This document is still under revision. All suggestions, critique, or comment gratefully received.

    This document assumes familiarity with Multivectors. Notations defined in that document are retained here. Note that we here use labels e₁,e₂,... to denote a typically fixed, "base", "universal", "fiducial" frame and h_ip  to denote tangent vectors. In much of the literature, e_i   represent tangent or otherwise "motile" vectors while s_i or g_i represent a "base frame" .
    This document makes extensive use of subscripts and superscripts to indicate dependencies usually "dropped" in conventional treatments and is, in consequnce, theoretically ambiguous. Does v_ip , for example, mean that v_i is defined over or dependant on p , or that v is a function of i_p? In practice, meanings will be clear in context.
    Tensors are traditionally a difficult concept but multivectors make them far easier to understand, manipulate, and generalise. They are fundamental to many applications so we address them here.

Notations
    Symbols such as d, d,¶, Ñ, and ð are used variously in the literature for various "differentiating" operators. We will introduce the unorthodox notation Ð^x_a for the "directed" derivative with regard to a multivector parameter x in a particular multivector "direction" a, and use Ñ to denote the un-directed ("splayed") derivatives traditionally denoted Ñ or ð . We will typically use d or d to denote a small scalar and d to denote a 1-vector interpreted as a (possibly large) "displacement". We will sometimes use dx to denote a a small change in multi-vector parameter x when ambiguity with multiplication by a scalar d cannot arise.

Multivector Functions as Tensors
    The traditional presentation of an N-dimensional tensor of integer rank r is a point-dependant N ^r element "array" or "matrix" defined with respect to a given N-dimensional coordinate frame, that transforms according to particular rules in accordance with transforms of the underlying coordinate frame. Multivectors provide an attractive alternative (and more general) formulation under which the conventional tensor product follows directly from the geometric product. More formal definitions of the following explicitly specify a "scalar source" from which to "build" linear combinations, but here we implicitly assume "real" scalars (from  or a (finite-precision) approximation thereof).

Fields
    A field is a function F: Â^p,q,r ® Â_p,q,r . In other words: a point-dependant multivector. If the function is (unit) k-vector valued, we have a (unit) k-field. A 0-field thus associates a scalar value with every point. A 1-field associates a 1-vector with every point.
    From a programmers' perspective, fields are functions having at least one 1-vector parameter. This "primary" parameter is usually interpreted as a point or position. When the primary argument is interpreted as a (scaled) direction rather than a point we will refer here to a directional k-field.

Tensors
    We regard an N-dimensional tensor of degree k as a point-dependant multilinear N-dimensional multivector-valued function of k N-dimensional 1-vectors. F_x  : (Â^p,q,r)^k ® Â_p,q,r where p+q+r = N . By multilinear we mean linear in each argument. If k=0 we have a point-dependant function taking no arguments and returning a (point-dependant) multivector, effectively a field. A (t;0)-tensor is thus a t-field, often refered to as an invariant tensor, though its "value" does in general vary with x,
    If F_x (a₁,a₂,..,a_k) = F_x (a₁,a₂,..,a_k)_<t>     (ie.. F_x  is t-vector valued) we say the tensor has type t and rank t+k   and refer to it here as a (t;k)-tensor .
    From a programmers' perspective, tensors are multivector-valued functions of at least one 1-vector argument, linear ("affine") in all but the primary argument.
    When t=k we refer to a k-tensor rather than a (k;k)-tensor . A k-tensor is thus a point-dependant k-vector-valued multilinear function of k 1-vectors. In particular, a 1-tensor is a point-dependant directional 1-field.
    The "scalar product" ¿ is a (0;2)-tensor, though we usually write a¿b in preference to ¿(a,b)   . The outter product Ù is a 2-tensor. The geometric product is a tensor of degree 2 but "mixed" type.

Forms
    If a (t;k)-tensor is skewsymmetric in its arguments so that F_x (a₁,a₂,..,a_k) =  L_x(a₁Ùa₂...Ùa_k) =  L_x(a_k) can be viewed as a function of a single k-blade rather than of k 1-vectors , then it is called a a skewsymmetric (t;k)-tensor or a (t;k)-multiform . When t=k we abbreviate to a k-multiform. If t=0 (ie. F_x  is scalar valued) then it is instead called a k-form. A 1-multiform is a 1-tensor. It can be shown [see Hestenes & Sobczyk] that any k-form can be expressed as L_x(a_k)=  u_k¿a_k where  u_k is a point-dependant k-vector .

    If a k-multiform maps any given k-blade to another k-blade (rather than to a k-vector) then we say the multiform is blade preserving. A 1-tensor is thus a blade-preserving 1-form since any 1-vector is a 1-blade. It can be shown that provided k¹½N , any blade preserving k-form is merely the outtermorphism of a 1-form. For k=½N, the gemetric dual prserves k-blades but is not an outtermorphism.

