revamping the teaching of the Calculus by using a geometrical explanation; Chord concept

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Topic:	Science > Physics
User:	""
Date:	12 Jan 2006 02:27:14 AM
Object:	revamping the teaching of the Calculus by using a geometrical explanation; Chord concept

I am guessing it was Fermat who came up with the chord concept to
obtain the slope of a arbitrary point on a smooth curve. Anyway, most
textbooks of Calculus omit the Chord concept and that is bad news
because the Chord concept is better than any algebraic concept to
understanding the derivative and how it is linked to the integral.
The old thread was getting too long and so this new thread.
I been meaning to ask a historical question about Newton and Leibniz
concerning the discovery of the Calculus. In Newton's Principia, if my
memory serves me, Newton made a mistake by calling energy as m*v, not
m*v^2. The v^2 is area and the integral is an area. And it was Leibniz
who claimed energy is m*v^2. So my question is, does this historical
facts indicate that Leibniz had a better grasp of the Calculus rather
than Newton's fluxions. Did Leibniz get the energy correct because of
Calculus?
So, now, what about 3rd dimension where a summation of cross-sections
of a object in 3rd dimension would be the volume of the object and so
the integral in 3rd dimension should be easy and strong, but now what
about derivative in 3rd dimension? If the integral is summation of
cross-sections what is the derivative? What is the generalization of a
chord in 3rd dimension.
So we seem to see Calculus falling apart in 3rd dimension. So why is
Calculus so weak as an enterprise that it is not able to generalize
into the 3rd dimension, and we only piecewise compute integration and
differentiation in 3rd dimension?
Is it because in Physics, velocity is linked to energy, but volume is
not linked to anything of physical forces. Can we say a Force is a
volume measure? So when we have F = m*a that the integral in 3rd
dimension is m*a and the derivative is acceleration?
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

User: "Proginoskes"
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	13 Jan 2006 01:48:00 AM

wrote:

I am guessing it was Fermat who came up with the chord concept to
obtain the slope of a arbitrary point on a smooth curve. Anyway, most
textbooks of Calculus omit the Chord concept [...]

Wrong. This is how I've seen it used in any Calculus book that _I've_
ever used.

I been meaning to ask a historical question about Newton and Leibniz
concerning the discovery of the Calculus. In Newton's Principia, if my
memory serves me, Newton made a mistake by calling energy as m*v, not
m*v^2.

Shouldn't that be mv^2 / 2?

The v^2 is area and the integral is an area. And it was Leibniz
who claimed energy is m*v^2. So my question is, does this historical
facts indicate that Leibniz had a better grasp of the Calculus rather
than Newton's fluxions. Did Leibniz get the energy correct because of
Calculus?
So, now, what about 3rd dimension where a summation of cross-sections
of a object in 3rd dimension would be the volume of the object

No, the _integral_ of _the area of_ the cross-sections of an object is
its volume.

and so
the integral in 3rd dimension should be easy and strong, but now what
about derivative in 3rd dimension? If the integral is summation of
cross-sections what is the derivative? What is the generalization of a
chord in 3rd dimension.

So we seem to see Calculus falling apart in 3rd dimension. So why is
Calculus so weak as an enterprise that it is not able to generalize
into the 3rd dimension, and we only piecewise compute integration and
differentiation in 3rd dimension?

Calculus does not "fall apart" in the 3rd dimension. It gets much more
complicated, yes, but it's still able to be handled. And things don't
really get worse in dimensions higher than that.

Is it because in Physics, velocity is linked to energy, but volume is
not linked to anything of physical forces. Can we say a Force is a
volume measure? So when we have F = m*a that the integral in 3rd
dimension is m*a and the derivative is acceleration?

I doubt it.
--- Christopher Heckman
.

User: "Ken Quirici"
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	13 Jan 2006 02:41:13 PM

Proginoskes wrote:

a_plutonium@hotmail.com wrote:

I am guessing it was Fermat who came up with the chord concept to
obtain the slope of a arbitrary point on a smooth curve. Anyway, most
textbooks of Calculus omit the Chord concept [...]

Wrong. This is how I've seen it used in any Calculus book that _I've_
ever used.

Shouldn't that be mv^2 / 2?

Are you possibly thinking of the distance travelled under
acceleration?
s = 1/2 at^2
Thanks.
Ken

So, now, what about 3rd dimension where a summation of cross-sections
of a object in 3rd dimension would be the volume of the object

No, the _integral_ of _the area of_ the cross-sections of an object is
its volume.

Calculus does not "fall apart" in the 3rd dimension. It gets much more
complicated, yes, but it's still able to be handled. And things don't
really get worse in dimensions higher than that.

I doubt it.

--- Christopher Heckman

User: ""
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	13 Jan 2006 02:55:48 PM

Ken Quirici wrote:

Proginoskes wrote:

a_plutonium@hotmail.com wrote:

I am guessing it was Fermat who came up with the chord concept to
obtain the slope of a arbitrary point on a smooth curve. Anyway, most
textbooks of Calculus omit the Chord concept [...]

Wrong. This is how I've seen it used in any Calculus book that _I've_
ever used.

Shouldn't that be mv^2 / 2?

Are you possibly thinking of the distance travelled under
acceleration?

s = 1/2 at^2

Thanks.

Ken

No need to reply. I just looked it up. You're right, 1/2 m v^2.
Thanks.
Ken

So, now, what about 3rd dimension where a summation of cross-sections
of a object in 3rd dimension would be the volume of the object

No, the _integral_ of _the area of_ the cross-sections of an object is
its volume.

Calculus does not "fall apart" in the 3rd dimension. It gets much more
complicated, yes, but it's still able to be handled. And things don't
really get worse in dimensions higher than that.

I doubt it.

--- Christopher Heckman

User: ""
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	13 Jan 2006 03:01:36 AM

Proginoskes wrote:

a_plutonium@hotmail.com wrote:

I am guessing it was Fermat who came up with the chord concept to
obtain the slope of a arbitrary point on a smooth curve. Anyway, most
textbooks of Calculus omit the Chord concept [...]

Wrong. This is how I've seen it used in any Calculus book that _I've_
ever used.

Well I do not see it in Ellis & Gulick nor in Strang, but I admit to
not reading every page and every sentence. I feel it is essential to
teach Calculus as purely geometrical explanations as much as possible,
not the algebraic heuristics. For the derivative and antiderivative say
they are inverses to the integration but never teaches us why they are
inverses. Calculus was borne of geometry and it should be taught 100%
as geometrical explanation and understanding. Teachers of Calculus have
failed to teach students why the integration is inverse to
differential, they just hide behind the stature of the Fundamental
Theorem of Calculus.

Shouldn't that be mv^2 / 2?

The 1/2 is superfluous in a discussion of "physical dimension" such as
the dimensions of force are kg*m/s^2 and what is energy but kg*m/s,
where the only difference is that force has seconds squared in
denominator. So 1/2 is superfluous to the physical-dimensions.

So, now, what about 3rd dimension where a summation of cross-sections
of a object in 3rd dimension would be the volume of the object

No, the _integral_ of _the area of_ the cross-sections of an object is
its volume.

I know what volume and cross-sections are. The question I wanted you to
address was what the derivative is in 3rd dimension. If it is a
tangent line at a point in 2nd dimension, what is the derivative with
respect to the integral in 3rd dimension. So if the integral is the
summation of line-segments in 2nd dimension as the Limit or Chord
approaches zero and the derivative is the tangent, then in 3rd
dimension the integral is the summation of cross-sections, then what on
earth is the derivative in 3rd dimension.
This is where I claim Calculus falls apart and why the Calculus does
not exist in 4th and higher dimensions, for it has a struggle in 3rd
dimension.

Calculus does not "fall apart" in the 3rd dimension. It gets much more
complicated, yes, but it's still able to be handled. And things don't
really get worse in dimensions higher than that.

Not really. It falls apart because it loses a geometrical meaning of a
derivative at a point. Perhaps what Martin is saying with the Secant,
that the Secant may be able to save the derivative in 3rd dimension.
And the integral is the summation of cross-sections which yields a
volume.

I doubt it.

--- Christopher Heckman

I suspect that because Physics stops with Force which is 3 dimensional,
then Calculus also stops in 3rd dimension and that a Calculus of 4th
dimension or higher is utterly meaningless, just as angels, witches
flying on broomsticks are imagination but never reality.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

User: ""
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	13 Jan 2006 12:10:43 PM

wrote:

Proginoskes wrote:

wrote:

I am guessing it was Fermat who came up with the chord concept to
obtain the slope of a arbitrary point on a smooth curve. Anyway, most
textbooks of Calculus omit the Chord concept [...]

Wrong. This is how I've seen it used in any Calculus book that _I've_
ever used.

Shouldn't that be mv^2 / 2?

So, now, what about 3rd dimension where a summation of cross-sections
of a object in 3rd dimension would be the volume of the object

No, the _integral_ of _the area of_ the cross-sections of an object is
its volume.

Lookup tangent plane in Mathworld, which defines the equation of
a tangent plane to a surface in terms of partial derivatives.
Also if you look up partial derivatives, it defines a 'total'
derivative -
I forget the exact name - in multiple dimensions (i think just 3 as
an example).
This might satisfy your requirements. Then again it might not.
Thanks.
Ken

So if the integral is the
summation of line-segments in 2nd dimension as the Limit or Chord
approaches zero and the derivative is the tangent, then in 3rd
dimension the integral is the summation of cross-sections, then what on
earth is the derivative in 3rd dimension.

This is where I claim Calculus falls apart and why the Calculus does
not exist in 4th and higher dimensions, for it has a struggle in 3rd
dimension.

Calculus does not "fall apart" in the 3rd dimension. It gets much more
complicated, yes, but it's still able to be handled. And things don't
really get worse in dimensions higher than that.

I doubt it.

--- Christopher Heckman

I suspect that because Physics stops with Force which is 3 dimensional,
then Calculus also stops in 3rd dimension and that a Calculus of 4th
dimension or higher is utterly meaningless, just as angels, witches
flying on broomsticks are imagination but never reality.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

User: ""
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	13 Jan 2006 02:02:54 PM

Ken Quirici wrote:
Lookup tangent plane in Mathworld, which defines the equation of
a tangent plane to a surface in terms of partial derivatives.
Also if you look up partial derivatives, it defines a 'total'
derivative -
I forget the exact name - in multiple dimensions (i think just 3 as
an example).
This might satisfy your requirements. Then again it might not.
A.P. writes:
I can visualize a tangent plane to say an object like a 3-d donut or
torus in 3 d Eucl geometry and I can visualize that tangent plant to a
point on the torus.
But then if the Calculus exists in the 3rd dimension and higher then
this tangent plane would be able to be the inverse of the
cross-sections of the torus. Where the volume of the torus is the
summation of cross-sections as the integral and the derivative would be
the tangent plane.
But it is impossible, as far as I know at this moment, to construe that
tangent plane to be the inverse of the cross sections.
That is why the Calculus breaks down and falls apart in 3rd dimension,
where it is able to deliver the integral as volume by means of
summation of cross-sections, but fails to connect differentiation with
those cross-sections. And of course if 3rd dimension is only partially
satisfied with Calculus that the 4th and higher have no Calculus
whatsoever.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

User: "Proginoskes"
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	14 Jan 2006 02:39:59 AM

wrote:

Ken Quirici wrote:
Lookup tangent plane in Mathworld, which defines the equation of
a tangent plane to a surface in terms of partial derivatives.
Also if you look up partial derivatives, it defines a 'total'
derivative -
I forget the exact name - in multiple dimensions (i think just 3 as
an example).

This might satisfy your requirements. Then again it might not.

A.P. writes:
I can visualize a tangent plane to say an object like a 3-d donut or
torus in 3 d Eucl geometry and I can visualize that tangent plant to a
point on the torus.

But then if the Calculus exists in the 3rd dimension and higher then
this tangent plane would be able to be the inverse of the
cross-sections of the torus. Where the volume of the torus is the
summation of cross-sections as the integral and the derivative would be
the tangent plane.

But it is impossible, as far as I know at this moment, to construe that
tangent plane to be the inverse of the cross sections.

