Maxwell Equations as axioms over all of physics and math #9 Textbook 2nd ed. : TRUE CALCULUS; without the phony limit concept

[Archimedes Plutonium]

Alright, I am learning more new things, for in this 2nd edition I have
an alternative to the picketfence model. I have the pure and straight
rectangle model and the pure and straight triangle. In the rectangle
model we fill the dx of 10^-603 width and the height is y itself. In
the pure triangle we have a right triangle on the leftside of the
point of the graph and the same triangle on the rightside with its
hypotenuse in the reverse direction as pictured like this:

/|
/ |
/ __|

unioned with this triangle

|\
| \
|__\

is the same area as the rectangle model of the point on the function
graph.

The problem, though, is that the angle of the hypotenuse does not like
like the slope or tangent to the point of that function graph. So I
need to see if that hypotenuse is related to the slope or tangent or
derivative at that specific point. If it is, then, clearly we see how
derivative is the inverse of integral, because both have the same area
and the triangle hypotenuse would be the derivative. So instead of
rectangles forming the integral we can take two triangles. So
hopefully I can work this out in the 3rd edition which I plan to start
in the next day or so.


Alright, this is the 10th page of the 2nd edition and the last page. I
want to devote the last page to showing how all this math is begot
from the Maxwell Equations.
Now on this last page I want to show how Calculus of its empty space
between successive numbers is derived from the Maxwell Equations as
the ultimate axiom set over all of mathematics. The Maxwell Equations
derives the Peano axioms and the Hilbert axioms. But I want to show
that the Maxwell Equations do not allow for the Reals to be a
continuum of points in geometry but rather, much like the integers,
where there is a empty space between successive integers.
The Reals that compose the x-axis of 1st quadrant are these:

0,  1*10^-603,  2*10^-603,  3*10^-603, 4*10^-603, 5*10^-603,
6*10^-603 . . on up to 10^603

Pictorially the Reals of the x-axis looks like this
...................>
and not like this
____________>

So in the Maxwell Equations we simply have to ask, is there anything
in physics that is a continuum or is everything atomized with empty
space in between? Is everything quantized with empty space in
between?

I believe the answer lies with the Gauss law of electricity, commonly
known as the Coulomb law. The negative electric charge attracts the
positive electric charge, yet with all that attraction they still must
be separated by empty space. If there was a continuum of matter in
physics, then the electron would be stuck to the proton. The very
meaning of quantum mechanics is discreteness, not a continuum.
Discreteness means having holes or empty space between two particles
interacting of the Maxwell Equations.
So if physics has no material continuum, why should a minor subset of
physics-- mathematics have continuums. If Physics does not have
something, then mathematics surely does not have it.

Now I end with reminders for the 3rd edition:

REMINDERS:
(1) First page talk about why Calculus exists as an operator of
derivative versus integral much the same way of add subtract or of
multiply divide because in a Cartesian Coordinate System the number-
points are so spaced and arranged in order that this spatial
arrangement yields an angle that is fixed. So that if you have an
identity function y = x, the position of points (1,1) from (2,2) is
always a 45 degree angle. So Calculus of derivative and integral is
based on this fact of Euclidean Geometry that the coordinates are so
spatially arranged as to yield a fixed angle. Numbers forming fixed
angles gives us Calculus.

(2) Somewhere I should find out if the picketfence model is the very
best, for it maybe the case that a rectangle model versus a pure
triangle model may be better use of the empty space of 10^-603 between
successive Reals (number points). The picketfence model is good, but
it never dawned on me until now that there is likely a better model
even yet-- pure rectangle versus two pure triangles. My glitch is to
get the hypotenuse related to the derivative. If I can solve that
glitch, I have a crystal clear understanding of the derivative,
integral and why they are inverses.

(3) I am really excited about that new method of arriving at the
infinity borderline of Floor-pi*10^603 via Calculus. The first number
which allows a half circle function to be replaced by a 10^1206
derivatives of tiny straight line segments and still be a truncated
regular polyhedra, is when pi has those 603 digits rightward of the
decimal point. The derivative of half circles of any number smaller
than Floor-pi*10^603 does not form a circle. And is that not what
Calculus is all about in the first place-- taking curves and finding
Euclidean straight line segments as derivative and area. Calculus is
the interpretation of curved lines into straight line segments. So,
onwards to 3rd edition.