Dyads
    A k-dyad is a k-multiform of the form D(a_k) = u_k(v_k¿a_k) where u_k, v_k are point-dependant k-blades. A k-multiform can be expressed in dyadic form as a sum of k-dyads. A 1-dyad is known as a dyad. A 0-dyad is the "succesive" multiplicative combination of two scalar fields D_x(a)=u_xv_x a

Multitensors

    We can generalise a (t;k)-tensor to a (t;k)-multitensor) being a point dependant multivector-valued function of k multivectors F_p(a₁,a₂,...,a_k) which is t-vector-valued when acting on k 1-vectors. , linear in all but the primary (point) argument .
    We will henceforth use the term tensor to refer to a multilinear multivector-valued function of k nonprimary 1-vector arguments and multitensor for a multilinear multivector-valued function of k nonprimary multivector arguments.
    We will typically restrict the grade of the nonlinear "primary" multivector argument p to 1 and consider it as a 1-vector "point" p . If a multitensor is t-vector valued, we can regard it as a sum of (t;k)-forms with k ranging from 0 to N.

Extended Fields

    Suppose now that we have k multivector fields a_ip=a_i(p). We can then extend a given k-multitensor F_p(a₁,a₂,...,a_k) with these fields to form an extended field   which we will also denote F_p mapping U^N×U_N^k ® U_N and defined by F_p = F_p(a_1p,a_2p,..,a_kp).

Outtermorphisms   and Determinants

    Let ¦ : Â^p,q,r ® Â^p,q,r be a linear transformation (ie. a 1-field over Â^p,q,r typically regarded as acting on and returning "points" rather than "vectors"). We can extend ¦ to a multivector field ¦^Ù over Â_p,q,r by defining
    ¦^Ù(a) º a     ; ¦^Ù(a) º ¦(a)     ; and ¦^Ù(aÙb) º ¦(a)Ù¦^Ù(b).
    This extension is known as the outtermorphism of ¦. Clearly ¦^Ù(a_<k>) = ¦^Ù(a)_<k> and in particular ¦^Ù(i) = |¦|i where scalar |¦| is the determinant of ¦ (nonzero iff ¦ invertible). We will henceforth consider all linear 1-fields (1-tensors) to be so extended and will frequently drop the ^Ù suffix   . We can similarly extend any k-tensor to be defined over k multivectors rather than k 1-vectors.
    Since ¯(cÙd,b) =  (¯_b(c))Ù(¯_b(d)) , ¯_b^Ù = ¯_b , and so ¯_b is an outtermorphism and we can write ¯_a^Ù =; ¯_a .
    It is worth explicitly noting that outtermorphisms preserve scalars.

Eigenblades
    We now generalise the concept of eigenvectors and associated eigenvalues. We say k-blade a_k is an left k-eigenblade of a general ¦: Â_p,q,r®Â_p,q,r with associated scalar eigenvalue a if ¦^Ñ(a_k)=aa_k . We say it is a right k-eigenblade if ¦^D(a_k)=aa_k .
    If a_k is both left and right eigenblade then the eigenvalue is common and we have a proper eigenblade
[ Proof :  a_Lefta_k² = a_k¿¦^Ñ(a_k) = ¦^D(a_k)¿a_k = a_Righta_k²  .]
    A proper 1-eigenblade is a conventional eigenvector. Scalars are 0-eigenblades of eigenvalue 1.
    i is an  N-eigenblade of ¦ with eigenvalue Det(¦) .

    If a_k and b_r-k are eigenblades with eigenvalues a,b then a_kÙb_r-k is either degenerate (zero) or an eigenblade of eigenvalue ab. We say an eigenblade is irreducable if it is not itself the join of two eigenblades. For a transformation ¦ with ¦(i)=|¦|i ,  "factorising" N-eigenblade i into irreducible "sub" eigenblades corresponds to decomposing the space spanned by i into subspaces invariant under ¦.

    If a and b are left and right eigenblades with eigenvalues a,b respectively then a¦^D(a¿b)=b(a¿b) and b¦^Ñ(aëb)=a(aëb) which is to say that the non-vanishing of contraction a¿b or b¿a is a right (or left) eigenblade having eigenvalue ab^-1 (or a^-1b).

    For any 1-vector a and linear outtermorphism ¦=¦^Ñ, the (k+1)-blade aÙ¦(a)Ù¦²(a)Ù...Ù¦^k(a) must vanish for some k £ N because all (N+1)-blades are degenerate. We then have ¦( aÙ¦(a)Ù...Ù¦^k-1(a) ) = l_a aÙ¦(a)Ù...Ù¦^k-1(a) for some scalar eigenvalue l_a of k-eigenblade a_k = aÙ¦(a)Ù...Ù¦^k-1(a) .  We say a has ¦-eigenicity k .
    But ¦ can also be expressed as a real N×N matrix which we know (from the characteristic polynomial methods of traditional matrix theory) has N eigenvectors, provided we allow complex vector coordinates and complex eigenvalues. Complex eigenvectors occur in conjugate pairs, say ¦(a+ib) = r(iq)^↑ (a+ib) and ¦(a-ib) = r(-iq)^↑ (a-ib) for real scalars r and q with q¹0.
    Taking real and imaginary parts we obtain ¦(a) = r( cos(q)a - sin(q)b) ; ¦(b) = r( cos(q)b + sin(q)a)       giving ¦(aÙb) = 2 cosq sin(q)(aÙb)   . = r² sin(2q)(aÙb) . The geometric interpreation of i in a Euclidean context is (aÙb)^~
    Thus we can choose a basis in which each elements e_i has an ¦-eigenicity of either one (when e_i is an eigenvector of ¦) or two (when e_iÙ¦(e_i) is a 2-eigenblade of ¦).

