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: