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Aug 9, 2008, 12:06:11 AM8/9/08

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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.

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

Derivative.

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

For more information see

http://www.octospace.com/files/Octonion_Algebra_and_its_Connection_to_Physics.pdf

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