Coordinate-based Tensor representations

    With regard to a given invertible frame {e₁,..,e_N}, we have an N ^r 0-field primary matrix representation of F_x  .
    F_x  ^m..q_i..l º e^q..m ¿ F_x (e_i,e_j,..,e_l)     [  with t suffix q..m and k suffix i..l ]
, ie. the component of e_m..q in F_x (e_i,e_j,..,e_l) .
    Alternate matrix representations are possible
    F_{x  m..q i..l} = e_q..m ¿ F_x (e_i,e_j,..,e_l)     giving the component of e^m..q in F_x (e_i,e_j,..,e_l) ;
    F_{x  m..q}^i..l = e_q..m ¿ F_x (eⁱ,e^j,..,e^l)     giving the component of e^m..q in F_x (eⁱ,e^j,..,e^l) ;   and so forth.
    Hence the alternate coordinate expressions v = å_i=1^N vⁱe_i = å_i=1^N v_ieⁱ for (1;0)-tensor v .
    With regard to orthonormal frames in Â_N , eⁱ=e_i and all these representations are identical.

    In particular, a 1-tensor (point-dependant 1-vector function of a 1-vector) has representations
    F_x ⁱ_j = eⁱ¿F(e_j) ; F_x ij = e_i¿F(e_j) ; F_x ^ij = eⁱ¿F(e^j) ; F_x i^j = e_i¿F(e^j) ;

    Note that the "height" of a suffix is often used in three related but alternate ways:

• As here, to indicate construction with regard to frames
• Distinguishing the k "input" suffixes from the t "output" suffixes, output suffixes appearing higher. This coincides with our primary matrix representation.
• To indicate "co" or "contra-variant" dependence on x with regard to a point-dependant inner product as explained later.

    We have the generalised (multivector) differential of v at a given x
    v^Ñ_x(a) º Ð^x_a(v(x)) º Lim_{e ® 0} ((v(x+ea)-v(x)) e^-1 )     [  e a scalar ] which we will see is linear in a ; and the generalised (multivector) centred differential of v at a given x
    v^oÑ_x(a) º ^oÐ^x_a(v(x)) º ½ Lim_{e ® 0} ((v(x+ea)-v(x-ea)) e^-1 ) .
    The centred differential has the advantage of sometimes being evaluable at x where v(x) is undefined but tends to be less applicable at boundary points. When v(x) is defined and the limit is well-defined being the same for e ® 0 from above as from below, the centralised differential is equivalent to the differential.

    Clearly Ð^x_a uxv   =   uav for any multivectors u,v independant of x and, in particular, Ð^a_pp = a which we can also write as 1^Ñ = 1.
    The differential of v(x) is thus the function which given a returns the a-directed derivative of v(x). A directed derivative can be regarded as a result of a particular evaluation of a differential.

    The full notation vÑxx(a) reminds us that the differential is both "with respect to" x and evaluated "at" a partcular x.  We will typically ommit at least one of these suffixes so that vÑxx(a) º vÑx(a) º vÑx(a) º vÑ(a) .
    We refer to Ða as the scalar a-directed derivative operator. By a "scalar" operator we here mean grade preserving in that Ða(v(x)<k>) = (Ða(v(x))<k> .

    We will frequently drop the brackets and write Ð_av_x for Ð_a(v(x)). We will use the notation =_( )= to indicate the mere addition or removal of brackets in accordance with our bracket conventions.

    Restricting a and x to be 1-vectors gives the 1-differential of v(x) at a given x
    v^Ñ_x(a) º Ð^x_a(v(x)) º Ð_a(v(x)) º Lim_{e ® 0} ((v(x+ea)-v(x)) e^-1 )
    We can outtermorphically extend the 1-differential to act on multivectors but it is important to recognise that even with 1-vector x=x, v^Ñ_x(a) = v^Ñ_x(a) is in general true only for 1-vector a.
    Ð_a obeys the product rule     Ð_a(b_xÙc_x) = (Ð_ab_x)Ùc_x + b_xÙ(Ð_ac_x) .
    Consequently Ð_a(u(x)Ùv(x)) ¹ u^Ñ(a)Ùv^Ñ(a) in general  .

    If v takes a 1-vector argument x (often interpreted as a "point") then, given an inverse frame, we can view v(x) as a multivector-valued function of N scalars   v(x¹,x²,..,x^N).
    We write Ð_x^k or Ð_ek for the partial derivative "scalar" operator Ð_x^k (v(x)) º ¶v(x)/¶x^k º Lim_{d ® 0} ((v(x+de_k)-v(x)) d^-1 ) º Ð_ek(v(x)) .
    The 1-differential at a given x of v(x) can then be expressed as
    v^Ñ_x(a)   º Ð_av(x) º å_k=1^N Ð_x^k ((a¿e^k)v(x)) = Lim_{d ® 0} ( (v(x+da)-v(x)) d^-1 ) .

