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Topic: unification of Green, Stokes, Divergence and keep the manifold #-107
Uni-textbook 7th ed.: TRUE CALCULUS; without the phony limit concept

Replies: 7   Last Post: Jul 26, 2013 2:07 AM

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 plutonium.archimedes@gmail.com Posts: 7,941 Registered: 3/31/08
unification of Green, Stokes, Divergence and keep the manifold #-107
Uni-textbook 7th ed.: TRUE CALCULUS; without the phony limit concept

Posted: Jul 24, 2013 3:53 PM

I was not expecting success in this unification, at least so very soon as this.

I had a mindblock, so to speak, about Green, Stokes, Divergence theorems. Especially Stokes, for nothing about it seemed to be intuitive.

And then last night, it all came together.

Actually, these three theorems are far more simple in New Math, then what the job of Stewart, Strang, Ellis & Gulick, Fisher and Ziebur struggle with when doing these 3 theorems in Old Math.

It is a breeze for me in New Math to teach these 3 theorems, all because of the concept of CELL. Where the derivative and integral all take place inside a CELL. In 10 Grid there are in total 100 cells, and in 100 Grid which most computer screens are based upon, there are 10^4 total cells. Every curved looking figure in a computer screen is due to a 10^4 total cells or 10^8 total pixels.

So, what is so fantastically easy and simple about Green, Stokes and Divergence theorems. So easy, that I am certain I can teach a High School student these three theorems all in one hour.

They are simple and easy because they are just two functions inside a Cell, rather than 1 function inside a cell.

Now here is my crude ascii art of y = 3 and y = 1/x in the cell of 1 to 1.1 in 10 Grid:

|-| 3
| |
| |
|-| .9

You see, the two vertical columns of the cell wall together with the one function as a floor and the other function as a ceiling cause these two functions to form a Closed Ring in Green theorem and a Closed Surface in Stokes and Divergence theorem.

So to a new student who approaches Green, Stokes, Divergence for the first time is wondering, what in the world am I faced with path and line integral and a normal and a surface and a curl of rotation, and what in the world and where in the world does all this come from?

In Old Math, it all comes out of the blue and makes no sense.

In New Math, it comes straight forward. The Calculus takes place in a Cell with its column walls and when you introduce 2 functions, simultaneously inside a Cell, you create a surface a line integral and even a manifold.

Now for me, it used to be that the Stokes theorem was the most difficult and non intuitive theorem. Today, the Divergence theorem is the most difficult. It is the most difficult because it has to delve with volume in 3rd dimension. For some reason, Calculus is basically a 2nd dimension operation and the 3rd dimension is sort of there to end all things of physics and math.

Green's theorem is just simply-- put two functions into a cell.

Stokes theorem is simply treat the the Green's theorem as if the two functions were surfaces (manifolds)
in 3rd dimension.

Divergence theorem wants volume.

The Divergence theorem is more tricky, now that we know 2 functions in a cell is the prime operation in all of this. The Divergence theorem asks us to generalize the CELL from 2nd dimension into 3rd dimension and that maybe tricky.

So far, I have suggested the generalization be that of making a CELL be a entire plane, rather than a column along the x-axis of a plane. In this viewpoint, volume would be the summation of cross sections of many planes. In the 10 Grid, there are 100 cells in 2nd dimension and there would be 100 planes as cells in 3rd dimension.

Is this generality of making a Cell in 3rd dimension sufficient to explain the Divergence theorem? I do not know yet and am looking at Stewart's page 1134 for the Divergence theorem. It looks good as a generalization of the Cell.

AP