)) )=0 ????  .]
   ^[Ñ¿¯] = 0     follows immediately from ^¯₁=0 with regard to 1-vectors.
4. I [ѯ] = I ×[ѯ] = I ×[ÑÙ¯]     abbreviating I_p×[ÑÙ¯](a) = I_p×[ѯ](a) = -[ѯ](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) = ([ѯ](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 :   ¯_Ip([ÑÙ¯](a)) º ¯ [ÑÙ¯](a) = 0 Þ [ÑÙ¯](a)=^([ÑÙ¯](a)) Þ [ÑÙ¯]([ÑÙ¯](a)) = 0 by 1.  .]
5. ¯₌[ѯ]     º     ¯[ѯ]¯     =     0        _{[ HS 4-2.45 ]}
   [ Proof : ¯([ѯ](¯(a))) = [ѯ](^(¯(a))) = [ѯ](0) = 0  .]   
   ^₌[ѯ]     º     ^[ѯ]^     =     0
   [ Proof : ^[ѯ]^ = ^¯[ѯ] = 0  .]   
6. ¯^Ñ(a,b) = ¯(a)×[ÑÙ¯](b) - ¯( a×[ÑÙ¯](b) ) = ¯(a)×[ѯ](b) - ¯( a×[ѯ](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)×[ѯ](b) .
   [ Proof : The projected product rule gives ¯(¯(a)[ÑÙ¯](b)) = ¯(a)¯([ÑÙ¯](b)) = 0 and as ¯(¯(a)Ù[ÑÙ¯](b)) = ¯(a)Ù¯([ÑÙ¯](b)) = 0 we must have ¯( ¯(a)×[ÑÙ¯](b) ) = 0  .]
7. [ÑÙ¯](b) = 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 geometrically 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.  .]
8. ¯^Ñ(a,b)=¯(a)¿[ÑÙ¯](b) - ¯(a¿[ÑÙ¯](b))     _{[ HS 4-2.27 ]}
   [ Proof :  a×b_<2> = a¿b_<2> combined with 5  .]
9. (¯^Ñ)²(F_p,a,b) =   ¯( [ѯ](b)×(¯(F_p)×[ѯ](a)) ) - ¯( F_p×[ѯ](a) )×[ѯ](b)
   [ Proof :  Applying Shape Property 6 twice while exploiting Shape Property 3
       ¯^Ñ(¯^Ñ(F_p,a),b) =   ¯(¯^Ñ(F_p,a))×[ѯ](b) - ¯( ¯^Ñ(F_p,a)×[ѯ](b) )
       = ¯(¯(F_p)×[ѯ](a) - ¯( F_p×[ѯ](a) ))×[ѯ](b)     -     ¯( (¯(F_p)×[ѯ](a) - ¯( F_p×[ѯ](a) ))×[ѯ](b) )
       = 0 - ¯²( F_p×[ѯ](a) ))×[ѯ](b) - ¯( (¯(F_p)×[ѯ](a) )×[ѯ](b) )   + 0     by the projected bivector commutation rule  .] If ¯(F_p)=F_p this reduces to
       ¯^Ñ(¯^Ñ(F_p,a),b) =   ¯( [ѯ](b)×(F_p×[ѯ](a)) )     _{[ HS 5-1.3 ]}   which we can abbrviate to
       ¯^Ñ2¯ = - ¯[ѯ]×[ѯ]ׯ .
10. [ѯ](I_p) = Ñ_pÙI_p = Ñ_pI_{p     [ HS 4-2.44 ]}
11. Pad -----
12. ¯_Ip(b)¿[ÑÙ¯](a) = ¯_Ip(a)¿[ÑÙ¯](b)     _{[ HS 4-2.28 ]}
13. ¯^Ñ_Ip(a,d) = ¯_Ip(b) ¿ [ÑÙ¯](d) - ¯_Ip(a¿[ÑÙ¯](d))     _{[ HS 4-2.27 ]}
14. ¯^Ñ_Ip([ÑÙ¯](a),d) = - ¯(I_p)([ÑÙ¯](a)×[ÑÙ¯](d))
15. Pad --++-------
16. a_*(¯_Ip^Ñ_p(d,b)) = b_*(¯_Ip^Ñ_p(d,a))
17. ¯_Ip^Ñ(d,aÙb) = ¯_Ip^Ñ(d,a) Ù ¯_Ip(b)   +   ¯_Ip(b) Ù ¯_Ip^Ñ_p(d,b)
18. ¯_Ip^Ñ_p(b,¯_Ip(a)) = ¯_Ip^Ñ_p(a,¯_Ip(b))
19. --- end list

