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From: Chip Eastham on 15 Jan 2010 20:02
On Jan 14, 9:15 am, "Robert H. Lewis" <rle...(a)fordham.edu> wrote:
> ...
>
>
>
> > Above, I'm working on getting a handle on 'what is
> > the problem'.
>
> > > No, it is not clear by any definition of
> > > derivative.
>
> > I'll need to clarify, that we're a bit deeper than
> > derivatives, specifically 'differential co-efficents'
> > then a bit beyond that, a respected mathematician
> > Herman Weyl was fond of.
>
> > > The casual treatment of the
> > > derivative dX/dT as a ratio of two values
> > > mixed with taking a square root in the
> > > numerator and denominator is lacking in
> > > logical justification.
>
> > Good enough!
> > Allow me to move to integration ($) and then pose,
>
> > dU = dV*dV , dV = sqrt(dU)
>
> > and then moving to integration, this is expected to
> > be a logical question, can we solve,
>
> > $ dU = $$ dV dV  ,   THE 2nd STEP.
> > .....
>
>   dU = dV * dV  makes no sense.  
>
>   Let me clarify: in the context of calculus, working with limits and defining derivatives, one learns to manipulate dx, dy, etc in certain ways.  The manipulations are valid because of the underlaying conceptual mathematics.   We call it "working with differentials."  Students often want to ignore the real, underlying mathematics because they have learned through their "high school"  "education" to concentrate on what will be on the test and ignore everything else.  Thus, they sometimes write things like you have written above, or, say, sin(x)/dx, a dx in the denominator.  It makes no sense.  They have taken the formal manipulations to a meaningless place.
>
>   I suspect that is what is going on in the OP.
>
>   Once in a great while, something weird turns out to have a valid meaning in a larger context, suitably redefined.  For example, generalized functions, in which the Dirac delta function makes sense, even though at first it seems to be nonsense.
>
> Robert H. Lewis
> Fordham University

Actually dV*dV makes sense, it's just zero.

There is a "calculus" of infinitesimals, and
in this setting multiplication is anticommutative.

Therefore any infinitesimal times itself must
be zero. This is related to the fact swapping
two rows (or columns) of a matrix changes the
sign.

regards, chip
From: Frederick Williams on 15 Jan 2010 20:55
"Ken S. Tucker" wrote:
>
> [...]

Work an example for us. What's

the integral from 0 to 1 of x sqrt(dx)

?

--
Pigeons were widely suspected of secret intercourse with the
enemy; counter-measures included the use of British birds of
prey to intercept suspicious pigeons in mid-air.
Christopher Andrew, 'Defence of the Realm', Allen Lane
From: Ken S. Tucker on 16 Jan 2010 10:15
Dr. Lewis, you're respond as an expert, granted,
I'm a serious student.

On Jan 14, 6:15 am, "Robert H. Lewis" <rle...(a)fordham.edu> wrote:
> ...
>
>
>
> > Above, I'm working on getting a handle on 'what is
> > the problem'.
>
> > > No, it is not clear by any definition of
> > > derivative.
>
> > I'll need to clarify, that we're a bit deeper than
> > derivatives, specifically 'differential co-efficents'
> > then a bit beyond that, a respected mathematician
> > Herman Weyl was fond of.
>
> > > The casual treatment of the
> > > derivative dX/dT as a ratio of two values
> > > mixed with taking a square root in the
> > > numerator and denominator is lacking in
> > > logical justification.
>
> > Good enough!
> > Allow me to move to integration ($) and then pose,
>
> > dU = dV*dV , dV = sqrt(dU)
>
> > and then moving to integration, this is expected to
> > be a logical question, can we solve,
>
> > $ dU = $$ dV dV , THE 2nd STEP.
> > .....
>
> dU = dV * dV makes no sense.
>
> Let me clarify: in the context of calculus, working with limits and defining derivatives, one learns to manipulate dx, dy, etc in certain ways. The manipulations are valid because of the underlaying conceptual mathematics. We call it "working with differentials." Students often want to ignore the real, underlying mathematics because they have learned through their "high school" "education" to concentrate on what will be on the test and ignore everything else. Thus, they sometimes write things like you have written above, or, say, sin(x)/dx, a dx in the denominator. It makes no sense.. They have taken the formal manipulations to a meaningless place.
>
> I suspect that is what is going on in the OP.
>
> Once in a great while, something weird turns out to have a valid meaning in a larger context, suitably redefined. For example, generalized functions, in which the Dirac delta function makes sense, even though at first it seems to be nonsense.
>
> Robert H. Lewis
> Fordham University

Following Robert's "conceptual mathematics",
Pardon me while I do a deviation, I use 3 vectors notated X> Y> Z>,
(standard Cartesian) and find a vector product,

dZ> = dX> x dY> , with a caveat, explain now...

