Wormhole theory

[Rich Grise]

Oh yeah, I was astounded too, when I realised calculus can be replaced
with algebra. And wondered why nobody ever mentioned it to me in that way.

Yeah! All you have to do is make delta infinitesimally small, and you're
pretty much there. Although you do have to introduce limits, cf. Zeno's
paradox.

Cheers!
Rich

 

[Robert]

But I need to dig into non-standard analysis and see. I'll get the
book you mention above and also some of the newer textbooks teaching
1st year calculus using non-standard analysis. But I doubt that
mathematicians would write these textbooks unless they had satisfied
themselves that it was a solid idea. Time will tell me, though.

Thanks, again.

Jon

Don't get me wrong. The book I mention has the details of why they had to
get away from the infinitesimal concept in math. Non-standard Analysis isn't
dealt with.

Robert

 

[Jonathan Kirwan]

Don't get me wrong. The book I mention has the details of why they had to
get away from the infinitesimal concept in math. Non-standard Analysis isn't
dealt with.

Understood. I'll start with non-standard analysis textbooks then and
see what that looks like, first.

Jon

 

[John Woodgate]

I read in sci.electronics.design that Terry Given <[email protected]>

Oh yeah, I was astounded too, when I realised calculus can be replaced
with algebra. And wondered why nobody ever mentioned it to me in that
way.

Just algebra by itself doesn't usually get you the answer. You need one
or two transforms as well. They are not easy to comprehend.

 

[John Woodgate]

You could use the infinitesimal to derive nonsense.

I can use ANY mathematical concept to derive nonsense. (;-)

 

[John Woodgate]

I read in sci.electronics.design that Robert Monsen

Now, on a separate but related subject, I wonder why there should be a
difference between inertial paths and accelerated paths.

I expect Kevin can explain this better, but it's not ANY acceleration
that matters.

What is an 'accelerated path' anyway? If I'm orbiting the earth, I
don't feel any acceleration, but I'm apparently aging a bit more
slowly.

In free orbit, you are an inertial observer - the water in your bucket
doesn't slosh about. The twin who travels in the spaceship and arrives
back younger than his stay-at-home twin cannot be an inertial observer.
As he accelerates and decelerates, the water in his bucket does slosh
about.

Your math pages link points out the idea that two orbits, one circular,
and one eliptical, will experience different 'proper time'. Neither can
tell they are accelerated unless they look out the window. Thus, by
comparing notes when they cross paths, they could violate the principle
of relativity, and figure out how fast they were going in relation to
one another.

That doesn't violate SR (or GR). Try telling a cop he can't tell you
were doing 65 MPH relative to his radar because that would violate SR!
(;-) Perhaps you wrote what you didn't mean.

 

[Frank Bemelman]

I can use ANY mathematical concept to derive nonsense. (;-)

....and getting better at it, every day

 

[Kevin Aylward]

I read in sci.electronics.design that Jonathan Kirwan
<[email protected]>) about 'Wormhole theory',


That concept is a lot older than the 1960s. You can see it in
l'Hopital's method for evaluating functions that have a limit of the
form 0/0 for some value of the independent variable.

Incidentally, I learned differential calculus the 'old' way. I'm no
mathematician, either, but I didn't find it difficult. I had a *good*
teacher. That makes a lot of difference, maybe all the difference.

f'(x) = lim h->0 f(x+h)-f(x)/h
g'(x) = lim h->0 g(x+h)-g(x)/h

Therefore:

f'(x)/g'(x) = lim h->0 (f(x+h)-f(x))/(g(x+h)-g(x))

if f(x)=g(x)=0 then:

f'(x)/g'(x) = lim h->0 f(x+h)/g(x+h)

Letting h->0, then:

f'(x)/g'(x) = f(x)/g(x)

for the condition f(x)=g(x)=0.


I pat myself on the back, as usual, for this proof as it was one that I
invented on my todd. I have yet to see it in any book, although I have
been informed after the fact that this approach is known about. I
impressed my math proff no end with this one as it was new to him as
well. He actually advised me to give up engineering and switch to math
instead because of this. Seriously. All the usually proofs of
l'Hopital's rule involve the theorem of the mean approach.

And of course, I can never resist pointing out that l'Hopital had
nothing to do with the rule. He was a Toff that paid someone to name the
rule after him.

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[John Woodgate]

I read in sci.electronics.design that Kevin Aylward

f'(x) = lim h->0 f(x+h)-f(x)/h
g'(x) = lim h->0 g(x+h)-g(x)/h

Therefore:

f'(x)/g'(x) = lim h->0 (f(x+h)-f(x))/(g(x+h)-g(x))

if f(x)=g(x)=0 then:

f'(x)/g'(x) = lim h->0 f(x+h)/g(x+h)

Letting h->0, then:

f'(x)/g'(x) = f(x)/g(x)

for the condition f(x)=g(x)=0.