That is why the Calculus breaks down and falls apart in 3rd dimension,

Or this is where your geometric analogy breaks down.

where it is able to deliver the integral as volume by means of
summation of cross-sections, but fails to connect differentiation with
those cross-sections. And of course if 3rd dimension is only partially
satisfied with Calculus

If physics is really king, then 3-dimensional Calculus _must_ work. For
if it doesn't give meaningful results to problems about force fields,
then it would never have been kept as a physics tool. And, in fact,
that's how things like flux and work are calculated. (When I first took
Calc III, I had the feeling that the last part of the course was too
"physics-oriented" for me; it had that sliminess that I've associated
with applied mathematics.)
If you look at the "Fundamental Theorems" of Calculus in higher
dimensions (FT of Line Integrals, Green's Theorem, The Divergence
Theorem, Stokes' Theorem), you will find a pattern of the form
(n-1)-dimensional integral of f over the boundary of S
= n dimensional integral of (some modification of f) over S
For instance, the FTOC fits into this mold by letting S = [a,b]; then
the integral over the boundary of S is
f(b) - f(a),
and the "modification of f" is f'(x). So this general form is saying
f(b) - f(a) = integral f(x), from a to b.
Also, the Fundamental Theorem for Line Integrals is a generalization of
FTOC, and Stokes' Theorem is a generalization of Green's Theorem. This
is the sort of thing I didn't discover until my Complex Analysis
course.

that the 4th and higher have no Calculus whatsoever.

Arguing from ignorance again, I see.
--- Christopher Heckman
.

User: "Proginoskes"
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	14 Jan 2006 02:24:22 AM

wrote:

Proginoskes wrote:

wrote:

I am guessing it was Fermat who came up with the chord concept to
obtain the slope of a arbitrary point on a smooth curve. Anyway, most
textbooks of Calculus omit the Chord concept [...]

Wrong. This is how I've seen it used in any Calculus book that _I've_
ever used.

Me too, which makes me wonder why BOTH of your teachers (in high school
and college) seem not to have taught it that way. Or maybe you were
absent on those days?

So, now, what about 3rd dimension where a summation of cross-sections
of a object in 3rd dimension would be the volume of the object

No, the _integral_ of _the area of_ the cross-sections of an object is
its volume.

I know what volume and cross-sections are.

Well, I wasn't sure whether you didn't know, or whether you were just
being sloppy.

The question I wanted you to
address was what the derivative is in 3rd dimension. If it is a
tangent line at a point in 2nd dimension, what is the derivative with
respect to the integral in 3rd dimension. So if the integral is the
summation of line-segments in 2nd dimension as the Limit or Chord
approaches zero and the derivative is the tangent, then in 3rd
dimension the integral is the summation of cross-sections, then what on
earth is the derivative in 3rd dimension.

This is a question that can be answered by consulting any decent
Calculus textbook, in the multivariate section.
Basically, if your function is from R (or some subset) to R^n, then you
have a curve in n-dimensional space, and the derivative is an
n-dimensional vector. If your function is from R^2 to R, then the graph
is a surface, and "tangent line" becomes "tangent plane". (There's lots
of stuff about partial derivatives here that I'm skipping.) If your
function is from R^n to R^n, then you have a vector field, and there is
no one nice derivative. You have things like divergence and curl, which
involve partial derivatives.
If you're doing an integral over a 2-dimensional region R, then you cut
R into little pieces R(1), R(2), ..., R(k), choose an x(i) in each R(i)
(which will actually be an ordered pair), then look at the Riemann sum
f(x(1)) * area(R(1)) + f(x(2)) * area(R(2)) + ... + f(x(k)) * area
(R(k)),
which will approximate the integral of f over the region R. If your
pieces R(i) have small diameter, you will get a better estimate for the
integral.
Integrals over 3-dimensional solids are done similarly, except "area"
is replaced by "volume".

This is where I claim Calculus falls apart and why the Calculus does
not exist in 4th and higher dimensions, for it has a struggle in 3rd
dimension.

Well, the big problem comes from force fields, and electro-magnetic
fields, where the curl and divergence come into play. So we could make
life easier for Calculus by throwing out the concept of force fields
from physics ...

Calculus does not "fall apart" in the 3rd dimension. It gets much more
complicated, yes, but it's still able to be handled. And things don't
really get worse in dimensions higher than that.

Not really. It falls apart because it loses a geometrical meaning of a
derivative at a point. [...]

The definition of differentiability for a function of two variables
seems to go "backwards": For a real-valued function of one variable,
you can define differentiability by looking at the appropriate limit,
and you have differentiability iff the limit exists. For a function of
two variables, you have two "partial" derivatives, but if these exist,
then it is not guaranteed that f is "differentiable".
Part of the confusion is because Calculus books don't give a decent
definition of a tangent line, to a curve other than a circle. Stuart
(the one ASU uses) in particular defines the line tangent to a curve
y=f(x) at the point x=a to be the line passing through (a,f(a)) and
which has slope f'(a).
A much better definition of the tangent line is the line passing
through P such that points on the line which are close to P are close
to points on the curve. This definition (which I prefer using when
teaching) shows how you can visually find the tangent line, without
having calculate derivatives, or even use Cartesian coordinates.
Then this concept will extend to a function of two variables by
considering the graph of z=f(x,y). The plane tangent to this surface at
a point P is one that passes through P and best approximates the
surface near P. You can move to functions of n variables with this
concept if you like.

I doubt it.

Higher dimensions may not have been in Newton's or Leibnitz's minds
when they worked out the theory behind Calculus, but that doesn't mean
that it can't be extended.
In fact, partial derivatives and integrals (as a real-valued function
of n variables) easily generalize to higher dimensions, and there are
FTOC-like results for every dimension. ("FTOC-like results" include the
Fundamental Theorem of Calculus, the Fundamental Theorem of Line
Integrals, Green's Theorem, The Divergence Theorem, and Stokes'
Theorem.)
Another example would be that a system of linear equations can have as
many variables as you would like; Linear Algebra isn't limited to 2 or
3 dimensions. Geometry of higher dimensions leads to Linear
Programming, which again is a useful type of optimization problem.
--- Christopher Heckman
.

User: ""
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	15 Jan 2006 02:27:37 AM

Proginoskes wrote:
(snip)

This is a question that can be answered by consulting any decent
Calculus textbook, in the multivariate section.

Basically, if your function is from R (or some subset) to R^n, then you
have a curve in n-dimensional space, and the derivative is an
n-dimensional vector. If your function is from R^2 to R, then the graph
is a surface, and "tangent line" becomes "tangent plane". (There's lots
of stuff about partial derivatives here that I'm skipping.) If your
function is from R^n to R^n, then you have a vector field, and there is
no one nice derivative. You have things like divergence and curl, which
involve partial derivatives.

If you're doing an integral over a 2-dimensional region R, then you cut
R into little pieces R(1), R(2), ..., R(k), choose an x(i) in each R(i)
(which will actually be an ordered pair), then look at the Riemann sum
f(x(1)) * area(R(1)) + f(x(2)) * area(R(2)) + ... + f(x(k)) * area
(R(k)),
which will approximate the integral of f over the region R. If your
pieces R(i) have small diameter, you will get a better estimate for the
integral.

Integrals over 3-dimensional solids are done similarly, except "area"
is replaced by "volume".

This is where I claim Calculus falls apart and why the Calculus does
not exist in 4th and higher dimensions, for it has a struggle in 3rd
dimension.

(snip)

Not really. It falls apart because it loses a geometrical meaning of a
derivative at a point. [...]

The definition of differentiability for a function of two variables
seems to go "backwards": For a real-valued function of one variable,
you can define differentiability by looking at the appropriate limit,
and you have differentiability iff the limit exists. For a function of
two variables, you have two "partial" derivatives, but if these exist,
then it is not guaranteed that f is "differentiable".

Part of the confusion is because Calculus books don't give a decent
definition of a tangent line, to a curve other than a circle. Stuart
(the one ASU uses) in particular defines the line tangent to a curve
y=f(x) at the point x=a to be the line passing through (a,f(a)) and
which has slope f'(a).

A much better definition of the tangent line is the line passing
through P such that points on the line which are close to P are close
to points on the curve. This definition (which I prefer using when
teaching) shows how you can visually find the tangent line, without
having calculate derivatives, or even use Cartesian coordinates.

Then this concept will extend to a function of two variables by
considering the graph of z=f(x,y). The plane tangent to this surface at
a point P is one that passes through P and best approximates the
surface near P. You can move to functions of n variables with this
concept if you like.

I doubt it.

Higher dimensions may not have been in Newton's or Leibnitz's minds
when they worked out the theory behind Calculus, but that doesn't mean
that it can't be extended.

In fact, partial derivatives and integrals (as a real-valued function
of n variables) easily generalize to higher dimensions, and there are
FTOC-like results for every dimension. ("FTOC-like results" include the
Fundamental Theorem of Calculus, the Fundamental Theorem of Line
Integrals, Green's Theorem, The Divergence Theorem, and Stokes'
Theorem.)

Another example would be that a system of linear equations can have as
many variables as you would like; Linear Algebra isn't limited to 2 or
3 dimensions. Geometry of higher dimensions leads to Linear
Programming, which again is a useful type of optimization problem.

--- Christopher Heckman

Well thanks for the overall survey. I do not know for certain whether
this statement is true for physics, but I have the hunch it is true for
physics. And if it is true then it would also prove that the Calculus
in mathematics is limited to only 2nd and 3rd dimension and is
meaningless, ie, nonexistent in other dimensions.
In physics, as I noted above that energy has the parameters of Kg
*distance^2/time^2
and force has the parameters of Kg*distance/time^2. Now in physics
there are other things such as pressure, momentum etc etc.
But the thing is, that given any assemblage of parameters they all
reduce to Force, energy, momentum.
Say someone does not like force and likes something of Kg^3*
distance^5/time^4.
Conjecture: no matter what wild assemblage of parameters one wishes to
consider, they all reduce to force, energy and momentum.
I think that conjecture is true. That underlying every wildly concocted
assemblage of parameters all reduce to the basic ones of force, energy,
momentum.
So what this tells me, is that the Calculus exists only in 2nd
dimension and 3rd dimension.
Now there should be a mathematical conjecture that says something to
the effect that if you can do it algebraically but not geometrically
then it is a false body of mathematics and vice versa that if you can
do something geometrically but not do it algebraically then it is a
false piece of mathematics. So that although the Calculus is thought to
be generalized into the 4th dimension and higher because it appears
algebraically feasible, is in fact a falsehood because it is not
geometrically possible.
I still have not seen how a derivative exists in 3rd dimension to a
integration of cross sections. Integration is okay and fine and dandy
in 3rd dimension but the inverse of integration is not possible and the
Fundamental Theorem of Calculus seems to be unable to generalize to
even the 3rd dimension.
So if that Physics datum is correct that all physics parameters are
reducible to Force, energy, momentum which are confined to 2nd and 3rd
dimension, implies that Calculus is nonsense in 4th dimension and
beyond. In other words, there is no 4th dimension or higher and are
just human imagination.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

User: "Proginoskes"
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	15 Jan 2006 11:45:19 PM

wrote:

Proginoskes wrote:

[summary of higher-dimensional calculus removed]
In fact, partial derivatives and integrals (as a real-valued function
of n variables) easily generalize to higher dimensions, and there are
FTOC-like results for every dimension. ("FTOC-like results" include the
Fundamental Theorem of Calculus, the Fundamental Theorem of Line
Integrals, Green's Theorem, The Divergence Theorem, and Stokes'
Theorem.)

Another example would be that a system of linear equations can have as
many variables as you would like; Linear Algebra isn't limited to 2 or
3 dimensions. Geometry of higher dimensions leads to Linear
Programming, which again is a useful type of optimization problem.

Well thanks for the overall survey. I do not know for certain whether
this statement is true for physics, but I have the hunch it is true for
physics.

Another instance why physics isn't the king of mathematics.

And if it is true then it would also prove that the Calculus
in mathematics is limited to only 2nd and 3rd dimension and is
meaningless, ie, nonexistent in other dimensions. [...]

Well, if you read my response, you'll see that you've reached a
contradiction ... And that means some assumption is false, namely that
mathematics is entirely contained within physics.

Only in your opinion. And there certainly isn't one if "mathematics" is
replaced by "pure mathematics".

So that although the Calculus is thought to
be generalized into the 4th dimension and higher because it appears
algebraically feasible, is in fact a falsehood because it is not
geometrically possible.