The implied concept is a differential of an infinitesmal Area .
What I choose to demo, is a single differential "dZ>" enabled
by a differential product "dX> x dY>", with the dZ> proportional
to 'differential variations' of X> and Y>.
By arbituary transformation (Poincare I think it's called) we do
the calculation at the origin, so X> and Y> are zero, and then
we can deal just with the differentials.
If that's ok, we can move to integrate dZ> from dX> x dY>.
Regards
Ken S. Tucker
From: Ken S. Tucker on 16 Jan 2010 21:40
On Jan 16, 7:15 am, "Ken S. Tucker" <dynam...(a)vianet.on.ca> wrote:
> Dr. Lewis, you're respond as an expert, granted,
> I'm a serious student.
>
> On Jan 14, 6:15 am, "Robert H. Lewis" <rle...(a)fordham.edu> wrote:
>
>
>
> > ...
>
> > > Above, I'm working on getting a handle on 'what is
> > > the problem'.
>
> > > > No, it is not clear by any definition of
> > > > derivative.
>
> > > I'll need to clarify, that we're a bit deeper than
> > > derivatives, specifically 'differential co-efficents'
> > > then a bit beyond that, a respected mathematician
> > > Herman Weyl was fond of.
>
> > > > The casual treatment of the
> > > > derivative dX/dT as a ratio of two values
> > > > mixed with taking a square root in the
> > > > numerator and denominator is lacking in
> > > > logical justification.
>
> > > Good enough!
> > > Allow me to move to integration ($) and then pose,
>
> > > dU = dV*dV , dV = sqrt(dU)
>
> > > and then moving to integration, this is expected to
> > > be a logical question, can we solve,
>
> > > $ dU = $$ dV dV , THE 2nd STEP.
> > > .....
>
> > dU = dV * dV makes no sense.
>
> > Let me clarify: in the context of calculus, working with limits and defining derivatives, one learns to manipulate dx, dy, etc in certain ways. The manipulations are valid because of the underlaying conceptual mathematics. We call it "working with differentials." Students often want to ignore the real, underlying mathematics because they have learned through their "high school" "education" to concentrate on what will be on the test and ignore everything else. Thus, they sometimes write things like you have written above, or, say, sin(x)/dx, a dx in the denominator. It makes no sense. They have taken the formal manipulations to a meaningless place.
>
> > I suspect that is what is going on in the OP.
>
> > Once in a great while, something weird turns out to have a valid meaning in a larger context, suitably redefined. For example, generalized functions, in which the Dirac delta function makes sense, even though at first it seems to be nonsense.
>
> > Robert H. Lewis
> > Fordham University
>
> Following Robert's "conceptual mathematics",
> Pardon me while I do a deviation, I use 3 vectors notated X> Y> Z>,
> (standard Cartesian) and find a vector product,
>
> dZ> = dX> x dY> , with a caveat, explain now...
>
> The implied concept is a differential of an infinitesmal Area .
> What I choose to demo, is a single differential "dZ>" enabled
> by a differential product "dX> x dY>", with the dZ> proportional
> to 'differential variations' of X> and Y>.
> By arbituary transformation (Poincare I think it's called) we do
> the calculation at the origin, so X> and Y> are zero, and then
> we can deal just with the differentials.
> If that's ok, we can move to integrate dZ> from dX> x dY>.
> Regards
> Ken S. Tucker

I think the integral operator "$" is reasonably applied this way,

$ dZ> = Z> = $ dX> x $ dY> = X> x Y> to get,

Z> = X> x Y>

In an other form of logic, is the $ applied to the equation,

$(dZ> = dX> x dY>).

I reason the above ok, however, can we extend that to the
scalar (dot) product such as dU = dV> . dV> ?
Regards
Ken S. Tucker



From: Ostap S. B. M. Bender Jr. on 17 Jan 2010 02:18
On Jan 13, 9:45 am, "Ken S. Tucker" <dynam...(a)vianet.on.ca> wrote:
> To Dr. Lewis et al,
>
> On Jan 13, 8:12 am, "Robert H. Lewis" <rle...(a)fordham.edu> wrote:
>
>
>
> > > > >${-oo,+oo} F(x) (dx)^.5   (1)
>
> > > > Where did you run across this? The notation makes very little
> > > > sense.
>

Maybe they meant

${-oo,+oo} F(x) d(x^.5)

?

>
> > > Jays doing just fine, David I think you need to take a basic
> > > calculus course, let me explain (using Jay's example), that I
> > > frequently encounter.
>
> > > V = dX/dT = Velocity.
> > > sqrt(V) = sqrt (dX/dT) , sqrt(dX) = sqrt(V)*sqrt(dT).
> > > That's SOP in calculus, from that a bit of algebraic massage
> > > produces *generally* a means to solve.
> > > Regards
> > > Ken S. Tucker
>
> >   SOP in Calculus?  Elementary course?   No way!
>
> >  I have been teaching calculus at all levels for close to 30 years, and I have never seen \sqrt(dx).  It is absurd on the face of it.  Perhaps some physicists have come up with this and found a coherent usage?
>
> Dr. Lewis, I too have taught and refined Calculus
>

Where would gentlemen, who claim to have 'refined calculus', teach?

>
> and very much
> respect the reasoning and notation conventions.
> I think it's reasonable to examine the 'differential coefficents',
>

What's a 'differential coefficent'?

You view the expression 'dX/dT' as a ratio of a number called 'dX' by
another number called 'dT'? LOL.

>
> apart from the ratio, let's start here,
>
> V=dX/dT, sqrt(V)= sqrt(dX/dT)
>
> that's an expression of V by differential coefficients,
>

No, it is an expression of sqrt(V) by differential coefficients,
whatever that means.

>
> The apparent confusion results from writing the above as,
>
> sqrt(V) = sqrt(dX)/sqrt(dT)   , THE STEP.
>

Why is sqrt(dX/dT) = sqrt(dX)/sqrt(dT)?

Wishful thinking?


>
> Here we ask, does THE STEP represent a clear definition
> of symbolic logic?
>

It's symbolic something, alright. But has nothing to do with human
logic.

>
> > Robert H. Lewis
> > Mathematics Department
> > Fordham University
>
> Regards
> Ken S. Tucker

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