    For 1-field v(x)=¦(x) the differential ¦^Ñ_x(a) is a 1-tensor with matrix representation  ¦^Ñ_xⁱ_j  =  ¶yⁱ/¶x^j ï_x      where y=¦(x).
[ Proof :  eⁱ¿¦^Ñ_x(e_j) = eⁱ¿(Ð_ej¦(x)) = eⁱ¿(¶¦(x)/¶x^j) = ¶yⁱ/¶x^j  .]
    If ¦: V^M ® U^N then  ¦^Ñ: V_M ® U_N can still be defined as Ð_a¦(x) for a,xÎV_M. Of course, if N<M then ¦^Ñ(i_M)=0 since any M-blade in U_N must be degenerate.

Linearity of the Differential
    That the differential is linear is surprising. One feels that one ought to be able to construct pathological functions, "directed bumps" which can "fool" a particular coordinate basis via a deceptive perfomance along the base axies. But one cannot do so without violating continuity assumptions for ¦.
    Consider as an example the 0-field v(x) = r sin2q = 2x₁x₂ defined over ². We write Ñv = 2(x₂e₁ + x₁e₂).
    Ð_e1v(x)=2x₂ while Ð_e2v(x)=2x₁ and both of these are zero at x=0.
    But Ð_e1+e2v(x)=x₁+x₂ is also zero at x=0 so linearity of v^Ñ survives there. Moving away from 0 to, say, point e₁, Ð_e2v(x) becomes non zero but v^Ñ is linear there too.
    Linearity survives at 0 by virtue of the r factor vanishing at 0, but only by having such a zeroing factor can we eliminate the discontinuity arising from q(x₁,x₂) being undefined at 0.
    We might attempt to cobble together something from splines with "flat areas" but anything so cobbled will require a discontinuity in a derivative of some order. If a function is flat to first order somewhere, it must be flat to first order everywhere, or face a second order discontinity at the "interface".
    One might think that continuously differentable "non-centred" functions are flat nowhere or everywhere, which would have crucial ramifications in physics since it implies that no truly continous fields can be entirely localised. Though we could damp function values with distance, there would always be theoretically detectable oscillations conceivably exploitable as an information channel. In non Euclidean spaces we can define inifnitely continous functions which become ever flatter as they approach the boundary of the null cone and are fully flat outside it, however.
    If a "particle" is modelled as a continous fluctation defined over relativistic timespace, that fluctaution must extend not only spacially, but temporally into the distant past and future of any observer.

    For linear ¦, ¦^Ñ=¦     [ Proof :  (Ð_a¦(x) = ¦(Ð_ax)=¦(a)  .] although note that ¦^Ñ is implicitly defined over all U_N even if ¦ is defined only over a subspace.  
    For small ex   , ¦(x+ex) » ¦(x)+¦^Ñ_x(ex) .
[ Proof :  ¦(x+ex) » ¦(x) + å_k=1^N ex^k¦^Ñ_x(e_k) = ¦(x) + å_k=1^N (ex¿e^k)¦^Ñ_x(e_k) = ¦(x)+¦^Ñ_x(ex)  .]

    The 1-differential ¦^Ñ_x0 can thus be thought of as the linear approximator to ¦(x)-¦(x₀) for x close to x₀.  If ¦^Ñ_x0(a) is constant "  unit a then ¦ is radially symmetric at _x0 (ie. can be expressed as a function of |x - x₀|).

    ¦^Ñ-1 = (¦^-1)^Ñ     when ¦^-1 exists, so we can denote both by ¦^-Ñ.
[ Proof :  ...  .]

    The outtermorphism extension of ¦^Ñ is denoted by ¦^{Ñ^} or just ¦^Ñ. Its determinant J_¦ º |¦^Ñ| º (¦^{Ñ^}(i))^* is the conventional Jacobian of ¦ at x.

    Of particular interest is the self-directed 1-differential or streamline derivative ¦^Ñ_x(¦(x)), which describes how a 1-field changes when it "follows itself".
    The composite 1-differential at x is ¦^ÑÑ_x(b) º ¦^Ñ_x^Ñ(b) = (Ñ_a¿b)¦^Ñ_x(a) . It is of limited usefulness.

Differentiating Exponentials
    We have Ð_d(x_*a)^↑ = Lim_{e ® 0} (e^(x+ed)_*a-e^x_*a)e^-1 = e^x_*a Lim_{e ® 0} (e^dd_*a-1)e^-1 = (d_*a) (x_*a)^↑ and more generally if Ð_dF_x commutes with F_x then Ð_d(F_x)^↑   =   (Ð_dF_x)(F_x)^↑ .
    If Ð_dF_x anticommutes with F_x then Ð_d(F_x)^↑   =   (Ð_dF_x)(1+F_x |²/3! + F_x⁴/5! + ...)   =   (Ð_dF_x)F_x^-1 sinh(F_x)

    Ð_de^(x_*a)b   =   (d_*a)e^(½p+(x_*a)b     provided b²=-1.

