Maxwell Equations as axioms over all of physics and math #10 TRUE CALCULUS; without the phony limit concept (textbook 1st ed.)

[Archimedes Plutonium]

Alright, this is a good 10 page textbook on Calculus and shows most of
the horrible mistakes of the Old Math Calculus.

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.

So if physics has no material continuum, why should a minor subset of
physics-- mathematics have continuums.

Now I leave this textbook with suggestions when I rewrite it for the
next edition. In the 2nd edition, I think it is wiser to start
Calculus not with the derivative but rather the integral. Old Math
generally started with the derivative due to the limit concept, but in
New Math, there is no limit but rather the picketfence model and it is
better to start with area as more intuitive than with slope.

I need a better example for dy/dx of slope that does not match the
derivative. In this edition I used y=x^2 when x = pi and a delta of pi
+1 and pi-1. I suspect there is a far better more simple example of
discordant values. Perhaps I should enlist trigonometry functions to
show where dy/dx does not match exactly the derivative value.

I need to find where the picket fence model arose in mathematics
history. Was it Leibniz and Newton circa 1675?

No chapters, since it is only 10 pages long.

Each edition should improve on better ascii art and more art.


--

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:

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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 21, 11:46 pm, Archimedes Plutonium

Now I am not going to count this as page 11, but rather a reply to
page 10 or an addendum to page 10.

Thinking about a better function to show where the limit value is
incongruent with the actual derivative value I should use the function
y= 1/x, and I should have realized this before that a log type of
function 1/x. So that in the case of x=2 the slope is valued by the
limit concept to be -1/3 with a delta as 1 unit of 1 and 3 with a dy/
dx as -(2/3)/2 whereas the true slope is -1/4 at x =2.

And for the first time the text Calculus by Ellis & Gulick, 1986 comes
to a rescue in a big way. On page 115, the authors describe how Newton
came to the derivative and more importantly how Leibniz came to the
derivative in a different manner. And although picket-fence is not
mentioned, what is mentioned is that Leibniz is motivated by small
triangles and when you think about it, Leibniz is using the small
triangle atop the picket fence but not mentioning the picket fence
structure. Quoting that passage:
"Motivated by the small triangles that appeared when he attempted to
find tangents to curves (Figure 3.13), he adopted the notation dy/dx
for the derivative. Here dy and dx signified small changes in y and in
x, respectively."

Also, I went back to one of my first books I read on mathematics in
Middle School from the Life Science Library titled MATHEMATICS by
David Bergamini, 1963 on page 109 shows a picture of the picket-fence
model and calls it "Picket-Fence Integrals". However, reading that
caption, the author thought the triangle portion was bad for calculus
saying that "..by making the pickets so thin the tops become
negligible." Sadly, if David had read this textbook, the entire
picket, both the triangle and rectangle portions are essential for the
triangle determines the derivative and the triangle with rectangle
determine the integral, all of which happens when the point has a hole
or gap on both right and left side.

Armed with that information, I am ready to do the 2nd edition of this
book, which should take me 2 or 3 days in toti.

[1treePetrifiedForestLane]

yay, trigona. I am not following your distinction
of the "trigonal & rectangular parts" of the deltum, er
sigmum, er areal infinetessimal.