You missed the part about cross-sections and the n-dimensional volume
formula that I mentioned.

I still have not seen how a derivative exists in 3rd dimension to a
integration of cross sections. Integration is okay and fine and dandy
in 3rd dimension but the inverse of integration is not possible and the
Fundamental Theorem of Calculus seems to be unable to generalize to
even the 3rd dimension.

Again, if you had read my post in its entirety, you wouldn't be
thinking such things. The Fundamental Theorem of Calculus does
generalize, to the Fundamental Theorem of Line Integrals, Green's
Theorem, The Divergence Theorem, and Stokes' Theorem, as well as
higher-dimensional analogs of these results. The basic pattern, if I
remember it correctly, is:
Integral over (the boundary of S) of f = Integral over S of df,
where df denotes the total differential of f. (Thank you, Gary
Meisters!)

So if that Physics datum is correct that all physics parameters are
reducible to Force, energy, momentum which are confined to 2nd and 3rd
dimension, implies that Calculus is nonsense in 4th dimension and
beyond. In other words, there is no 4th dimension or higher and are
just human imagination.

Who's to say that the whole universe (including physics) isn't "just
human imagination"?
--- Christopher Heckman
.

User: ""
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	15 Jan 2006 11:19:51 AM

Kyle wrote:

On Jan 15, 2006 2:27 AM CT, Archimedes Plutonium wrote:

[...]

Conjecture: no matter what wild assemblage of
parameters one wishes to consider, they all reduce to
force, energy and momentum.

I think that conjecture is true.

I think your conjecture is clearly biased because it
assumes that hyperdimensional space is nonexistant. Of
course these parameters will reduce to force, energy,
and momentum; clearly this is just algebraic manipulation.
However, you're ASSUMING these quantites have no meaning
in higher dimensions.

That underlying every wildly concocted assemblage of
parameters all reduce to the basic ones of force,
energy, momentum.

Yes they will, but how do you know for certain that these
"wildly concocted" parameters have no meaning in higher
dimensions? Nobody confined to this manifold can answer
that with 100% certainty.

So what this tells me, is that the Calculus exists
only in 2nd dimension and 3rd dimension.

Didn't you say earlier that calculus fell apart in the
3rd dimension? ^_^ You're not changing your position
mid-argument are you?

Now there should be a mathematical conjecture that
says something to the effect that if you can do it
algebraically but not geometrically then it is a false
body of mathematics and vice versa that if you can
do something geometrically but not do it algebraically
then it is a false piece of mathematics. So that
although the Calculus is thought to be generalized into
the 4th dimension and higher because it appears
algebraically feasible, is in fact a falsehood
because it is not geometrically possible.

Just because you cannot fathom hyperdimensional space
doesn't mean it cannot exist. I'm not saying it *does*
exist, my point is that you cannot conclude for certain
that it cannot exist.

I still have not seen how a derivative exists in 3rd
dimension to a integration of cross sections.
Integration is okay and fine and dandy in 3rd dimension
but the inverse of integration is not possible and the
Fundamental Theorem of Calculus seems to be unable to
generalize to even the 3rd dimension.

Please take a course on calculus or read a multivariable
calculus text before you go bashing the subject.

So if that Physics datum is correct that all physics
parameters are reducible to Force, energy, momentum
which are confined to 2nd and 3rd dimension, implies
that Calculus is nonsense in 4th dimension and
beyond.

That is nonsense; the mathematics in higher dimensions is
completely valid. If anything, only the validity of
*physical concepts* in higher dimensions is questionable.

In other words, there is no 4th dimension or higher and
are just human imagination.

I can partially agree with you here. Possibly higher
dimensional do not exist and it's purely human imagination
at work. However, nobody can say with 100% certainty that
these higher dimensions do not exist.

At the risk of speaking beyond myself here, have you
considered the nature of fermions? Why is such an
exotic algebra needed to describe such physics? Isn't
it possible that some physics seems so bizarre because
it's taking place on a higher dimensional manifold?

Regards,
Kyle

Kyle, there is a idea in Physics and the sciences that answers your
question and is the cornerstone, or hallmark of science. It says if it
is never possible to measure, observe or experiment with an idea, then
it is not science. Mathematics has tried to be above science by
claiming it is purely abstraction and not an experimental science. That
was allowable in the Big Bang theory, but in the Atom Totality theory,
mathematics is pulled into being just a small subset of Physics and
since it is inside of Physics, mathematics, like all the other sciences
is experimental and is a pure result of whatever Physics is.
So, dream on, dream about 4th dimensional Euclidean geometry. Dream
about Cantor transfinites. For which none of them exists. They are
dreams and hyper imagination. There are no witches or fire breathing
dragons and there are no supernatural because they are not science, and
likewise there are no higher dimensions than 3rd.
Three dimensions suffice for fermion physics because the StrongNuclear
Force when coupled with the WeakNuclear Force becomes merely a Coulomb
force confined to the nuclear region.
I do not know if Abian was the first to say that "imagination is the
key and all the rest is detail" or words to that effect. But many
times, imagination is flat wrong and that all we needed was a careful
examination of what we presently have, and that we are failing to see
it as sufficient to lead us to the correct theory.
We fail to see that 3 dimensions yields "time" and that we do not need
a 4th dimension for time. We fail to see that a unification of forces
of physics means they are all "one" force and that would mean less
imagination for new things, but rather instead, coupling the old forces
into the one perfect force which is Coulomb force. So imagination often
hinders scientists and mathematicians from finding the truth.
One of the warning signals that should have popped up that there are no
dimensions beyond 3rd is the axes problem. You can have a perpendicular
to 1 dimension to get 2nd dimension and then a perpendicular to 2nd to
obtain 3rd. But a perpendicular to 3rd, well it never works does it.
Yet mathematicians blithely sail on and never think that they are on
nothing but grotesque imagination gone awry.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

User: "Proginoskes"
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	16 Jan 2006 12:02:26 AM

wrote:

[...]
Kyle, there is a idea in Physics and the sciences that answers your
question and is the cornerstone, or hallmark of science. It says if it
is never possible to measure, observe or experiment with an idea, then
it is not science. [...]