--
More than 90 percent of AP's posts are missing in the Google
newsgroups author search archive from May 2012 to May 2013. Drexel
University's Math Forum has done a far better job and many of those
missing Google posts can be seen here:

http://mathforum.org/kb/profile.jspa?userID=499986

Archimedes Plutonium
http://www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

On May 25, 12:43 am, Archimedes Plutonium

I am not going to count this as the 11th page of a 10 page textbook,
but rather as a reply. I found out tonight that these hypotenuse of
right triangles of points on the graph of a function are related to
the slope or derivative of the function at that point.

So in my previous graph of the function y=x^2 in 10 grid:

.    .    .    .    .    . 
                         x 
.    .    .  
 .    .    . 
                    x 
.    .    .    x    .    .
.    .    .    .    .    . 
0  .1  .2  .3  .4   .5

at x=.3, y=.09
at x=.4, y =.16 
at x=.5, y =.25
Now each of those intervals of .1 width has 2 pure
triangles as these two

     /|
/  |

/ __|
unioned with this triangle

|\
|  \
|__\

So in the interval between .3 and .4 of a dx of .1 sits two triangles
where their hypotenuse cross one another and intersect at a point and
the same is true of the next dx =.1 interval of two triangles
intersecting and if we draw a line between the two intersections we
end up with the derivative. Sort of reminds me of the projective
geometry Desargues theorem.

But I need to confirm all of this.

The importance of this is that the picketfence model gets thrown out
and replaced by the pure 2 triangles aside each point of the graph of
a function and the 2 triangles determine the derivative and the
integral and it is easy to see how the derivative is the inverse of
integral.

[Archimedes Plutonium]

I was hoping I could replace the picketfence model by two pure
triangle model.

Remember the sawtooth function of F(x) = 0 for even numbered x and
F(x) = 10^603 for odd numbered x. Here we can do the integration, not
by summation of thin picketfences but rather by summation of two thin
pure triangles along the leftside and rightside of each point of the
function graph.
From that sawtooth function I was hoping to eliminate the rectangle
portion of the picketfence model, but I cannot. So that leaves me with
the-- having to prove that Calculus best model is the picketfence and
it is irreplaceable for the calculus.
Proof: I would use the identity function and obviously the picketfence
model gives the derivative of 1 and the integral of 1/2x^2, but, can
I get a derivative of 1 via 2 pure triangles? So far, no. If the only
way to get the derivative and integral of the identity function via
picketfence model, then it is irreplaceable.

[Archimedes Plutonium]

Alright, I am just about to start the 3rd edition but let me make
special notes of a few things.

In the box function of y = 3 the derivative is 0 and the integral is
Int = 3x and in this function we see the picketfences as rectangles
only with no triangle atop.
Now in the sawtooth function of F(x) = 0 for even numbered x and 
F(x)
= 10^603 for odd numbered x. We have no rectangle in the picketfence
but just pure triangles and the derivative dy/dx is 10^603/10^-603 =
10^1206 and the integral in the interval 0 to 10^603 is 0.5*10^1206.
Now the function y = 3 can be sawtooth also by this manipulation F(x)
= 0 for even numbers and F(x) = 3 for odd numbers, so that instead of
a flat line across y = 3, we have a sawtooth pattern. And the
derivative dy/dx is 3/10^-603 which is 3*10^603 and the integral in
the interval 0 to 10^603 is 1/2(3*10^603). So for the function y=3 its
area from 0 to 10^603 is 3*10^603 and a sawtooth pattern is 1/2 of
that area.

So in the 3rd edition I must realize that sometimes the function has
only a rectangle for integration, and sometimes only triangles, but
most often it has a combination of rectangle and triangle atop that is
a picketfence.

So what the Fundamental Theorem of Calculus concerns itself with is
why is Euclidean geometry the only geometry to have the Calculus and
that it turns all curves into straight lines for derivative and for
integral. So that when we graph the half-circle and find its
derivative and integral the circle curve at the 10^-603 is no longer a
curve but a tiny straight line segment. That is the essence of what
the Calculus does to the function, replaces its curves with straight
line segments. And that is why Euclidean geometry is the only one of
the three geometries to have a Calculus for we end up replacing the
curves of Elliptic and Hyperbolic geometry with straightline segments.

That is why the borderline of infinity cannot be a number smaller than
Floor-pi*10^603 because only there is a semicircle function able to be
a truncated regular polyhedra.

Alright, I am ready to do the 3rd edition.