    [ѯ]  = Ñ_Þ ¯^Ñ enables us to grade extend [ѯ] from the grade extension of_prl0g() as
    [ѯ](aÙb) = Ñ_c¯^Ñ(aÙb,c) = Ñ_c(¯^Ñ(a,c)Ù¯(b)+¯(a)Ù¯^Ñ(b,c))     so we immediately have [ѯ](^(a)Ù^(b))=0     _{[ HS 4-2.40c ]}.  Furthermore
    [ѯ](¯(a_j)Ù¯(b)) = ([ѯ](¯(a_j))Ù¯(b) + (-1)^j¯(a_j)Ù([ѯ]¯(b))     _{[ HS 4-2.40a ]}
[ Proof :  [ѯ](¯(a_j)Ù¯(b_k)) =  [ѯ]¯(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)
    = ([ѯ](¯(a_j))Ù¯(b_k) + (-1)^j¯(a_j)Ù([ѯ]¯(b_k))  .]
    In particular, [ѯ]¯(aÙb)  = [ѯ]¯(a)Ù¯(b)-[ѯ]¯(b)Ù¯(a) .

    [ѯ](^(a_j)Ùb) =   [ѯ](^(a_j)Ù¯(b)) =   [ѯ](^(a_j))Ù¯(b) + (-1)^j¯_Ñ(^(a_j))Ù(Ñ¿b)     _{[ HS 4-2.40b ]}
[ Proof :  [ѯ](^(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
    = ([ѯ]^(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-1p = Ð_b(n_pi) = (Ð_bn_p)i = n_p^Ñ(b)i so Shape Property 7 gives
    [ÑÙ¯](a) = i_N-1p^-1( Ð_¯(a)i_N-1p) = 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     [ѯ]²

    The shape tensor [ѯ] raises the grade of ¯_Ip(a) while lowering the grade of ^_Ip(a), preserving neither grade nor tangency (containment within I_p) but the squared shape or squape 1-tensor [ѯ]² preserves both with [ѯ]²  = ^₌([ѯ]²)  + ¯₌([ѯ]²) .
[ Proof :   Follows from ¯₌[ѯ] = ^₌[ѯ] = 0 and [ѯ]^ = ¯[ѯ] since
    [ѯ]² = (¯+^)[ѯ](¯+^)[ѯ](¯+^) = ^[ѯ]¯[ѯ]^  + ¯[ѯ]^[ѯ]¯ = ^[ѯ]²^²  + ¯[ѯ]²¯  .]
    We also have [ѯ]²^ = ^[ѯ]²   = [ѯ]¯[ѯ] º [ѯ]₌¯ . _{[ HS 4-2.46b ]}

    The squape tensor decomposes as [ѯ]_<2> + [ѯ]_<1> + [ѯ]_<0>

Ricci 1-tensor       ₍[ѯ]²¯₎
    The intrinsic squape 1-multitensor is the projection of the squape ¯[ѯ]²   = [ѯ]²¯ =   ¯₌([ѯ]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
    [ѯ]² ¯   =   (ÑÙÑ) ¯   =   (ÑÙÑ) ׯ     _{[ HS 5-1.28 5-1.29 ]} abbreviating
    [ѯ]²(¯(a_p)) = (ÑÙÑ)(¯(a_p)) = (ÑÙÑ)×(¯(a_p))     " a_p .
    Hence ÑÙÑ is grade-preserving.

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

    ¯[ѯ]² = -Ñ[ÑÙ¯] = -Ñ¿[ÑÙ¯]     abbreviating ¯[ѯ]²(b) = -Ñ_p[ÑÙ¯](b) = -Ñ_p¿[ÑÙ¯](b) . _{[ HS 5-1.19 ]}
[ Proof : Ñ_a   ¯[ѯ]^×(a,b) = -Ñ_aÐ_a[ÑÙ¯](b) = -Ñ_p[ÑÙ¯](b)     by Curvature Identity 2   below  .]