One of the warning signals that should have popped up that there are no
dimensions beyond 3rd is the axes problem. You can have a perpendicular
to 1 dimension to get 2nd dimension and then a perpendicular to 2nd to
obtain 3rd. But a perpendicular to 3rd, well it never works does it.
Yet mathematicians blithely sail on and never think that they are on
nothing but grotesque imagination gone awry.

Well, if you're going to be consistent, then you'll realize that
Riemann geometry and Lobachevsky geometry don't exist, either, and that
your equation "RG + LG = EG" is as meaningless as 4-dimensional
geometry and also "grotesque imagination gone awry".
--- Christopher Heckman
.

User: "Proginoskes"
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	16 Jan 2006 12:08:20 AM

wrote:

Actually it's not an idea in science; it's an idea in _the philosophy
of_ science.
--- Christopher Heckman
.

User: ""
Title: Re: revamping the teaching of the Calculus	16 Jan 2006 01:01:21 AM

Kyle wrote:
(snip)

"Mathematics is the queen of the sciences and number
theory is the queen of mathematics." - Gauss

I guess Gauss's comment is reasonable and accurate if we
say Physics is the king of science and where math is a tool
for physics.
This reminds me of a quote from Hardy in his book
A Mathematicians Apology where he says words to the
effect-- whomever finds a connection between Physics
and Mathematics will have found the ultimate--
Now I wonder, Kyle, whether in the history of mathematics,
whether that idea was original to Hardy, that math and
physics had some sort of separateness and that a link or
connection between the two subjects is a great accomplishment.
Because, due to string theory and other exotica, the mood
and direction of physics and mathematics by college professors
is one diametrically opposite of the view of Hardy, where they
believe physics reduces to mathematics and math is king, and
in this viewpoint math is not separate from physics but that
physics reduces to some math formula.
I think someone ought to examine the history of physics and
math with this central idea of Hardy's of a link and connection
between Physics and Mathematics.
Because with the Big Bang, mathematics can be king and physics
the lesser subject. But with the Atom Totality theory, physics is
king and mathematics a mere tool, because atoms are numerous
creates arithmetic and because atoms have shape creates
geometry.
Now according to Gauss's statement, Gauss would have accepted the
Atom Totality theory rather than the Big Bang.
So I wonder if Hardy is the first person to have focused on this very
big question. Call it an insight. Perhaps one of the most important
questions of all time. And an entire book should be written around this
question of the connection and link-up of Physics with Mathematics.
Because the Big Bang treats this question very much differently from
the Atom Totality theory.
(snip)

That's quite a noble quest you have there AP. Good luck
on debunking centuries of mathematics.

Regards,
Kyle

It sounds as if you are bowing out of this conversation. I wanted to
ask you a question but if you withdraw, someone else will answer it. I
am thinking of Fourier Transforms and how the sine, cosine seem to be
able to describe all functions. What is the Fourier Transform theory
again?
I was thinking of this in connection with all physics assemblage of
parameters reduces to energy or force or momentum. So if we take force
as sine and energy as cosine can we prove that all physics assemblages
reduce to either energy or force?
Thanks for the comment of noble quest. I do not do this stuff just to
be contrary or revolting against the establishment. I do it because in
1990 I came to a new idea of the Atom Totality and if true, changes all
of physics and mathematics. So I must explore that idea against the
established ideas. If I had never had that idea of Atom Totality,
no-one on the Internet in the sci newsgroups would have ever heard of
me or had a post of mine.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

User: ""
Title: Hardy's page 123 Re: revamping the teaching of the Calculus	16 Jan 2006 11:05:31 PM

wrote:

Kyle wrote:

(snip)

"Mathematics is the queen of the sciences and number
theory is the queen of mathematics." - Gauss

I should be precise as to what Hardy said:
--- quoting page 123 A MATHEMATICIAN'S APOLOGY, Hardy, 1940 ---
Some hold that it is 'mental' and that in some sense we construct it,
others that it is outside and independent of us. A man who could give a
convincing account of mathematical reality would have solved very many
of the most difficult problems of metaphysics. If he could include
physical reality in his account, he would have solve them all.
--- end quoting Hardy ---
The Atom Totality Theory fully explains Hardy's call. This theory
connects physics with mathematics and makes mathematics a minor subset
of physics.
The Big Bang theory is deaf dumb and silent as to Hardy's call.
However, it dismisses Hardy's subject of Metaphysics because it is a
nonexistant subject. There is no Metaphysics in the Atom Totality, and
in fact God is the Atom Totality where Science is god and god is
Science.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

User: ""
Title: Re: revamping the teaching of the Calculus	17 Jan 2006 02:40:04 AM

Kyle wrote:

I guess Gauss's comment is reasonable and accurate if
we say Physics is the king of science and where math is
a tool for physics.

If this is true and physics is the king of science, I
still do not see mathematics being a mere tool; surely
a king would not think of his queen as a simple tool.

In the sense that because atoms are numerous creates numbers
and arithmetic and because atom have shapes creates geometry.
Because the Atom Totality is 231Pu which has in the 5f6, in rational
form 22 subshells inside 7 shells and only 19 subshells filled at any
instant of time, and thus creating a pi of 22/7, and e of 19/7 in
rational
irrational form if you have uncollapsed wavefunction.
What I am saying is that mathematics is a fallout of Physics. Physics
creates everything that is in mathematics. Every mathematical theorem
is a consequence of Physics of atoms.
(snip)

Again, I haven't studied the work of Fourier formally,
but this proposition does not seem worthwhile. You are
suggesting that we consider the force and energy of a
system as the imaginary and real parts of the Fourier
transform, respectively. I don't see how this would be
beneficial in anyway.

