#010 Lines come in varieties in True Geometry, such as dashed line

#010 Lines come in varieties in True Geometry, such as dashed line 8th ed.: TRUE CALCULUS

Now this is the last post of True Geometry, 10 posts in all and the main purpose is to warn and introduce the facts of what is wrong in current day High School and College geometry as it is taught in school and in books.

The huge error of not placing a border between finite and infinity allows there to be a line with length composed only of points that have no length, and a limit concept that is unable to connect derivative with integral. So by not placing a border between finite and infinity, Geometry cannot become True Geometry in which a finite point in a coordinate system (Grid) has empty space between it and the next neighboring finite point and no other finite points lie between those two finite points.

I cited various major consequences of this True Geometry, perhaps the most important is that no curves exist and that Calculus cannot exist unless there is empty space between successive finite points.

But let me cite one more consequence which I have not yet digested. The idea that a line can come in a variety of types of line since it is composed of successive finite points with empty space in between. The empty space are infinity-points.

In Old Math they had just one type of line-- which I call a full line as seen by this __________________

That line is full because it is solid black line.

But in True Geometry we can have dashed lines like this:

-------------------------

Where the solid black is the connecting of every other pairs of finite points. For example in 10 Grid if we connect 0 to .1 then skip the empty space between .1 and .2 and then connect .2 to .3 and do that over and over again..

So in True Geometry, we have a possibility of a variety of lines not possible in Old Math.

Now this possibility looks to be of importance to physics for it allows us to treat waves whether longitudinal or transverse waves, to treat then as a line. So that a dashed line is a longitudinal wave where the rarefaction is the empty space and the compression is the solid black line.

Or, where the transverse wave of destructive interference is the empty space then solid black line.

So in physics, we have an instant major use for lines being of varieties and types.

In mathematics, we seem to not have any use for various different types of lines. Perhaps that is because we have an axiom that says two points determine a line.
So that in the Calculus, the sawtooth function has just two finite points and the entire rest of the line is empty space. The x-axis and y-axis line in 10 Grid can be a solid black line or can be a dashed line of 50 empty spaces with 50 dashes to compose 1 dashed line.

So far I see no application, no use for varieties of lines in mathematics. I do see a huge use of types of lines for physics, and since physics is the superior subject and math is a subset of physics, that I know varieties of lines is important to mathematics.

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Drexel's Math Forum has done an excellent search engine for author posts as seen he
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Now, the only decent search for AP posts on Google Newsgroups, is a search for for it brings up posts that are mostly authored by me and it brings up only about 250 posts. Whereas Drexel brings up nearly 8,000 AP posts. Old Google under Advanced Search
for author, could bring up 20,000 of my authored posts but Google is deteriorating in quality of its searches, likely because AP likes an author search and Google does not want to appear as satisfying to anything that AP likes. If AP likes something, Google is quick to change or alter it.

So the only search engine today doing author searches is Drexel. Spacebanter is starting to do author archive lists. But Google is going in the opposite direction of making author archived posts almost impossible to retrieve.

All the other types of Google searches of AP are just top heavy in hate-spam posts due to search-engine-bombing practices by thousands of hatemongers who have nothing constructive to do in their lives but attack other people.

Now one person claims that Google's deteriorating quality in searches of science newsgroups is all due to "indexing". Well, that is a silly excuse in my opinion, because there is no indexing involved when one simply asks for a author search. No indexing involved if one wants only the pure raw complete list of all posts by a single author. And Google is called the best search engine of our times, yet I have to go to Drexel to see 8,000 of my posts of which I had posted 22,000 to 36,000 posts from 1993 to 2013. It is a shame that Drexel can display 8,000 while Google has a difficult time of displaying 250 of my authored posts. Where the premiere search engine of Google is outclassed by Drexel and even by Spacebanter.

Archimedes Plutonium

An interesting question arose today as to knowing how many lines altogether exists in 10 Grid.

Now the 10 Grid in 1st quadrant only, has 101 finite points along the x-axis (includes 0) and 101 along the y-axis. So the total number of finite coordinate points in the plane is 101 x 101 = 10,201 coordinate points. Now each of those coordinate points with a different coordinate point forms a line in that plane.

Now in this problem I focus only on full solid lines and not any dashed lines or line segments, because in dashed lines is seen line segments.

So, of those 10,201 coordinate points since two points determine a line, the question is how many such lines does that make?