The Directed Chain Rule

    If ¦(x)=g(h(x)) then we have the Chain Rule ¦^Ñ(a)=g^Ñ_h(x)(h^Ñ_x(a)) .
    When g and h are linear this reduces to ¦^Ñ(a)=g^Ñ_h(x)(h(a)) = (h(a)¿Ñ_x)g(x) = Ð^x_h(a)g(x) .

Primary Differential
    Let us switch from x to p and suppose we have an extended field F_p = F(p,a_1p,a_2p,...,a_kp) = F_p(a_1p,a_2p,...,a_kp) .
    Ð_dF_p = Ð_d(F_p(a_1p,a_2p,...,a_kp)) = Lim_{e ® 0}e^-1( F(p+ed,a_1p+ed,a_2p+ed,...,a_kp+ed)-F(p,a_1p,a_2p,...,a_kp))
    We might consider regarding F_p as a multivector field by holding the k linear parameters a_ip constant at their p values throughout a neighbourhood of p and then take its d-directed derivative, defining
    Ð_ßd F_p(a_1p,a_2p,...,a_kp) º Lim_{e ® 0}e^-1( F(p+ed,a_1p,a_2p,...,a_kp) - F(p,a_1p,a_2p,...,a_kp) )
but this raises difficulties if the a_ip are restricted in some manner and unable to hold their "at p" values away from p.
    A better definition for the d-directed primary derivative operator is
    Ð_ßd F_p(a_1p,a_2p,...,a_kp) º (Ð_dF_p)(a_1p,a_2p,...,a_kp)
    º Ð_d (F_p(a_1p,a_2p,...,a_kp)) - F_p(Ð_da_1p,a_2p,...,a_kp)) - F_p(a_1p,Ð_da_2p,...,a_kp)) - ... - F_p(a_1p,a_2p,...,Ð_da_kp)) .
    Ð_ßd F_p(a_1p,...)  is then a well-defined point dependant multilinear function of k multivector arguments known as the d-directed primary derivative of F.
    The choice of the _ß symbol is here intended to suggest of the "lowering" of the "scope" of Ð_d to apply only to the "low-suffixed" primary p.

    We have discussed a 1-vector-directed primary derivative. Generalising to a multivector "point" p we have the obvious a-directed primary derivatives for general multivector a.
    In particular, we have the traditonal derivative of a multivector-valued function of a scalar ¦ : ®U_N as Ð₁¦(x) = ¶¦(x)/¶x = ¦'(x)

Second Primary Differential
    The first differential ¦^Ñ_p(a) = ¦^Ñ(a) can be extended via a given 1-field a_p=a(p)ºa into a field whose b-directed primary derivative is given by
    Ð_ßb ¦^Ñ_p(a_p) = (Ð_b¦^Ñ_p)(a_p) = Ð_b (¦^Ñ_p(a_p)) - ¦^Ñ_p(Ð_ba_p) = Ð_bÐ_ap¦(p) -  ¦^Ñ_p(Ð_ba_p) = Ð_bÐ_ap¦(p) -  Ð_Ðbap¦(p)
    This provides a bilinear second differential <1;2>-tensor ,
    ¦^Ñ2_p(a,b) º Ð_ßb(¦^Ñ_p(a)) = (Ð_b¦^Ñ_p(a)) -  ¦^Ñ_p(Ð_ba) = Ð_bÐ_a¦(p) -  ¦^Ñ_p(Ð_ba)
    If the second differential ¦^Ñ2 is symmetric we say ¦ satisfies the integrability condition which we can consequently express as
    Ð_bÐ_a¦(p) -  ¦^Ñ_p(Ð_ba) =  Ð_aÐ_b¦(p) -  ¦^Ñ_p(Ð_ab) Û  (Ð_b×Ð_a)¦(p) = ½¦^Ñ_p(Ð_ab - Ð_ba) º ½¦^Ñ_p(a_Äb) .
    Provided Ð_ba = Ð_ab (which we can also denote a_Äb=0) the commutability of Ð_ßa and Ð_ßb is thus equivalent to the commutability of Ð_a and Ð_b ; and this is trivially true in the particular case Ð_ba = Ð_ab = 0 corresponding to "constant" a and b.
    ¦^Ñ2_p(a,b) is the directed derivative at p in direction b of the a-directed derivative ¦^Ñ_p(a). It is maximised when b is normal to the surface ¦^Ñ_p(q) = ¦^Ñ_p(a) .

    Consider direction dq at point p + dp. ¦ is approximated near p as
    ¦(p+dp+dq) » ¦(p) + ½( ¦^Ñ_p(dp) + ¦^Ñ_p+dp(dq) + ¦^Ñ_p(dq) + ¦^Ñ_p+dq(dp))
     » ¦(p) +  ¦^Ñ_p(dq)  + ¦^Ñ_p(dp) + ½(¦^Ñ2_p(dq,dp)  + ¦^Ñ2_p(dp,dq)) = ¦(p)  +  ¦^Ñ_p(dq)  + ¦^Ñ_p(dp) +  ¦^Ñ2_p(dq,dp) .

