text:
Hamilton: two mysteries solved

23 views

John Baez

Nov 4, 2005, 1:13:13 PM11/4/05
to
Thanks to Dirk Schlimm, I've finally found an online map showing
the bridge where Hamilton carved his famous formula defining the
quaternions:

You can see where Broombridge Road crosses the Royal Canal -
that's the place. For pictures and bus directions, try this:

http://math.ucr.edu/home/baez/dublin/index.html#brougham_bridge

And speaking of Hamilton, Theron Stanford has sent me an answer to
one of life's persistent questions:

why is momentum denoted by the letter p?

Stanford writes:

While Googling various subjects, I came across the following from
your Quantum Gravity Seminar notes from 2001:

Again Oz was overcome with curiosity, so mimicking Toby's voice,
he asked, "Why do we call the momentum p?"

The Wiz glared at Toby. "Because m is already taken -- it stands
for mass! Seriously, I don't know why people call position q and
momentum p. All I know is that if you use any other letters,
people can tell you're not a physicist. So I urge you to follow

Well, I have an answer. Hamilton, the first physicist to actually
understand the importance of the concept of momentum, chose pi to
stand for momentum (actually, it's not the usual pi, but what TeX
calls varpi, a lower-case omega with a top, kinda like the top
of a lower-case tau). Jacobi changed this to p in one of his
seminal papers on the subject; he also used q in the same paper
to stand for position. In the 1800s (I want to say 1850s, though
it might have been a decade or two later) Cayley presented a paper
to the Royal Academy in which he says (and I paraphrase), "Well,
it seems that p and q are pretty well established now, so that's
what I'm going to use."

So, now we need to hold a seance to see why Hamilton chose the letter
"varpi" for momentum. In fact, we should have done this on Hamilton's
200th birthday, earlier this year!

Lee Rudolph

Nov 5, 2005, 6:35:52 AM11/5/05
to
ba...@galaxy.ucr.edu (John Baez) writes:

...

>And speaking of Hamilton, Theron Stanford has sent me an answer to
>one of life's persistent questions:
>
> why is momentum denoted by the letter p?
>
>Stanford writes:

...

> Well, I have an answer. Hamilton, the first physicist to actually
> understand the importance of the concept of momentum, chose pi to
> stand for momentum (actually, it's not the usual pi, but what TeX
> calls varpi, a lower-case omega with a top, kinda like the top
> of a lower-case tau).

...

>So, now we need to hold a seance to see why Hamilton chose the letter
>"varpi" for momentum. In fact, we should have done this on Hamilton's
>200th birthday, earlier this year!

Be careful. Unless there is evidence (not cited above) that Hamilton
intended the glyph that TeX calls "varpi" to be, in fact, a variant
of pi, it remains possible--and I think likely--that he intended that
glyph to be what a Russian Orthodox priest of my acquaintance (at a
time long before he was a priest; actually, at a time when he, I,
high school together) told me was called "ot": a ligature of "tau"
and "omega" (actually, a superposition of tau and omega, but apparently
that counts as a ligature in paleography if not in typography).
So: is there any Greek, English, Latin, or Gaelic word or phrase
that might plausibly suggest--or be suggested by--the word or concept
of momentum, and which could be abbreviated by "o" followed or preceded
by (or superposed with) "t"? (Same question for "p", of course.)

Lee Rudolph

John Baez

Nov 5, 2005, 11:20:19 AM11/5/05
to
In article <dki5eo$mdv$1...@panix2.panix.com>,
Lee Rudolph <lrud...@panix.com> wrote:

>>So, now we need to hold a seance to see why Hamilton chose the letter
>>"varpi" for momentum. In fact, we should have done this on Hamilton's
>>200th birthday, earlier this year!

>Be careful. Unless there is evidence (not cited above) that Hamilton
>intended the glyph that TeX calls "varpi" to be, in fact, a variant
>of pi, it remains possible--and I think likely--that he intended that
>glyph to be what a Russian Orthodox priest of my acquaintance (at a
>time long before he was a priest; actually, at a time when he, I,
>high school together) told me was called "ot": a ligature of "tau"
>and "omega" (actually, a superposition of tau and omega, but apparently
>that counts as a ligature in paleography if not in typography).

That's possible; unlike the LaTeX "varphi", which is an alternative
version the Greek phi, I'm not sure the LaTeX "varpi" is really
supposed to be a variant of "pi".

Let's look around....

There are a number of nonstandard Greek letters, of which
Wikipedia lists four: digamma, san, qoppa and sampi:

http://en.wikipedia.org/wiki/Greek_alphabet

The sampi is the only one here with any resemblance to the "varpi"...

