Why Octonions 4: The Octonion Ensemble Derivative

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Aug 9, 2008, 12:06:11 AM8/9/08
There are alternate definitions for gradient, divergence
and curl that involve a limiting process of the ratio of
integral over enclosing surface to integral over enclosed
volume as the volume approaches zero. The differential
surface normal vector multiplies the scalar function for
gradient. For divergence, the inner product of differential
surface normal and function vector is used. For curl, the
cross product of differential surface normal and function
vector is used.

If we back down division algebras to the Quaternions, the
forms for gradient, divergence and curl are all present in
the singular Quaternion product of differential operator
and function Quaternion. A natural extension of the
integral definitions for gradient, divergence and curl
mentioned above would be to first replace the 3D
differential surface normal vector with the 4D Quaternion
differential surface normal. Next form a singular
definition for differentiation that is the limit as the
enclosed 4D volume approaches zero for the ratio of
integral of the product of Quaternion differential surface
normal and Quaternion function divided by the integral of
the enclosed 4d differential volume. This form covers
the traditional forms for gradient, divergence and curl as
an ensemble, which is why I call it the Ensemble

It is a straight shot from this Quaternion definition to
one for the Octonions. We just need to substitute the 8D
definition for differential surface normal, Octonion
function, Octonion product and 8D differential volume.

The beauty of the integral definition for differentiation
is the ability to easily move to a description using an
alternate variable set. If a functional relationship exists
between the variable sets, Jacobian formalism can be used
to cast the differential surface normal and differential
volume in terms of the new set of variables, Jacobians, and
cofactors of the Jacobian matrix.

I take the integral definition for differentiation as its
fundamental definition, not just an alternate form. By
defining differentiation fundamentally using a
diffeomorphism between the intrinsic system attached to the
algebra directly and an alternate, the transformation
properties for differential equations are intrinsic to the
definition itself, not an afterthought or something tacked
on. There is more here than say, a conversion from
rectilinear coordinates to spherical-polar, although this
is certainly covered. When the system is a hypercomplex
one, as with the Quaternions and Octonions, the door is
open to allow one of the variables to represent time, and
the diffeomorphism to represent a velocity transformation.
As with any curvilinear system, the velocity components
present in the Jacobian matrix may freely be functions of
time, in other words accelerated frames of reference will
be covered as simply curvilinear in time.

The task is to define the Ensemble Derivative form at a
single point in the coordinate space. The volume in
question always includes this single point as an interior
point. The surface in question always surrounds the single
point without ever contacting it. The limit process allows
the surface to come arbitrarily close to the single point.

Since we have an ASCII character format here, please take
d/dri as the i'th partial derivative with respect to the
fundamental basis coordinate system directly attached to
the Octonion basis units. In other words, ri ui sum i from
0 to 7 is the fundamental Octonion position vector. Take
d/dvi as the i'th partial derivative with respect to the
diff-morphed system of coordinates. Then dri and dvi are
the i'th differentials of their respective systems.

The volume always includes the point in question as an
interior point. The differential volume can therefore be
expressed in the v system as

J dv0 dv1 dv2 dv3 dv4 dv5 dv6 dv7

Here J is the Jacobian dr/dv of the diffeomorphism from r
to v, evaluated at the point where we wish to define the
derivative at by mean value arguments. Since it is
evaluated at a single point, its variation about the
coordinate neighborhood of this point is not in issue.
This Jacobian thus can thus be brought outside the integral
as a constant scaling factor 1/J on the eventual form for
the Ensemble Differentiation.

The simplest form for the differential surface normal is
the Octonion form (summation over all i)

dNi = J dvi/drj uj dv0 dv1 dv2 dv3 dv4 dv5 dv6 dv7 / dvi

Unlike the Jacobian in the differential volume element,
both J and dvi/drj here are evaluated off the single point
at which we wish to define the Ensemble form. Their
variation in the coordinate neighborhood of the point in
question is very much an issue in the definition of the
Ensemble Derivative form.

If we take F(v) as the Octonion function to differentiate,
the limit process will yield the following for the Ensemble
derivative E of F(v)

E(F) = 1/J d/dvi [ J dvi/drj uj*F ] sum ij

Here "*" is Octonion multiplication of F by basis unit uj
as defined by the algebra representation of choice. We may
write F in terms of its connection to the fundamental
Octonion basis units as

F(v) = Fk drl/dvk ul

Then the Ensemble form may be written as

E(F) = 1/J d/dvi [ J dvi/drj drl/dvk Fk ] uj ul sum ijkl

The fundamental basis units are constants, so may be
brought outside the differentiation.

The Ensemble Derivative with respect to v on F(v) can be
associated with its equivalent function G(r) simply by
equating r to v. Then the Jacobian is real unity, and
dvi/drj is non zero unity only for i=j, and drl/dvk is non
zero unity only for k=l. The Ensemble Derivative of (Gl ul)
sum l with respect to r is then

E(G) = dGl/drj uj ul sum jl

Equating, sum ijkl on

dGl/drj uj ul = 1/J d/dvi [ J dvi/drj drl/dvk Fk ] uj ul

F and G are related by

Gi(r) ui = Fj(v) dri/dvj ui sum j for any i

Both sides of E(G(r)) = E(F(v)) may be equated for fixed
jl, since Octonion result product histories are identical.
Looking closely at E(G(r))=E(F(v)), the chain rule may be
written as

d/drj = 1/J d/dvi [ J dvi/drj ..... sum i

This will be equivalent to the classic chain rule

d/drj = dvi/drj d/dvi sum i

only if J and dvi/drj are constant in v. Perhaps the
classic chain rule is not so general. Let me put the above
form out as a better choice.

If we take the case of v=r a little further, we could
define an Octonion differential operator D as

D = ui Di = ui d/dri sum i

This differential operator multiplies like any other
Octonion by the rules of the selected algebra
representation. The partial differentiation is
applied after algebraic multiplication as a scalar
operation on the remaining product term components.

One very important thing to keep in mind is that for all
diffeomorphisms to alternate coordinate system v, we never
lose the fundamental basis units ui. They are always
present and always define the operation of multiplication
the same way for any choice of v. This implies Algebraic
Invariance is coordinate system invariant.

Rick Lockyer


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