I pat myself on the back, as usual, for this proof as it was one that I
invented on my todd. I have yet to see it in any book, although I have
been informed after the fact that this approach is known about.

It's the justification of de l'Hopital's method that I was taught. It
isn't rigorous, which is why it's called 'justification', not 'proof'.

A justification based on Taylor series expansions is in Lowry H V and H
A Hayden, 'Advanced [= 'not advanced' in mathematician-speak]
Mathematics for technical students - Part 2',pp. 1-2, Longmans, Green
and Co (London, 1955) (No ISBN)

I impressed my math proff no end with this one as it was new to him as
well. He actually advised me to give up engineering and switch to math
instead because of this. Seriously. All the usually proofs of
l'Hopital's rule involve the theorem of the mean approach.

And of course, I can never resist pointing out that l'Hopital had
nothing to do with the rule. He was a Toff that paid someone to name
the rule after him.

That was by no means unusual at the time, and I think it still goes on
in France. However, this site:

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/De_L'Hopital.h
tml

indicates that you do him an injustice:

Guillaume De l'Hôpital served as a cavalry officer but resigned because
of nearsightedness. From that time on he directed his attention to
mathematics. L'Hôpital was taught calculus by Johann Bernoulli from the
end of 1691 to July 1692.

L'Hôpital was a very competent mathematician and solved the
brachystochrone problem. The fact that this problem was solved
independently by Newton, Leibniz and Jacob Bernoulli puts l'Hôpital in
very good company.

L'Hôpital's fame is based on his book Analyse des infiniment petits pour
l'intelligence des lignes courbes (1696) which was the first text-book
to be written on the differential calculus. In the introduction
L'Hôpital acknowledges his indebtedness to Leibniz, Jacob Bernoulli and
Johann Bernoulli but L'Hôpital regarded the foundations provided by him
as his own ideas.

In this book is found the rule, now known as L'Hôpital's rule, for
finding the limit of a rational function whose numerator and denominator
tend to zero at a point.

 

[John Woodgate]

I read in sci.electronics.design that Frank Bemelman

...and getting better at it, every day

Actually, my limited ability comes and goes. If it's in the 'gone'
phase, and I have an important problem to solve, it takes about 3 days
to come back.

 

[Kevin Aylward]

I read in sci.electronics.design that Kevin Aylward

f'(x) = lim h->0 f(x+h)-f(x)/h
g'(x) = lim h->0 g(x+h)-g(x)/h

Therefore:

f'(x)/g'(x) = lim h->0 (f(x+h)-f(x))/(g(x+h)-g(x))

if f(x)=g(x)=0 then:

f'(x)/g'(x) = lim h->0 f(x+h)/g(x+h)

Letting h->0, then:

f'(x)/g'(x) = f(x)/g(x)

for the condition f(x)=g(x)=0.


I pat myself on the back, as usual, for this proof as it was one
that I invented on my todd. I have yet to see it in any book,
although I have been informed after the fact that this approach is
known about.

It's the justification of de l'Hopital's method that I was taught. It
isn't rigorous, which is why it's called 'justification', not 'proof'.

A justification based on Taylor series expansions is in Lowry H V and
H A Hayden, 'Advanced [= 'not advanced' in mathematician-speak]
Mathematics for technical students - Part 2',pp. 1-2, Longmans, Green
and Co (London, 1955) (No ISBN)

I have seen this approach, but its not as neat as mine is it

Each step can be justified fully rigorously with theorems on limits, or
so my proff told me when he examined it.

That was by no means unusual at the time, and I think it still goes on
in France. However, this site:

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/De_L'Hopital.h
tml

indicates that you do him an injustice:

Maybe


In this book is found the rule, now known as L'Hôpital's rule, for
finding the limit of a rational function whose numerator and
denominator tend to zero at a point.

Interesting that this last quote neatly side steps the issue of where
the proof came from. I did a search and found this:

http://concise.britannica.com/ebc/article?tocId=9370139

"It states that when the limit of f(x)/g(x) is indeterminate, under
certain conditions it can be obtained by evaluating the limit of the
quotient of the derivatives of f and g (i.e., f¢(x)/g¢(x)). If this
result is indeterminate, the procedure can be repeated. It is named for
the French mathematician Guillaume de L'Hôpital (1661-1704), who
purchased the formula from his teacher the Swiss mathematician Johann
Bernoulli (1667-1748)."

So, there you go.