But - aha! - "varpi" is called "pomega" here:

http://www.ams.org/STIX/stixfullr/stixfull-06.html

It plays a role in astronomy, where it stands for the "longitude of
the pericenter" (whatever the heck *that* is):

http://scienceworld.wolfram.com/physics/Pomega.html

I don't know how "pomega" is related to "ot".

>So: is there any Greek, English, Latin, or Gaelic word or phrase
>that might plausibly suggest--or be suggested by--the word or concept
>of momentum, and which could be abbreviated by "o" followed or preceded
>by (or superposed with) "t"?

Momentum says where a particle is going TO! :-)

PiSigma

Nov 5, 2005, 3:44:55 PM11/5/05
to

"John Baez" <ba...@galaxy.ucr.edu> wrote in message
news:dkim43$4sa$1...@glue.ucr.edu...

> In article <dki5eo$mdv$1...@panix2.panix.com>,
> Lee Rudolph <lrud...@panix.com> wrote:
>
> >>So, now we need to hold a seance to see why Hamilton chose the letter
> >>"varpi" for momentum. In fact, we should have done this on Hamilton's
> >>200th birthday, earlier this year!
>
> >Be careful. Unless there is evidence (not cited above) that Hamilton
> >intended the glyph that TeX calls "varpi" to be, in fact, a variant
> >of pi, it remains possible--and I think likely--that he intended that
> >glyph to be what a Russian Orthodox priest of my acquaintance (at a
> >time long before he was a priest; actually, at a time when he, I,
> >high school together) told me was called "ot": a ligature of "tau"
> >and "omega" (actually, a superposition of tau and omega, but apparently
> >that counts as a ligature in paleography if not in typography).
>
> That's possible; unlike the LaTeX "varphi", which is an alternative
> version the Greek phi, I'm not sure the LaTeX "varpi" is really
> supposed to be a variant of "pi".

Actually, it's a variant of "pi": see
http://omega.enstb.org/yannis/pdf/boston99.pdf

Lee Rudolph

Nov 5, 2005, 7:13:37 PM11/5/05
to
"PiSigma" <psi...@tee.gr> writes:

...
>> Lee Rudolph <lrud...@panix.com> wrote:
...

>> >Be careful. Unless there is evidence (not cited above) that Hamilton
>> >intended the glyph that TeX calls "varpi" to be, in fact, a variant
>> >of pi, it remains possible--and I think likely--that he intended that
>> >glyph to be what a Russian Orthodox priest of my acquaintance (at a
>> >time long before he was a priest; actually, at a time when he, I,
>> >high school together) told me was called "ot": a ligature of "tau"
>> >and "omega" (actually, a superposition of tau and omega, but apparently
>> >that counts as a ligature in paleography if not in typography).
>>
>> That's possible; unlike the LaTeX "varphi", which is an alternative
>> version the Greek phi, I'm not sure the LaTeX "varpi" is really
>> supposed to be a variant of "pi".
>
>Actually, it's a variant of "pi": see
>http://omega.enstb.org/yannis/pdf/boston99.pdf

That the glyph in question can be (and is) used as a variant
of pi does not rule out that it can be (and has been) used as
a ligature of omega and tau. And, in fact, as my friend told
me and Google has now verified, that is the case: "ot" is the
name (obviously, transcribed into Latin letters...) of just
such a ligature in the alphabet used for Church Slavonic.

The Orthodox Church Slavonic alphabet is shown in [4, 5].
There are the following differences between Church Slavonic
and Old Slavonic (ustav) writings:
...
* The order of the alphabet is changed, some letters changed
their names, some letters changed their shape. Some letters
became obsolete and are excluded from the alphabet, but as
a compensation the new letter _ot_ appears which is the
ligature between O (Greek omega) and T with three stems.
(In Old Slavonic the name _ot_ was reserved for Greek
\omega which is called _omega_ in Church Slavonic.)

(From http://www.uni-giessen.de/partosch/eurotex99/berdnikov2.pdf .
Unfortunately the scanned images of glyphs from actual Church
Slavonic documents, referenced above as "[4, 5]", were at URLs
that now seem to be inoperative; and the hacked-together glyph
included in a table at the end of that PDF document--which looks
like it wasn't even produced with Metafont, but is simply one standard
TeX glyph overstruck on another--is a very bad representation of
_ot_.)