Third Primary Differential
    The second differential can itself be primary differentiated Ð_ßb(¦^Ñ2(a_1 p,a_2 p)) = Ð_b(¦^Ñ2_p(a_1 p,a_2 p)) -  ¦^Ñ2_p(Ð_ba₁,a_2 p) -  ¦^Ñ2_p(a_1 p,Ð_ba_2 p)

Secondary Differential
    We here define the secondary directed differential by
    Ð_Þd F_p(a,b,...) º Ð_d F_p(a_Ð,b,...) º Lim_{e ® 0} e^-1 (F_p(a+ed,b,...)-F_p(a,b,...)) .
    a_Ð here denotes the scope of the differentiaon implicit in Ð applying only to parameter a.
    Thus "secondary derivative" refers to differentiation with respect to the second (first nonprimary) parameter, whereas "second derivative" usually refers to the combination of two successive primary derivatives,
    More generally, we have the (i+1)^ary directed differential
    Ð_Þⁱd F_p(a,b,...) º Ð_d F_p(a,b,.g_Ð,..) where g is the i^th non-primary parameter.

    If F_p has k nonprimary parameters we have Ð_a(F_p(a₁,....)) = (Ð_ßa + å_i=1^k Ð_ÞⁱaF_p .  

    Let F_p = F_p(a₁,...a_k)  be a tensor taking k non primary parameters . We can form
    F_p^Ñ = F_p^Ñ(a₁,..,a_k,d) º Ð_ßdF_p(a₁,..,a_k) º (Ð_dF_p)(a₁,..,a_k) º Ð_d (F_p^Ñ(a₁,..,a_k))) -  F_p(Ð_da₁,..,a_k) - ... -  F_p(a₁,..,Ð_da_k) .

Lie Product
    Having defined a directed derivative operator Ð_ap we define the skewsymmetric bilinear Lie product  by
    a_pÄb_p º Ð^p_apb_p - Ð^p_bpa_p º Ð_apb_p - Ð_bpa_p
    This is often known as the Lie Bracket and denoted [a_p,b_p] but we will favour the _Ä product notation here.

Undirected Derivatives
" Here, I'd like to introduce you to a close personal friend of mine. M-41A 10mm pulse-rifle, over and under with a 30mm pump-action grenade launcher."
    Corporal Dwayne Hicks, "Aliens".

    "Undirected derivatives" can be thought of as "splayed out" directed derivatives, or as   "embodying" derivatives in multiple directions.

1-derivative     Ñ

    We define the 1-vector del-operator (aka. nabla) or 1-derivative (aka. vector derivative)
    Ñ = Ñ_x º å_k=1^N e^kÐ^x_ek =  å_k=1^N Ð_eke^k
so that Ñv(x) = å_k=1^N e^kÐ_ek(v(x)) = å_k=1^N Ð_ek(e^kv(x)) = å_k=1^N (¶/¶x^k)(e^kv(x)) .
    Note that this definition remains consistant for all frames {e_i} including nonorthonormal ones.

    If v(x)=v(x) is scalar valued, Ñv(x) = å_k=1^N eⁱÐ_x^k v(x¹,..,x^N) is the conventional gradient with regard to Euclidean Â^N.

    Applying Ñ as a geometric product gives Ñ v(x) = å_k=1^N Ð_x^k (e^k v(x)) = å_k=1^N Ð_x^k (e^k¿v(x) + e^kÙv(x)) = Ñ¿v(x) + ÑÙv(x).

    If v(x)=v(x) is 1-vector valued, the scalar ¿ term is just å_k=1^N Ð_x^k (v_k) which for a Euclidean space (v^k=v_k) is the traditional divergence of v(x), while the bivector Ù term is known as the curl of v(x). For N=3, this is dual to (minus) the conventional curl Ñ×v(x).
    We have thus essentially unified and generalised the three conventional differential operators grad, div, and curl. It is possible to define Ñ independantly of an inverse coordinate frame as the limit of a surface integral, as discussed briefly under Tangential Derivative below.
    We refer to Ñ_xv(x) as the 1-derivative of v with respect to x.

    We have Ð_a = (a¿Ñ)    = (a_*Ñ)         with the brackets here emphasising "precedence" rather than specifying the "scope" of the Ñ which should be thought of as extending rightwards from the expression.
    Conventionally, a leftward, rightward, or double-headed horizontal arrow above the Ñ is used to indicate the direction(s) of differential scope, but this technique is typographically unavailable here.
    Ñ_a(¦^Ñ_x(a)) = Ñ_a((a¿Ñ_x)¦(x)) = (Ñ_a(a¿Ñ_x))¦(x) = Ñ_x¦(x)        so we have the operator identity
    Ñ_a(a¿Ñ_x) = Ñ_a Ð^x_a = Ñ_x which we can abbreviate as Ñ_aÐ_a = Ñ .