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[Kevin Aylward]

I read in sci.electronics.design that Kevin Aylward

I found this as well

http://homepages.compuserve.de/thweidenfeller/mathematiker/bernoullijo.html

"The well known de l'Hôpital's rule is contained in this calculus book
and it is therefore a result of Johann Bernoulli. In fact proof that the
work was due to Bernoulli was not obtained until 1922 when a copy of
Johann Bernoulli's course made by his nephew Nicolaus(I) Bernoulli was
found in Basel. Bernoulli's course is virtually identical with de
l'Hôpital's book but it is worth pointing out that de l'Hôpital had
corrected a number of errors such as Bernoulli's mistaken belief that
the integral of 1/x is finite. After de l'Hôpital's death in 1704
Bernoulli protested strongly that he was the author of de l'Hôpital's
calculus book. It appears that the handsome payment de l'Hôpital made to
Bernoulli carried with it conditions which prevented him speaking out
earlier. However, few believed Johann Bernoulli until the proofs
discovered in 1922. "

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[John Woodgate]

I read in sci.electronics.design that Kevin Aylward
<[email protected]>) about 'Wormhole theory',

I found this as well

http://homepages.compuserve.de/thweidenfeller/mathematiker/bernoullijo.h
tml

I am concerned that this facility for finding obscure information on the
web may indicate that you are turning into 'Robert' (aka FB?). (;-)

 

[Rich The Newsgropup Wacko]

f'(x)/g'(x) = f(x)/g(x)

I pat myself on the back, as usual,

Imagine my surprise. You and what's-his-name ought to get along famously.

for this proof as it was one that I
invented on my todd.

What does "on my todd" mean? Is it anything like "on the head of
a pin?" I wanted to make a joke about whacking on my part, but "todd"
is just too obscure. >:->

Thanks!
Rich

 

[Rich Grise]

f'(x) = lim h->0 f(x+h)-f(x)/h
g'(x) = lim h->0 g(x+h)-g(x)/h

Therefore:
f'(x)/g'(x) = lim h->0 (f(x+h)-f(x))/(g(x+h)-g(x))
if f(x)=g(x)=0 then:
f'(x)/g'(x) = lim h->0 f(x+h)/g(x+h)
Letting h->0, then:
f'(x)/g'(x) = f(x)/g(x)
for the condition f(x)=g(x)=0.

I pat myself on the back, as usual, for this proof as it was one that I
invented on my todd. I have yet to see it in any book, although I have

Are you sure it wasn't from your arithmetic teacher? I ask, because my
arithmetic teacher was the one that showed me something very much like
this about a half a century ago.

But I thank you for presenting it, because I was thinking this very thing
as a way to introduce ~~SciGirl~~ to calculus, gently - I just didn't
remember the exact formulae.

Thanks!
Rich

 

[John Woodgate]

I read in sci.electronics.design that Rich The Newsgropup Wacko
<[email protected]>) about 'Wormhole theory',

What does "on my todd" mean? Is it anything like "on the head of a
pin?" I wanted to make a joke about whacking on my part, but "todd" is
just too obscure. >:->

'on one's own'. Rhyming slang, from 'Tod Sloane', a US horse-jockey.
http://www.worldwidewords.org/qa/qa-tod1.htm

 

[Kevin Aylward]

Are you sure it wasn't from your arithmetic teacher? I ask, because my
arithmetic teacher was the one that showed me something very much like
this about a half a century ago.

But I thank you for presenting it, because I was thinking this very
thing as a way to introduce ~~SciGirl~~ to calculus, gently - I just
didn't remember the exact formulae.

That's because you didn't invent it!

As soon as I read Johns post, I typed it of the cuff. One don't forget
ones rare moments of er.. genius...

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[~~SciGirl~~]

I saw this very demonstration in animation on teevee about fifty
years ago. It was a Disney/Bell Labs thing, I think. And it didn't
go "click", it went "boink." ;-) The one time I did get to go to
Disneyland, Tomorrowland was closed for repairs or some such. )-;
(in retrospect, given the timeframe, they were probably transforming
it to punkland or gothland or something.)

Thanks!
Rich

Hmm... gothland. Sounds creepy.

What does "the particle can't think" have to do with the uncertainty
principle? If the uncertainty principle wasn't true, photons still wouldn't
be able to think

 

[John Woodgate]

I read in sci.electronics.design that ~~SciGirl~~ <[email protected]>

What does "the particle can't think" have to do with the uncertainty
principle? If the uncertainty principle wasn't true, photons still
wouldn't be able to think

Besides, we only have one experiment that appears to show that they
can't the double-slit experiment, they can't think which slot to go
through, so they all go through both.

 

[Rich Grise]

Rich Grise wrote: ....

That's because you didn't invent it!

As soon as I read Johns post, I typed it of the cuff. One don't forget
ones rare moments of er.. genius...

My moments of genius always come the next day. Once, at a job interview,
they gave applicants a test. One of the questions was to write a program
that prints out the primes up to 1000. I screwed it up - probably because
I don't work well under pressure. It's hard to write spaghetti code in C,
but that's about what I accomplished. I didn't get the job. The next
day, I wrote a Sieve of Eratosthenes in three lines on the back of an
envelope at the bus stop.

Cheers!
Rich