This text is shaping up excellently as far as tying up loose ends of previous editions. In this edition, I finally solve the famous formula
Euclidean geometry = Elliptic unioned with Hyperbolic geometries
A proof comes from the idea that given any curve of a circle as function, that a curve of a logarithmic spiral can counteract and the two curves cancel to leave a straight line segment remaining, )|(.
Another way of writing the formula above is this:
| = ) unioned (
Let me spend this post in commenting on the progress made so far and what to expect in the last posts of this text, for I am nearly at 30 to 40 pages.
I covered the derivative and I need more graphing points, although the y = 1/x was good, I should expand that even more. Learning is in the doing, not just reading. So I want students to do a lot of graphing.
Now the Fundamental Theorem of Calculus is not so important in New Math and is rather a breeze to prove in New Math, for that we have integrals as picketfences with the hypotenuse of the right triangle atop the rectangle of the picketfence.
|\ || ||
So to prove the fundamental theorem in New Math is simply the noting that the area of the picketfence, the trapezoid is dependent on that hypotenuse. So in essence, the derivative is that hypotenuse and the integral is the area of the trapezoid. Specify the hypotenuse and it delivers a integral, or specify a area and it delivers a hypotenuse.
Now I should review the area of a trapezoid (picketfence) and the angle of the hypotenuse, because I anticipate that we can do the entire derivative and integral teaching, just by focusing on how the area relates to the hypotenuse and its angle. Of course, the width of these trapezoids is always the cell width of 1*10^-603. So give me a function and I focus on any specific cell, and I can easily determine the y values of two successive x values and thus I can determine the area inside the cell and the derivative of the hypotenuse.
But now one of the harder problems to solve is how the Maxwell Equations agree and concur with the notion that curves do not exist in mathematics, but rather are a bunch of tiny straightline segments glued together to give the appearance of a "curve". So do the Maxwell Equations concur with the notion that curves are nonexistent and they are really a collection of tiny straight lines glued together? I think I answered this before by saying that a light wave goes in a straightline unless it enters a different medium and is refracted. And in the refraction, light just changes angle and goes on its merry way in a new straightline. Light never travels in a curve, but only straightlines. But what about a proton or electron that experiences a attraction or repulsion force, and whether it travels in a "curve path". So does it travel in a curve or does it travel in a sequence of tiny straight line segments? And this harkens to the Coulomb force law that is inverse square with distance. Is it a force of "curved lines" or a force of tiny straightlines which overall looks like a curve?
If the Maxwell Equations cannot get rid of curved lines, then curved lines are here to stay and I lost this argument. But if every motion of the Maxwell Equations are tiny straightlines, then I won the argument. And I have to run this argument through the Maxwell Equations for they are the final axioms over all of physics and mathematics.
Apart from light waves always traveling in a straightline and never a curve, is that good enough to explain that a electron and proton also travel in straightline segments that only gives the appearance of a curve. So this problem is tough, and tougher than the Fundamental Theorem of Calculus. And if I am not careful, it may end up being the toughest proof in this textbook.