    It is customary to abbreviate Ñ(a_p) by Ña_p, treating Ñ as a "left-multiplier" but we loose associativity in that (Ña)b ¹ Ñ(ab) in general. We can use Ñ as a right-multiplier eg. aÑ provided we understand the "scalar" Ð_x^k to apply "leftwards" as well as "rightwards".
    The expression     abÑcd     is usually interpreted (defined) as ab(Ñc)d but could (perhaps more properly) be considered to mean ab(Ñ(cd)) + ((ab)Ñ)cd .
    We will retain the traditonal "rightward-only scope" for Ñ here but when we include Ñ in a list of operators fÑgh this should be thought of as abbreviating the composite operation fÑgh(a_p) º f( Ñ(gh(a_p)) ) º f( Ñ( g(h(a_p)) ) ) . The scope of Ñ thus extends rightwards to encompass all following symbols unless contraindicated by brackets.
    We will use ( ) to denote the extent of Ñ s whenever possible but this becomes complicated by brackets expressing product precedence. When we wish the derivative aspect of a Ñ to "hop over" intervening terms or to move leftwards rather than rightwards or just wish to emphasise that the default applicability we will add a _Ñ suffix to the term to which the Ð_ei "apply". The e^k act geometrically on "intervening" terms irrespective of any _Ñ's.

    In general we will here assume the "differentiating scope" of Ñ to extend rightwards but not leftwards. Thus (Ñ_p¿a_p) and  (a_p¿Ñ_p) are distinct scalar operators since
    (a_p¿Ñ_p)F_p = (a_p¿Ñ_p)F_pÑ whereas
    (Ñ_p¿a_p)F_p = (Ñ_p¿a_pÑ)F_p + (Ñ_p¿a_p)F_pÑ = (Ñ_p¿a_pÑ)F_p + (a_p¿Ñ_p)F_pÑ = (Ñ_p¿a_pÑ)F_p + (a_p¿Ñ_p)F_p . Only if (Ña_pÑ)_<0> = 0 (eg. if a_p=a independant of p) are they equivalent.

    We have the geometric product rule Ñ(a¨b) =  Ñ(a_Ѩb) + Ñ(a¨b_Ñ)     where ¨ denotes any bilinear multivector product (¿,Ù,., geometric,etc. ) and _Ñ denotes the differentating scope scope of the Ñ.
    As long as we remember the geometric product rule, we can derive many equations involving Ñ simple by reference to its 1-vector nature. a¿(a¿b)=0, for example, gives Ñ¿(Ñ¿b)=0.

    However, care must be taken with Ñ.  If can readily be verified that Ñx = N and that Ñ(x²)=2x . Derivations such as Ñ(x²) = Ñx_Ñx + Ñxx_Ñ = 2(Ñx)x = 2Nx = 2Nx are erroneous. We cannot commute x with x while we are varying one of them, because the variation may not commute with x.

Useful Ñ results

    ѺÑ_x ; y^↑ º e^y ; _* denotes scalar product throughout :

Ñ x = Ñ¿x = N	Ñ Ù x = 02   = 0
Ñx(xb) = (Ñxx)b = Nb	Provided Ñxb=0. This grade decomposes into
Ñ(x¿bk) = (bk.Ñ)x = kbk in particular: Ñ(x¿a) = (a¿Ñ)x = a	Ñ(xÙbk) = (bkÙÑ)x = (N-k)bk in particular: Ñ(xÙa) = (N-1)a
Ñ(bkx) = (-1)k(N-2k)bk	in particular: Ñ(ax) = Ñ(2(x¿a)-xa) = (2-N)a
Ñx(x*a)↑ = (åi=1N ei(ei*a))(x*a)↑ = a<1> (x*a)↑	and so: Ñx2ex*a = a<1>2 ex*a
Ñx((x*a)b)↑ = a<1> ((½p + x*a)b)↑	and so: Ñx2e(x*a)b = -a<1>2 e(p+x*a)b     provided b2=-1
Ñ(lx2b)↑ = 2lNx(lx2b)↑b
Ñ(¦(x)b)↑ = (Ѧ(x))(¦(x)b)↑b for central ¦(x)
Ñ(g(x)F(x)) = (Ñ(g(x))F(x) + g(x)(ÑF(x))	so Ñ(f(x)~) =   Ñ(|f(x)2|-½) f(x) + |f(x)2|-½ Ñf(x)     = -½|f(x)2|-3/2Ñ(|f(x)2|) f(x) + |f(x)2|-½ Ñf(x)     = |f(x)2|-½ ( -/+ ½|f(x)2|-1Ñ(f(x)2) f(x) + Ñf(x) ) according to sign of f(x)2.
Ñ(F(x)G(x)) = (ÑFÑ(x))G(x) + (Ñ¿F(x))G(x)Ñ + (ÑÙF(x))G(x)Ñ	so Ñ(f(x)~) =   Ñ(|f(x)2|-½) f(x) + |f(x)2|-½ Ñf(x)     = -½|f(x)2|-3/2Ñ(|f(x)2|) f(x) + |f(x)2|-½ Ñf(x)     = |f(x)2|-½ ( -/+ ½|f(x)2|-1Ñ(f(x)2) f(x) + Ñf(x) ) according to sign of f(x)2.
According as x2 is ± :
Ñ(|x|m) = ±m|x|m-2x = ±m|x|m-1x~	Ñ(x|x|m) = (N+m)|x|m     " mÎÂ Þ Ñx(x~) = (N-1)|x|-1
Ñ ¦(|x|) = ±¦'(|x|)x~	Ñ(¦(|x|)x~) = ¦'(|x|) + (N-1)|x|-1¦(|x|)
Ñ x2k = 2kx2k-1	Ñ x2k+1 = Ñ¿x2k+1 = (2k+N)x2k
Ñ((lx)↑) = l(lx)↑ + (N-1)x-1¿((lx)↑)	Only for N=1 do we have Ñ((lx)↑) = l(lx)↑  .
Ñ (¦(|x|)x~)↑ = ¦'(|x|)(¦(|x|)x~)↑ + sin(¦(|x|))(N-1)|x|-1 for x2 < 0.
Ñx((x-a)|x-a|-N)   =   Ñx¿((x-a)|x-a|-N)   =   oN = |dSN-1| at x=a and 0 elsewhere.
Ñx((x¿a)m) = m(x¿a)m-1a
Ñbk(x¿a2) = 2(-1)k(bkÙa2-bk.a2)	k³2
Ñ((x¿a2)¿bk)=a2×bk + 2a2¿bk ;	k³2
(bk.Ñ)(x¿a2) = bk×a2 + 2a2¿bk ;	k³2
Ñ((x¿a2)Ùbk) = a2×bk + 2a2Ùbk
(bkÙÑ)(x¿a2) = bk×a2 + 2a2Ùbk
Ñ(axb)↑ = (Ñax)(axb)↑ b
bx¿(Ñax) = (bx¿Ñ)axÐ	Holds only for scalar ax. We cannot, in general, retrieve (b¿Ñx)ax from b and Ñxax .
Ñ¿(axcx) = ax(Ñ¿cx) + (Ñax)¿cx	ÑÙ(axcx) = ax(ÑÙcx) + (Ñax)Ùcx
Ñ¿(axÙcx) = (ax¿Ñ)cxÑ + (Ñ¿axÑ)cx - axÙ(Ñ¿cxÑ) - axÑÙ(Ñ¿cx) In particular Ñ¿(aÙbx)    = (a¿Ñ)bx - a(Ñ¿bx) [ Proof :    a¿(bÙc) = (a¿b)c - bÙ(a¿c) with a=Ñ  .]	ÑÙ(ax¿cx) = (ax¿Ñ)cxÑ + cx(Ñ¿axÑ) - ax¿(ÑÙcxÑ) - (cxÙÑ).axÑ In particular Ñ(a¿bx) = (a¿Ñ)bx - a¿(ÑÙbx) [ Proof :   a¿(bÙc) = (a¿b)c - bÙ(a¿c) with b=Ñ  .]
Ñ x~(x~+a)~   =   2-½(1±x~¿a)-½|x|-1 (a(N-3/2) + ½x~)	when x~2 = a2 = ±1
Ñ |x|½x~(x~+a)~   =   2-½(1-x~¿a)-½|x|-½ (a(N-1)+x~)	when x~2 = a2 = ±1

 

Monogenic Functions
    We say v(x) is monogenic (aka. analytic) if Ñ_xv(x) = 0 " x . We say v(x) is meromorphic if it is monegenic at all x except some welldefined poles x₁,x₂,...,x_k at which we have Ñ_xv(x) ï_xi   =   - o_M R_i where multivector R_i is the residue at pole x_i and o_M is the boundary content of a unit radius (M-1)-sphere.

    Monogenic functions are fundamental in theoretical physics, particularly (in nonrelativistic central potential theory) spherical monogenics Y_x of the form Y(x) = x^l y(x^~) = r^l y(q,f) for N=3 . Monogenity condition Ñ_xY_x=0 requires (xÙÑ_x) y(q,f) = l y(q,f) interpreted as Y having constant scalar integer "angular-momentum operator" eigenvalue l, known as the angular quantum number . [  The brackets around (xÙÑ_x) denote the precendece of the Ù ; the differentiating scope of the Ñ_x acting rightwards over the y. ]. For l<0 we have a single pole at 0.

Laplacian     Ѳ
    Since Ñ_xv(x) = Ñ_av^Ñ_x(a) = Ñ_a((a¿Ñ_x)v(x))) we have Ñ_x²v(x) = Ñ_x(Ñ_av^Ñ_x(a)) = Ñ_bÑ_av^Ñ2_x(a,b) = (Ñ_b¿Ñ_a +  Ñ_bÙÑ_a)v^Ñ2_x(a,b) .
    If v obeys the integrability condition, the symmetry of second differential v^Ñ2_x(a,b) causes the Ù term to vanish [   (Ñ_bÙÑ_a)v^Ñ2_x(a,b) = -(Ñ_aÙÑ_b)v^Ñ2_x(a,b) = -(Ñ_aÙÑ_b)v^Ñ2_x(b,a) ] and so Ñ_x²   = Ñ_x¿Ñ_x is a grade-preserving ("scalar") operator known as the second derivative or Laplacian or D'Alembertian operator.
    Consequently Ñ_x² = Ñ_x¿</