    ¯[ѯ]² is symmetric (self-adjoint).
[ Proof : ???  .]

Curvature 2-tensor     [ѯ]^×
    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 [ѯ]^×(a,b) º [ѯ](a)×[ѯ](b) = [ÑÙ¯](a)×[ÑÙ¯](b)
satisfies [ѯ]×(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
    [ѯ]^×   =   [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     [ѯ]^×Ñ

    This enables us to express the primary differential of the curvature:
    [ѯ]^×Ñ(a,b,c) º ([ѯ]^×(a,b))^Ñ(c) º Ð_ßc([ѯ]^×(a,b)) º (Ð_c([ѯ]^×))(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 ([ѯ]^×)^Ñ(a,b,c)  º  S_←abc Ð_ßc([ѯ]^×)(a,b)   =   0     _{[ HS 5-1.39 ]} which we can also write as ([ѯ]^×)^Ñ_→¿ = 0 . The curvature thus has vanishing exterior 1-differential.

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

    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
    [ѯ]^×(aÙb) º [ѯ]×(a,b) . It is natural to extend this to a multivector argument via [ѯ]^×(a) = [ѯ]^×(aÙ1) = [ѯ]×(a,1) = 0 . º [ѯ]×(a,b) . It is natural to extend this to a multivector argument via [ѯ]^×(a)=[ѯ]^×(a)=0

Intrinsic curvature 2-tensor       ¯[ѯ]^×

    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
    ¯( [ѯ](a)×[ѯ](b)) = ¯([ÑÙ¯](a)×[ÑÙ¯](b) )     and denote it by
    ¯[ѯ]^×     =     ¯[ÑÙ¯]^×
 with ¯[ѯ]^×(a,b) º ¯( [ѯ](a)×[ѯ](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       ¯₌[ѯ]^×

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

    (¯[ѯ]^×)^Ñ_→¿(a,b,c) = S_←abc ¯([ѯ]^×^(aÙb))×[ѯ]¯(c)     with the immediate consequence (¯₌[ѯ]^×)^Ñ_→¿(a,b,c) = 0 , ie. the Riemann curvature has vanishing exterior 1-codifferential, which is the traditional second Bianchi identity.
[ Proof :  Ð_ßc ¯[ѯ]^×(aÙb) º (Ð_c ¯[ѯ]^×)(aÙb) = ¯^Ñ , c)) + ¯( Ð_ßc[ѯ]^×(aÙb))
    Þ S_←abc (Ð_c ¯[ѯ]^×)(aÙb) = S_←abc ¯([ѯ]^×(aÙb))×[ѯ](c) .     Hence
    S_←abc (Ð_c ¯[ѯ]^×)(aÙb) = S_←abc ¯^Ñ , c)  + ¯( S_←abc Ð_ßc[ѯ]^×(aÙb)) = S_←abc ¯^Ñ , c)  + ¯( 0)
    = S_←abc (¯([ѯ]^×(aÙb))×[ѯ](c)   - ¯([ѯ]^×(aÙb) ×[ѯ](c) )       by Shape Proprety 6
    = S_←abc ¯([ѯ]^×(aÙb))×[ѯ](c)   - ¯(S_←abc (([ѯ](a)×[ѯ](b))×[ѯ](c) ) = S_←abc ¯([ѯ]^×(aÙb))×[ѯ](c)       by Jacobi Identity .  .]
    Hence S_←abc (Ð_c ¯[ѯ]^×)(aÙb) = S_←abc ¯( Ð_¯(c) ¯[ѯ]^×)(aÙb) ) = S_←abc ¯( ¯([ѯ]^×(aÙb))×[ѯ](¯(c)) )
    = S_←abc ¯( [ѯ]^×(aÙb))×[ѯ](¯(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  ¯( [ѯ](a)×[ѯ](b)) solely in terms of the intrinsic squape multitensor via
    ¯[ѯ]^×(aÙb) = ½( [ѯ]²(a)Ùb + aÙ[ѯ]²(b) - [ѯ]²(aÙb) )     " a,b Î I_{p     [ HS 4-2.48 ]}.

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

    The Riemann curvature is protractionless
    Ñ_aÙ¯[ѯ]^×(¯(aÙb))   =   0     _{[ HS 5-1.11 ]} .