Now here we are going to have to trim away all those lines that held more than 2 coordinate points, such as all the flat lines like y = 4 or the diagonal lines like y = x.

So, for the first time in the history of geometry, we actually can answer a question as to how many lines are possible to exist. Before, such a question had no meaning because infinity was not precisely defined.

And also, there is another curious problem that needs a answer. When geometry is finite points having empty space between successor points, the distance to macroinfinity is 10 in 10 Grid, but a diagonal line of y = x , the distance from (0,0) to (10,10) is more than 10 and thus an infinity distance. So this may poise as a problem or a challenge, and a possible solution is that the True Geometry Grid is not a square and that 3rd dimension true geometry is not a cube, but rather a circle and sphere as the Grid.

AP

Alright, I was waylaid or made a detour in having to add on a True Geometry text to this 4 text book.

So now I have this textbook as 4 books in one:

1. True Geometry for High School and College; 10 pages
2. True Calculus for High School; 10 pages
3. True Calculus for Uni (University); 50 pages
4. Advanced Calculus; 40 pages

All these 4 books when combined should be around 100 to 110 pages and no more. So all 4 books wrapped as 1 book is a thin book of no more than 110 pages.

Now to finish off the Uni text, I think I need to do several more integrations of specific functions, then do the chain rule on composite functions and then discuss some special functions like the complimentarity function. Finally I should talk about the Fundamental Theorem of Calculus.

The only new idea not discussed previously in a earlier edition is the chain rule and composite functions. If memory serves me, I recall that Stewart had the best write up of composite functions rather than Strang, Fisher & Ziebur, Ellis & Gulick.

So rather than more integral examples, let me start page #37 with the chain rule.


--
Drexel's Math Forum has done an excellent search engine for author posts as seen he
http://mathforum.org/kb/profile.jspa?userID=499986

Now, the only decent search for AP posts on Google Newsgroups, is a search for for it brings up posts that are mostly authored by me and it brings up only about 250 posts. Whereas Drexel brings up nearly 8,000 AP posts. Old Google under Advanced Search
for author, could bring up 20,000 of my authored posts but Google is deteriorating in quality of its searches, likely because AP likes an author search and Google does not want to appear as satisfying to anything that AP likes. If AP likes something, Google is quick to change or alter it.

So the only search engine today doing author searches is Drexel. Spacebanter is starting to do author archive lists. But Google is going in the opposite direction of making author archived posts almost impossible to retrieve.

All the other types of Google searches of AP are just top heavy in hate-spam posts due to search-engine-bombing practices by thousands of hatemongers who have nothing constructive to do in their lives but attack other people.

Now one person claims that Google's deteriorating quality in searches of science newsgroups is all due to "indexing". Well, that is a silly excuse in my opinion, because there is no indexing involved when one simply asks for a author search. No indexing involved if one wants only the pure raw complete list of all posts by a single author. And Google is called the best search engine of our times, yet I have to go to Drexel to see 8,000 of my posts of which I had posted 22,000 to 36,000 posts from 1993 to 2013. It is a shame that Drexel can display 8,000 while Google has a difficult time of displaying 250 of my authored posts. Where the premiere search engine of Google is outclassed by Drexel and even by Spacebanter.

Archimedes Plutonium

Chain Rule with Cell concept #37 Uni-text 8th ed.: TRUE CALCULUS without the phony limit concept

Alright, I did some chain rule derivatives today to see whether the Cell theory is going to make the Chain Rule much easier, or whether, the Cell theory has no impact on the Chain Rule. It appears it has little impact on the Chain rule.

Now in Stewart's book Calculus, 5th ed, 2003, page 219, he uses the composite function F(x) = sqrt(x^2 +1).

And by using the chain rule he determines the derivative of (fg)'(x) = f'(g(x))g')(x) to be that of x/sqrt(x^2+1)

Now the Cell theory can easily come in and determine the derivative dy/dx of any function, whether composite or not composite, by simply doing the raw calculations of dy and dx of a given cell.

So for y = sqrt(x^2 +1) in 10 Grid of the interval
x= 2 then y = sqrt 5 = 2.2
x = 3 then y = sqrt10 = 3.1

So the derivative of the function is dy/dx which is 3.1-2.2= 0.9 and 3-2=1 so we have 0.9/1
for a derivative of 0.9.

Now Stewart used the Chain Rule to get a generalized derivative of x/sqrt(x^2+1). So let us see if it gives 0.9 in that interval. And we have 3/sqrt10 and that is 3/3.1 which is 0.9 in 10 Grid. So we have a match.

However, the generalized formula is much easier to work with since we just plug in numbers, but the Cell theory would have us test each cell for the derivative of that cell.

So, new True Calculus finds nothing wrong with the Chain Rule and is correct whether we have the cell concept or the limit concept. However, the proof of the Chain Rule cannot be made with the phony limit concept. The proof has to come from Cell theory.

Now Stewart gives a preliminary proof of the Chain Rule using the limit concept and he warns the reader that delta u can be 0 and we cannot divide by 0. Whereas in the case of the Cell theory, there is never any worry of division by 0 since the empty space cannot be 0.

So I think I need to include the Chain Rule in True Calculus and note that the only major change is that there was no valid proof in Old Math of the Chain Rule whereas in True Calculus and the cell theory there is a valid proof of the Chain Rule.

The rule itself remains the same.

Now if only I could find a way of calculating the derivate in two different cells and then be able to use those to determine what the general formula is, would be a major improvement. I could always guess at the general formula and then see if the two cells confirm the guess. For in that manner, we do not have to be splitting up a function into two composites and determine the derivative. All we have to do is take two cells, find the derivative with the "whole function" and then determine its general derivative formula. If I can do that, then I would have made the Chain Rule a bit obsolete-- or better said, I would have given an alternative approach to the Chain Rule..

Looks like I will not be spending much time on the Chain Rule, unless I can find a quick formula via the cell theory.

--
Drexel's Math Forum has done an excellent search engine for author posts as seen he
http://mathforum.org/kb/profile.jspa?userID=499986

Now, the only decent search for AP posts on Google Newsgroups, is a search for for it brings up posts that are mostly authored by me and it brings up only about 250 posts. Whereas Drexel brings up nearly 8,000 AP posts. Old Google under Advanced Search
for author, could bring up 20,000 of my authored posts but Google is deteriorating in quality of its searches, likely because AP likes an author search and Google does not want to appear as satisfying to anything that AP likes. If AP likes something, Google is quick to change or alter it.

So the only search engine today doing author searches is Drexel. Spacebanter is starting to do author archive lists. But Google is going in the opposite direction of making author archived posts almost impossible to retrieve.

All the other types of Google searches of AP are just top heavy in hate-spam posts due to search-engine-bombing practices by thousands of hatemongers who have nothing constructive to do in their lives but attack other people.

Now one person claims that Google's deteriorating quality in searches of science newsgroups is all due to "indexing". Well, that is a silly excuse in my opinion, because there is no indexing involved when one simply asks for a author search. No indexing involved if one wants only the pure raw complete list of all posts by a single author. And Google is called the best search engine of our times, yet I have to go to Drexel to see 8,000 of my posts of which I had posted 22,000 to 36,000 posts from 1993 to 2013. It is a shame that Drexel can display 8,000 while Google has a difficult time of displaying 250 of my authored posts. Where the premiere search engine of Google is outclassed by Drexel and even by Spacebanter.

Archimedes Plutonium

#37.1 Chain Rule with Cell concept #37.1 Uni-text 8th ed.: TRUE CALCULUS without the phony limit concept

Now I need to fill in more of the composite function which Stewart uses in his book Calculus, 5th ed, 2003, page 219, he uses the composite function F(x) = sqrt(x^2 +1). Because students are not going to be carrying around other textbooks while reading my textbook.

So I need to fill in the dy/dx = dy/du (du/dx) of the chain rule and of that specific function the f'(u)
= 1/2 u^-1/2. You see, I have to write this book so the student need not have to refer to these other texts if not available to them. So I have to fill in more of the prior post of #37.

Now Stewart defines limit on page 93 and then starts a proof using limits of the Chain Rule on page 218.

Now I do not know if the definition of limit and the proof of the Chain Rule is the first proof in most of these Old Math Calculus books of Stewart, Strang, Fisher & Ziebur, Ellis & Gulick.

In Fisher and Ziebur, they define limit on page 59 and on page 94 do the Chain Rule.

In Strang, he defines limit on page 82 and does the Chain Rule on page 156.

In Ellis & Gulick, they define limit on page 57 and do the Chain Rule on page 137.

So the question is, whether the Chain Rule is the first proof that uses the limit concept in most of these Old Math textbooks which requires the phony limit concept.

Now one thing that caught my attention of Fisher & Ziebur on page 94 is that the Chain Rule in essence is nothing more than a double application of the Power Formula, so that the Chain Rule is basically multiple applications of the Power Formula for in the example above of Stewart's F(x) = sqrt(x^2 +1)
is a double application of the Power Formula, first on f(u) = sqrt(u) which is 1/2 u^-1/2 and then the second Power Formula on x^2 + 1 which has the derivative 2x. And when you assemble those two Power Formula applications as 1/ 2(sqrt (x^2+1)) multiply by 2x we arrive at the final derivative formula of x/(sqrt(x^2+1)).

As I said in the prior post, the Chain Rule is true, whether you learn it from the phony limit concept or whether you learn it from the true concept of Cell theory.

However, what the Chain Rule exposes to our attention is the fact that all Calculus proofs based upon the phony limit concept are all invalid fake proofs.

In Old Math, their derivative was dy/dx. And their concept was the limit. Trouble is when dx is 0 for you then have division by 0 which is undefined. So in Old Math, all their derivatives end up being dy/0. What they called infinitesimally close in the limit application.

So, in Old Math, they failed with the integral by applying the Limit because they ended up with rectangles so thin by the limit application that their rectangles had no interior area.

They failed in the derivative with the application of the Limit because their dy/dx as x approaches the limit infinitesimally close that dx is 0. The integral is empty of interior area because dx is 0 and the derivative is a division by 0.

Here we see how and why the Cell theory saves Calculus, because once you define the border of finite with infinite, you establish instantly a microinfinity border which is the smallest nonzero finite number and thus is the empty space between successive number points in a coordinate system grid.

In Cell theory all dx in the dy/dx are positive numbers that are never 0 unless the function is a flat function such as y = 2. In true calculus the dy is one leg of a right triangle and the dx is the other leg where the hypotenuse is the derivative itself. So that in the expression dy/dx we are asking for a hypotenuse of a right triangle.

I know it is hard, extremely hard for people to read the above after their minds were polluted with fake garbage of Old Math of its silly and phony limit concept. It is hard to imagine so many people being suckers of fakery, but then again, nearly everyone before Copernicus and Galileo and others showed that the Earth is round and revolves around the Sun and is not flat and stationary. So if most people can buy into a flat Earth that has the Sun moving around Earth, then most people can buy into a silly idiotic limit that is division by 0 in dy/dx and is a summation of rectangles that have no interior area.

People only wisen-up when they see alternatives to a question or problem. Until 2011, no-one had an alternative to Limit, so no matter how rotten and phony the limit is, if you are the only thing around, then all the faults are overlooked. But once you give an alternative-- Cell theory, you tend to immediately compare and then the blemishes quickly come forth.

AP

Now I should stick around with the Chain Rule for a bit longer rather than speed off onto a different topic. Because the Cell theory does give us immediately the Power formula and the Chain Rule is basically two Power formulas in succession.

So do you remember how in the High School text that the Power formula was derived purely from the Cell theory? So here is a refresher of that derivation.

In the Cell theory, the derivative is the hypotenuse of a right triangle and the dy is the one leg and the dx the other leg of that right triangle. Usually the right triangle comes fixed atop a rectangle and we have to ignore the rectangle for the derivative but not ignore it for the integral, since the integral is the area of both the rectangle with the right triangle atop it.

So if we take the simple function of y = x, the identity function, in its first cell in 10 Grid from 0 to .1 for a dx of .1-0 = .1 and its dy of .1-0 = .1 we have a derivative of dy/dx = .1/.1 = 1. Now in the second cell of y = x, from .1 to .2 we have to deal with a rectangle and then the triangle atop, but the triangle atop is the very same right triangle as in the first cell where both legs equal each other as a isosceles right triangle.

So what is the area in the first cell of y = x and it is 1/2 base times height of a isosceles right-triangle such that the area is 1/2 b^2. Now what is the integral in the first cell of the function y = x using the Power formula? The Power formula on x is 1/2x^2.

So here we have the situation of where the Cell provides us with deriving the Power Formula itself. The cell also provides the derivative of y = x in that the Power formula on x is 1x^0 = 1.

Now the limit concept could never, and can never derive the Power formula. The Cell concept can and does derive the Power formula.

So now, the Chain rule is a succession of Power formulas. Can the Cell theory derive the Chain Rule?
Probably, but not as easily as the Power formula.

AP