Regards,
Kyle

Yes I am suggesting that. I am suggesting that notice how messy the
derivative as inverse to a integral in 3rd dimension becomes. So if
Physics stops with 3rd dimension and the 4th dimension is nonexistant
in physics, then, shifting over to mathematics, we should find that
Calculus becomes messy in 3rd dimension and nonexistant in 4th or
higher.
Likewise, if Physics parameters of energy and force can account for any
and all parameters in physics, then there should be some analogous
mathematics that follows the mirror image track and it would be Fourier
Transform Theory where the sine and cosine are manipulated to account
for any and every function. So we simply say, energy in physics is sine
in Fourier Transform mathematics and force in physics is cosine in
Fourier Transform.
So if something is true in physics, then it has its equal counterpart
in mathematics and vice versa. And if something is false in physics
such as 4th dimension, then it is false and nonexistant in mathematics.
If Physics has only one type of infinity-- the infinity of counting
numbers, then mathematics has only one type of infinity and Cantor
transfinites is a fakery.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

User: ""
Title: Re: revamping the teaching of the Calculus	17 Jan 2006 02:46:45 AM

Now there is one way to convince me that 4th dimension and higher
exists or that Cantor transfinites exist. If you can convince me that
an atom, any atom needs 4th dimension or higher, then I will admit to
being wrong. And if you can convince me
that atoms need more than one type of infinity, then I will also admit
Cantor was
correct.
So if you can show me that atoms need more than 3 dimensions and that
atoms
need more than one type of infinity. Then I will accept that as true.
But from where I sit or stand at this moment, I do not see where atoms
need the 4th dimension nor where atoms need transfinite-infinities.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

User: "Proginoskes"
Title: Re: revamping the teaching of the Calculus	17 Jan 2006 04:20:31 PM

wrote:

I would like to ask another question: Why does an atom need _3rd_
dimension or higher?
--- Christopher Heckman
.

User: ""
Title: Re: revamping the teaching of the Calculus	18 Jan 2006 01:00:13 AM

Proginoskes wrote:

a_plutonium@hotmail.com wrote:

I would like to ask another question: Why does an atom need _3rd_
dimension or higher?

--- Christopher Heckman

Forces are 3rd dimensional, as is energy is 3rd dimensional and you
cannot do physics without forces and energy. The Schrodinger Equation
and Dirac Equation are 3rd dimensional.
The nucleus of an atom can be represented as 3rd dimension tiny sphere
inside the
larger sphere of the electron cloud, where the nuclear Coulomb force is
the union of the
StrongNuclear and WeakNuclear forces. So combining of StrongNuclear
with WeakNuclear yields a Nuclear-Coulomb of a small sphere nested
inside the larger sphere of the electron cloud.
So 4th dimension is never needed, and everything is 3rd dimensional.
Time is the arrangement and rearrangement of the atoms inside the Atom
Totality so time never becomes a 4th dimension, but a byproduct of the
atoms rearranging in space.
In String theory, extra dimensions are added as curled dimensions. But
an atom does not need curled dimensions. The nucleus is not a curled
dimension. The nodes in atoms are not curled dimensions. The
Schrodinger and Dirac Equations are never needing curled dimensions.
There is not a single case to be made where atoms need the 4th
dimension or higher.
I am going to take a vacation from the Internet for about a week. I
have to write up a report on Darwin Evolution which is going to absorb
my attention for a week.
I have not forgotten about the piece of mathematics I wanted to
conquer. That piece where the Native Points of Lobachevsky geometry are
some form of adics and where the Native Points of Riemann geometry are
some form of adics in a equation:
Riem geom + Loba geom = Eucl geom
which translates into
form of adics + another form of adics = Reals
On PBS some years ago was a TV show on dogs and where they were testing
the smartness of dogs of throwing a stick into a thick brush and to
test dogs for how long they stayed focused on recovery of the stick,
with some planned distractions and other devices. And the really smart
dog, just remembers where the focus was all along. So I am not a dumb
dog, as I meander through scores of other subjects, always remembering
of what it is I want foremost to know.
I think the trick is going to be to put a negative sign on Adic
Rationals, and for the Riem geom they are simply the Adic Integers. I
will need help from Dik Winter when I finally get to the point of
sewing this thing together.
So that this emerges:
Adic Integers union the -(Adic Rationals) = Reals
Riem geom + Loba geom = Eucl geom
And the geometrical picture which looks promising is that of the
Magnetic Field of a current ring as seen on page 366 of BERKELEY
Physics Course volume 2, Electricity and Magnetism 1963. Where half of
the lines of force are Riem geom and the other half as Loba geom and
the two together makes the Eucl plane.
But so much more work needs be done. If it was easy, someone would have
discovered this 300 years ago, instead of now.
See you after a week.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

User: "Proginoskes"
Title: Re: revamping the teaching of the Calculus	18 Jan 2006 08:25:25 PM

wrote:

Proginoskes wrote:

wrote:

I would like to ask another question: Why does an atom need _3rd_
dimension or higher?

Forces are 3rd dimensional, [...]

Forces are vectors, which in 3-dimensional space are 3 dimensional.
You're going around in circles here.

I have not forgotten about the piece of mathematics I wanted to
conquer. That piece where the Native Points of Lobachevsky geometry are
some form of adics and where the Native Points of Riemann geometry are
some form of adics in a equation: [...]

By your own admission you're dealing with two types of geometry
(Riemann, Lobachevsky) which don't exist, since the universe is
Euclidean, and you _claim_ that any mathematics which doesn't
correspond to physics is fantasy. That makes your "equation" a
phantasm, by your own logic.

On PBS some years ago was a TV show on dogs and where they were testing
the smartness of dogs of throwing a stick into a thick brush and to
test dogs for how long they stayed focused on recovery of the stick,
with some planned distractions and other devices. And the really smart
dog, just remembers where the focus was all along. So I am not a dumb
dog, as I meander through scores of other subjects, always remembering
of what it is I want foremost to know. [...]

Evidently Calculus isn't listed among the things you "need to know",
because your posts clearly indicated you didn't remember basic
concepts. Linear Algebra also seemed to have slipped your mind since
college, as well as the definition of a field and the difference
between Peano Axioms and Peano Arithmetic.
Hmmm.....

But so much more work needs be done. If it was easy, someone would have
discovered this 300 years ago, instead of now.

Maybe someone _did_ discover it 300 years ago ... and abandoned it.
--- Christopher Heckman
.

