Re: [teslafy] Determinant Actions
--- On Sat, 3/6/10, Harvey D Norris <harvich@...> wrote:
From: Harvey D Norris <harvich@...>
Subject: [teslafy] Determinant Actions
Date: Saturday, March 6, 2010, 1:16 AM
It would seem that I should go back to school and repeat my assignments. This comes as a useful endeavor for repetition of knowledge and keeping the name sake. The subject of determinants at first seems mystical, and I had lost all my knowledge about this, or actually I didn't study it when I should have, so a repetition is in order.
Let us say x + y = 7.
An x-y 2D graph shows the equation's solutions as a straight line; this is a linear relationship.
Now let us say x + y = 9.
There is of course no two numbers that will add up to both seven and nine. When each equation is graphed out on the (x,y) axis we find that they are parallel, therefore they have no intersection pt for a single set of numbers represented on the x/y axis that will satisfy both equations. We now turn to a somewhat obscure subject known as tensor analysis. Here I will quote from Joseph P Farrell's book; "Secrets of the Unified Field". Chapter 2 is entitled Gabriel Kron and Einsteins Unified Field Theory.
The theme of Kron's work: The Tensor Analysis of Electrical Machines
Before the advent of Kron's work on the application of tensor analysis to electrical machines, the most use tensors had in "practical" physical applications was in the geometry of gravitational fields in the General Theory of Relativity. Tensor analysis seemed to be applicable to the physics of the very large; to planets, stars and galaxies. Kron's analysis of electrical machines was based in part on the very "incomplete" Unified Field Theory of Albert Einstein that the later had begun to outline in his 1928 paper, the first to include torsion as a principle component... that theory was inspired by the "higher dimensional strategy" of Kaluza's five dimensional theory. In short Kron's work from his 1934 paper, to his first books in tensor analyis of electrical machines, was maintaining that only a higher dimensional theory, incorporating some form either of curved geometries or torsion, could account for certain observed and well- known features of electrical machinery, particularly electrical machinery involving rotation, or that was connected in complex networks.
The basic problem of tensor analysis is how one type of geometric or physical object, described in one system of mathematical coordinants, can be transformed into another object, or alternatively, into the same object, but viewed from a different frame of reference or coordinant system. This is the basic problem tensor analysis was invented to handle, and the process of transforming from one system to another is known as a transformation tensor. The transformation tensor in turn can describe the transformation of an object in, say five dimensions, to the same object in four dimensions, and so on.
The notion of torsion is that it involves the folding or pleating of local space that results when rotating systems are coupled in certain ways. The "spiralling" of the vectors of force that result when torsion occurs means that when torsion is extreme enough, the vectors of normally perpendicular forces, such as electricity and magnetism, may become almost "parallel"; they undergo a kind of "distant parallelism" brought about by such extreme warping of space. Depending upon the amount of torsion. those vectors may not necessarily be perpendicular any more, but deviate slightly or greatly, from such perpendicularity.
Now returning to the subject of matrices, and the solving of simultaneous equations by such means, what we actually have here is a mathematical method to describe higher dimensional objects, that cannot be visualized by the mind itself using the three dimensions of space itself as a reference point.
In the previous example the equations are represented in two dimensions, using two unknowns having a relationship to a constant value, and furthermore having two such arrangements where it is seen that no set of numbers can be members of both sets; therefore each linear equation represented as a line MUST be in a parallel relationship with each other. To then extend upwards in dimension formerly we had two unknowns in relation with a constant value, so now we merely make three unknowns in relationship with a constant value, so adapting this to the previous equations we may also venture to see if a solution is possible to satisfy both equations where;
x + y + z = 7
x + y + z = 9
It is of course silly to assume that three numbers can both sum to seven or nine, so we know that no 3D coordinate intersecting both lines in space exists for those equations. But in the 2D case we concluded because no simultaneous solution existed, the lines must be parallel. The question now becomes do our 3D equations as lines in space that do not intersect also now reveal themselves to be parallel in a three dimensional sense? Note here that in the expansion of the dimensional unknown quantities from two to three, new laws of relationship have come into being, and now two lines sharing the same space do not have to be parallel for them to have no simultaneous solution, and in fact those lines could be drawn so that they might even appear perpendicular to one another in a three dimensional sense, and yet not touch each other. One may wonder if the term "distant parallelism" exists here as an analogy.
I will return next with some 3D vector problems, which is one of the first thing a college physics course teaches.
Pioneering the Applications of Interphasal Resonances http://tech.groups.yahoo.com/group/teslafy/