The challenge is now to show that Hamilton was acquainted with
Church Slavonic. That isn't (perhaps) quite as absurd as it sounds;
he was already a polyglot at a very young age (according to the
biography, title long forgotten, from which I extracted the
information in ll. 40 _et seq._ of "Frobenius: A Sesquilogue",
http://black.clarku.edu/~lrudolph/poetry/frobenius/index.html,
to wit, that he knew Hebrew, Latin, and Greek by the age of 5,
and Sanskrit, Arabic, and Persian by the age of 10).

Lee Rudolph

Dik T. Winter

Nov 5, 2005, 8:14:49 PM11/5/05
to
In article <dkim43$4sa$1...@glue.ucr.edu> ba...@galaxy.ucr.edu (John Baez) writes:
> In article <dki5eo$mdv$1...@panix2.panix.com>,
> Lee Rudolph <lrud...@panix.com> wrote:
>
> >>So, now we need to hold a seance to see why Hamilton chose the letter
> >>"varpi" for momentum. In fact, we should have done this on Hamilton's
> >>200th birthday, earlier this year!
>
> >Be careful. Unless there is evidence (not cited above) that Hamilton
> >intended the glyph that TeX calls "varpi" to be, in fact, a variant
> >of pi, it remains possible--and I think likely--that he intended that
> >glyph to be what a Russian Orthodox priest of my acquaintance
...

> That's possible; unlike the LaTeX "varphi", which is an alternative
> version the Greek phi, I'm not sure the LaTeX "varpi" is really
> supposed to be a variant of "pi".

I have also looked around. It appear that the distinction "phi" and "varphi"
just depends on font. But, there appears to have been quite some discussion
about the variant symbols in Unicode. See for instance:
<http://omega.enstb.org/yannis/pdf/amendments2.pdf>
where the status of "varpi" is said to be uncertain, but where it is stated
to be an alternative for the standard pi (in that document it is also not
suggested not to use it in Greek text, which is done for quite a few other
symbols). And:
<http://ptolemy.tlg.uci.edu/~opoudjis/unicode/letters.html>
states that "varpi" dates from the 8-th century and was used alongside
the standard form until the mid 20-th century.

> There are a number of nonstandard Greek letters, of which
> Wikipedia lists four: digamma, san, qoppa and sampi:
>
> http://en.wikipedia.org/wiki/Greek_alphabet

I can not view that article properly, but those are archaic letters, not
used in Greek texts. (San has been added with Unicode 4.0, as Sho,
which Wikipedia does not mention.) And another not mentioned letter
is the stigma...

> The sampi is the only one here with any resemblance to the "varpi"...

There is no relation.

> But - aha! - "varpi" is called "pomega" here:
> http://www.ams.org/STIX/stixfullr/stixfull-06.html

To me that appears to be techspeak.

I think that Hamilton used the pi as appeared frequently in hand-written
material. "varpi" was a cursive form (i.e. handwritten). That it
deviates quite far from the standard form is not so very surprising.
In the Latin script we have the a and g with in general deviations
between handwritten and printed material. In Cyrillic it is quite a
bit stronger. The lower case d, which in print looks like a small delta
with two descenders on both sides, is written as "g". And it took me
some time when I was in Bulgaria to determine that some neon advertisement
that wrote "onmuka" actually was displaying what in a normal transcription
would be "optika".
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

john_r...@sagitta-ps.com

Nov 5, 2005, 10:08:56 PM11/5/05
to

Dik T. Winter wrote:
>
> > http://en.wikipedia.org/wiki/Greek_alphabet
>
> I can not view that article properly, but those are archaic letters, not
> used in Greek texts. (San has been added with Unicode 4.0, as Sho,
> which Wikipedia does not mention.) And another not mentioned letter
> is the stigma...

There's also the digamma

The page http://ptolemy.tlg.uci.edu/~opoudjis/unicode/numerals.html
includes he following:

"If you're using letters of the alphabet as numerals, you want to have
some
device for differentiating numbers from words. The main such device in
antiquity and the middle ages was the overbar, which was also used to
cite words."

Hero

Nov 6, 2005, 2:47:01 PM11/6/05
to

John Baez wrote:
> Thanks to Dirk Schlimm, I've finally found an online map showing
> the bridge where Hamilton carved his famous formula defining the
> quaternions:
>
>
> You can see where Broombridge Road crosses the Royal Canal -
> that's the place. For pictures and bus directions, try this:
>
> http://math.ucr.edu/home/baez/dublin/index.html#brougham_bridge
>
> And speaking of Hamilton, Theron Stanford has sent me an answer to
> one of life's persistent questions:
>
> why is momentum denoted by the letter p?
Thanks to David R. Wilkins Hamilton's writings are online at:
http://www.maths.tcd.ie/pub/HistMath
May be one can find a hint to the letter p in it.