User: ""
Title: Re: revamping the teaching of the Calculus	16 Jan 2006 01:16:09 AM

Now let us play with the Fourier Transform Theory for just a little
bit. If it says what I think it says, that every function can be
written as a sine and cosine mix, is almost
identical in nature to the idea that the energy and force parameters
cover all parameters
in physics.
So let us ask the next question, about dimensions. Since Force and
Energy are a mix of 2nd and 3rd dimensions, then no higher dimensions
exist. So does that also agree with the Fourier Transform that if every
function can be written as a mix of sine and cosine, that if you have
4th dimension or higher, does that contradict the Fourier
Transform Theory, because in one of those dimensions, this Fourier
Transform
breaks apart.
It breaks apart not because the higher dimension is valid, but because
there are no higher dimensions than 3rd.
Now, another aspect of the Fourier Transform Theory is the light wave
which is a transverse wave of its electrical component and its magnetic
component perpendicular. Now if there exists a 4th dimension or higher,
is light able to travel in those dimensions? I would say not. Not
because of anything wrong with light, but rather the fault is that 4th
dimension and higher are nonexistent. This is a puzzling path and have
to think more on this.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

User: "Proginoskes"
Title: Re: revamping the teaching of the Calculus	17 Jan 2006 04:28:49 PM

wrote:

Now let us play with the Fourier Transform Theory for just a little
bit. If it says what I think it says, that every function can be
written as a sine and cosine mix, [...]

User: ""
Title: Re: revamping the teaching of the Calculus	18 Jan 2006 01:12:39 AM

wrote:

Now let us play with the Fourier Transform Theory for just a little
bit. If it says what I think it says, that every function can be
written as a sine and cosine mix, [...]

That's not what it says. For one thing, the function you're
representing has to be periodic. Second, if the function is not
continuous at x=a, the Fourier series might not converge to f(a).
And it's not called "Fourier Transform Theory", just a Fourier
transform.
http://en.wikipedia.org/wiki/Fourier_transform
--- Christopher Heckman
When we consider the Universe at large it is continuous, at least it is
a good assumption. As to whether the Universe is periodic would have to
be defined better. So the Universe at large obeys the Fourier Transform
theory. So we can replace sine with energy and cosine with force,
likewise replace the photon electric component with sine and the
magnetic component with cosine of its transverse wave nature. So this 2
dimensional requirement is filled in a 3 dimensional space, and thus
never a need for 4th dimension. So the Fourier Transform is a
supporting evidence that 4th dimension does not exist.
If 4th dimension exists then there would have to be a Fourier Transform
theory where 3 items, not just sine and cosine are necessary.
I am going to take a week break from Internet posting in order to write
up a Darwin Evolution paper. I shall be back by 25 January. See you
then.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

User: "Proginoskes"
Title: The Universe is 2-dimensional; was: Re: revamping the teaching of the Calculus	18 Jan 2006 08:32:10 PM

wrote:

[...]
When we consider the Universe at large it is continuous, [...]

You're mixing definitions of "continuous" here. In order to use the
_mathematical_ definition, you need to know that the Universe is a
function, its domain is the real numbers, and its codomain is the real
numbers.
What I think you meant to say is that the Universe is homogeneous.

If 4th dimension exists then there would have to be a Fourier Transform
theory where 3 items, not just sine and cosine are necessary.

Well, you don't really need sine and cosine. You can get away with e^(i
n x), just one family of functions, as opposed to two.
By your logic you've proven THE UNIVERSE IS NOT 3-DIMENSIONAL, BUT HAS
ONLY TWO DIMENSIONS!!!!! You're a shoe-in for the Nobel Prize this
year.

I am going to take a week break from Internet posting in order to write
up a Darwin Evolution paper. I shall be back by 25 January. See you
then.

I can't imagine what you would say about Darwin's theory. Maybe I'll
check out your website.
--- Christopher Heckman
.

User: "Proginoskes"
Title: Re: The Universe is 2-dimensional; was: Re: revamping the teaching of the Calculus	18 Jan 2006 08:37:19 PM

Proginoskes wrote:

a_plutonium@hotmail.com wrote:

[...]
When we consider the Universe at large it is continuous, [...]

You're mixing definitions of "continuous" here. In order to use the
_mathematical_ definition, you need to know that the Universe is a
function, its domain is the real numbers, and its codomain is the real
numbers.

What I think you meant to say is that the Universe is homogeneous.

If 4th dimension exists then there would have to be a Fourier Transform
theory where 3 items, not just sine and cosine are necessary.

Well, you don't really need sine and cosine. You can get away with e^(i n x),
just one family of functions, as opposed to two.

Where n is an integer, not just a natural number, of course.
--- Christopher Heckman

By your logic you've proven THE UNIVERSE IS NOT 3-DIMENSIONAL, BUT HAS
ONLY TWO DIMENSIONS!!!!! You're a shoe-in for the Nobel Prize this
year.

I am going to take a week break from Internet posting in order to write
up a Darwin Evolution paper. I shall be back by 25 January. See you
then.

I can't imagine what you would say about Darwin's theory. Maybe I'll
check out your website.

User: ""
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	13 Jan 2006 03:09:15 AM

correction: energy of course is kg * m^2/s^2, and the difference
between force and energy is the squaring of length in numerator. I
should proofread before sending sometimes.
.

User: "Proginoskes"
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	14 Jan 2006 02:25:08 AM

wrote:

correction: energy of course is kg * m^2/s^2, and the difference
between force and energy is the squaring of length in numerator. I
should proofread before sending sometimes.

You should proofread before sending ALWAYS.
--- Christopher Heckman
.

User: "Hero"
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	12 Jan 2006 01:59:09 PM

wrote:

.........

So we seem to see Calculus falling apart in 3rd dimension. So why is
Calculus so weak as an enterprise that it is not able to generalize
into the 3rd dimension, and we only piecewise compute integration and
differentiation in 3rd dimension?

.......

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

You reasoning sounds good ( for a start). Differ between geometry
(without time) and dynamical geometry 3D-Space + time.
Start with studying Archimedes "The method".
Take Your time.
I'm eager to read the results of Your thinking about this.
Hero
.

User: "Proginoskes"
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	13 Jan 2006 01:49:17 AM

Hero wrote:

[to Archimedes Plutonium]
Take Your time. [...]

You don't know him too well, do you?
--- Christopher Heckman
.

User: "Proginoskes"
Title: Re: revamping the teaching of the Calculus by using a geometrical explanation; Chord concept	13 Jan 2006 02:00:17 AM

Hero wrote:

[to Archimedes Plutonium]
Take Your time. [...]

You don't know him too well, do you?
--- Christopher Heckman
.