    We also have
    Ñ_ßÙ¯[ѯ]^ׯ   =   0     _{[ HS 5-1.13a ]}
[ Proof : See _{[ HS p191 ]}  .]

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

1. ¯[ѯ]^×   =   -¯^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 -¯( [ѯ](a)×[ѯ](b) )  .]
2. ¯[ѯ]^×   =   [ÑÙ¯]^Ñ = (Ð [ÑÙ¯])     abbreviating ¯[ѯ]^×(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.  .]
3. ¯[ѯ]^×ׯ    =   -2(¯^Ñ)^2ׯ     abbreviating ¯([ѯ]^×(a,b))ׯ(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) =   ¯( [ѯ](b)×(¯(F_p)×[ѯ](a)) ) -   ¯( [ѯ](a)×(¯(F_p)×[ѯ](b)) )
       =   ¯( ¯(F_p)×([ѯ](b)×[ѯ](a)))     by Jacobi identity
       =    ¯(F_p)ׯ([ѯ](b)×[ѯ](a)))     by projected bivector commutation rule  .]
4. ¯[ѯ]^×ׯ   =   -2Ð^ׯ     abbreviating ¯([ѯ]^×(a,b))ׯ(F_p) = - 2(Ð_a×Ð_b)¯(F_p)     _{[ HS 5-1.4 ]}
   [ Proof :   Follows immediately from (¯^Ñ)^2ׯ = Ð ^ׯ and the previous identity.  .]
       c_pab)×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 b_2p = 0 for any bivector field b_2p
       This enables us to also regard the Riemann curvature as a pointer-dependant bivector-specific 1-tensor
       c_paÙbc) 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.

5. ¯[ѯ]²   =   Ñ_Þ ¯[ѯ]^×   =   Ñ_Þ ¿ ¯[ѯ]^×   =   Ñ_Þ ¿ ¯[ѯ]^×     provides the more traditional definition of the Rici 1-tensor as the contraction of the Riemann curvature 2-tensor . _{[ HS 5-1.17 ]}
   [ Proof :  ¯[ѯ]([ѯ](b)) = [ѯ]([ѯ](¯b) ) = [ѯ]([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) .  .]
6. ¯₌[ѯ]^×(aÙb)   =   (b¿Ñ_p) [ÑÙ¯](a)   =   Ð_b[ÑÙ¯](a)     _{[ HS 5-1.5c ]}
   expresses ¯₌[ѯ]^×(aÙb) as the b-directed primary coderivative of [ÑÙ¯](a), the geometric operator that sends I_p to Ð^p_aI_p .
   [ Proof :  ???  .]
7. ¯₌[ѯ]^×(aÙb)   =   ¯^Ñ_Ip([ÑÙ¯](b),a)   =   Ñ_uÙ¯^Ñ_aÑ_v ¯^Ñ(¯^Ñ(v,b),a)     _{[ HS 5-1.5b ]} .
   [ Proof :  ???  .]
8. ¯₌[ѯ]^×(aÙb)   =    (Ñ_uÙÑ_v)(¯^Ñ_a(u)¿¯^Ñ_b(v))     _{[ HS 5-1.5d ]} .
   [ Proof :  ???  .]
9. Ñ_ßÙ¯₌[ѯ]^×    =   0 abbreviating (ÑÙ¯₌[ѯ]^×)(aÙb)=0     _{[ HS 5-1.13a ]} .
   [ Proof :  ???  .]
10. ¯₌[ѯ]^×(Ñ_ÜÙb)   =   ÑÙ₍[ѯ]²¯₎(b)     _{[ HS 5-1.22 ]}.
    [ Proof : ¯₌[ѯ]^×_Ñ(ÑÙb) º ¯(¯₌[ѯ]^×_Ñ(ÑÙb)) = ¯((Ñ_Þ¿Ñ)¯₌[ѯ]^×(aÙb))     by the tangential gradifying substition rule
        = ¯( Ñ_Þ¿(ÑÙ¯₌[ѯ]^×(aÙb) + ÑÙ(Ñ_Þ.¯₌[ѯ]^×(aÙb)) )     via a¿(bÙc)=(a.b)c-bÙ(a.c)
        = ¯( Ñ¿0 + ÑÙ₍[ѯ]²¯₎(b)) ) º ÑÙ₍[ѯ]²¯₎(b)     by the Bianchi identity.  .]