> So, now we need to hold a seance to see why Hamilton chose the letter
> "varpi" for momentum. In fact, we should have done this on Hamilton's
> 200th birthday, earlier this year!
David R. Wilkins wrote an article, celebrating this event. And he shows
the bridge too.
By the way, he was talking to his wife about knitting a pullover, and
she explained to him Aaran knitting style.
Hero

Dave Rusin

Nov 6, 2005, 4:42:16 PM11/6/05
to
In article <IpIE4...@cwi.nl>, Dik T. Winter <Dik.W...@cwi.nl> wrote:

>And it took me
>some time when I was in Bulgaria to determine that some neon advertisement
>that wrote "onmuka" actually was displaying what in a normal transcription
>would be "optika".

And you have to get used to going to the pectopah for something to eat.

Dik T. Winter

Nov 6, 2005, 9:46:51 PM11/6/05
to
In article <dkjhrh$nf$1...@panix2.panix.com> lrud...@panix.com (Lee Rudolph) writes:
...
> >> Lee Rudolph <lrud...@panix.com> wrote:
> >> >Be careful. Unless there is evidence (not cited above) that Hamilton
> >> >intended the glyph that TeX calls "varpi" to be, in fact, a variant
> >> >of pi, it remains possible--and I think likely--that he intended that
> >> >glyph to be what a Russian Orthodox priest of my acquaintance (at a
> >> >time long before he was a priest; actually, at a time when he, I,
> >> >and John Baez's future undergraduate thesis advisor were in junior
> >> >high school together) told me was called "ot": a ligature of "tau"
> >> >and "omega" (actually, a superposition of tau and omega, but apparently
> >> >that counts as a ligature in paleography if not in typography).
...

> That the glyph in question can be (and is) used as a variant
> of pi does not rule out that it can be (and has been) used as
> a ligature of omega and tau.

Can be true, of course. I distrust the reference you give:
> http://www.uni-giessen.de/partosch/eurotex99/berdnikov2.pdf
but it will not work with the version of acrobat reader I have.
But I will look into Peter T. Daniels monumental work on world
scripts.

> The challenge is now to show that Hamilton was acquainted with
> Church Slavonic.

There are quite a few reasons to suspect a Greek source. "varpi"
has been used in Greek from about the 8-th century to well into
the 20-th century. The "omega" was still part of Cyrillic in 1708,
but was gone quite early (btw, the oldest Cyrillic dates from the
10-th century, and indeed, that one had an omega-like symbol (or
w-like) called "otu"). And while a short-hand 't' might have been
written as a tilde (in cursive form it looks like an 'm'), I would
expect such ligatures to be uncommon. Anyhow, your quote:

> * The order of the alphabet is changed, some letters changed
> their names, some letters changed their shape. Some letters
> became obsolete and are excluded from the alphabet, but as
> a compensation the new letter _ot_ appears which is the
> ligature between O (Greek omega) and T with three stems.
> (In Old Slavonic the name _ot_ was reserved for Greek
> \omega which is called _omega_ in Church Slavonic.)

is unreliable. In 10-th century the symbol was called "otu" and
in 1708 the same symbol was called "omega".

John Baez

Nov 7, 2005, 1:09:39 AM11/7/05
to
In article <dkjhrh$nf$1...@panix2.panix.com>,
Lee Rudolph <lrud...@panix.com> wrote:

>"PiSigma" <psi...@tee.gr> writes:

>>> Lee Rudolph <lrud...@panix.com> wrote:

>>> >Be careful. Unless there is evidence (not cited above) that Hamilton
>>> >intended the glyph that TeX calls "varpi" to be, in fact, a variant
>>> >of pi, it remains possible--and I think likely--that he intended that

>>> >glyph to be what a Russian Orthodox priest of my acquaintance [...]

>>> >told me was called "ot": a ligature of "tau" and "omega"

>>Actually, it's a variant of "pi": see
>>http://omega.enstb.org/yannis/pdf/boston99.pdf

>That the glyph in question can be (and is) used as a variant
>of pi does not rule out that it can be (and has been) used as
>a ligature of omega and tau.

For the purposes of figuring out how "p" got to be used as the
standard symbol for "momentum", the question is what Hamilton
intended by using the character "varpi" (or "ot") to stand for
momentum - and what the people who read it thought they were reading!

Since Jacobi changed this symbol to "p", it seems he considered
it to be a version of the letter "pi", not the letter "ot".

And, remark #5 on page 8 here:

http://omega.enstb.org/yannis/pdf/boston99.pdf

suggests that Hamilton could easily have used it as a version of "pi".
To quote:

The "round" pi (varpi) (U+03D6 in UNICODE). This letter has
been a variant of pi from the very first days of Greek typography:
in Figure 2, the reader can see an excerpt of [Pol], a book
printed in 1489 (!). In this excerpt both versions of pi have
been used, apparently randomly.

Once again, [Imp, p. 95] sugggests that the round pi be used only
at the beginning of a word. According to [Tre], this rule has
been applied in certain French journals (for example the Memoires
de la Societe de Linguistique in the late 19th and early 20th
century); [Tre] appliesit in all of his writings [....]

[Imp] again, but in a different chapter (the one on the typesetting
of mathematics, p. 108) calls this letter "Doric pi".

From this it seems quite possible that Hamilton wrote (or at least
typeset) the letter pi in this "rounded" or "Doric" form.
It seems less plausible to me that he would have used the character
"ot" from Orthodox Church Slavonic in his mathematical writings.
Even if he knew this language (and you're right, he was a famous
polyglot), why would he have used it in his mathematical writings?
Or: what evidence do we have for this?

What's really tantalizing is the reference above to appearance of
the round pi in "the typesetting of mathematics"! Can anyone get
ahold of this and see what it says:

Lexique des regles typographiques en usage a l'Imprimerie nationale,
Impremerie Nationale, 1990, page 108. [Look for "Doric pi", or something
like that in French.]

?

Peter T. Daniels

Nov 7, 2005, 9:06:32 AM11/7/05
to
[from sci.lang]

Have you tried looking in Cajori's *History of Mathematical Notations*?
(2 vols., 1928-29; Dover reprint in 1 vol., 1993.)

--
Peter T. Daniels gram...@att.net

Timothy Murphy

Nov 7, 2005, 6:44:01 PM11/7/05
to
John Baez wrote:

> For the purposes of figuring out how "p" got to be used as the
> standard symbol for "momentum", the question is what Hamilton
> intended by using the character "varpi" (or "ot") to stand for
> momentum - and what the people who read it thought they were reading!

My colleague David Wilkins, who is our resident expert on Hamilton,
and has probably spent more hours poring over his works,
both hand-written and printed, than anyone else -
but who unfortunately is not a devotee or sci.math -
writes as follows:

"Hamilton tended to be meticulous with regard to typography, proof-reading
etc. of his mathematical papers. He would have used the typefaces available
and in regular use by printers of academic works, and regularly used for
mathematical works.

I would suggest that the glyph \varpi was not infrequently
used in published mathematical works in the nineteenth century to denote
pi. For example, note Riemann's major paper on the Theory of Abelian
functions. When reprinted in the standard Weber 19th century edition
of Riemann's works, the printers (Teubner, if I remember correctly off
the top of my head) used \omega in the latter parts of the paper to
denote quantities denoted by \pi in the earlier part of the same paper.
But if one looks at the original printed text of this paper, published
by the Berlin Academy, one finds that the printers used \varpi to
denote pi in the latter part of the paper, and this seems not to
have been noticed by the typesetters of the collected works.

I myself would take it for granted that Hamilton might well use
\varpi as representing the Greek letter pi. I think it unlikely
that he would think in terms of some glyph in handwritten Greek
like the 'ot' thing that has been mentioned. I am not aware of any interest
in early manuscripts (as opposed to printed texts) on the part
of Hamilton.

(To help settle the matter, one might look at a variety of nineteenth
century publications of standard Greek texts, to see how often the
glyph \varpi is in regular use by academic printers at that time.)

In any case, Hamilton would, I am sure, give particular attention
on the *printed* appearance of his papers, and the systematic use
of Greek, roman, italic and small capitals fonts within them.
(On the occasions when he attempted to introduce new symbols, he
did this by rotating existing ones through one or two right angles.)

For my transcription of relevant Hamilton papers, see

Hamilton uses \varpi in conjunction with \eta, see in particular pp. 4-5
of the first of these papers.

...

Hamilton probably didn't think of his quantities \varpi_1, varpi_2 etc.
as 'momenta' but as partial derivatives of his characteristic and
principal functions; analogous to the quantities \sigma, \tau and
\upsilon that appear, in the analogous optical context, in his
famous Third Supplement (the paper containing the prediction of
conical refraction)."

--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland

John Baez

Nov 9, 2005, 2:57:29 AM11/9/05
to
In article <7FRbf.18344\$R5....@news.indigo.ie>,
Timothy Murphy <t...@birdsnest.maths.tcd.ie> wrote:

>John Baez wrote:

>My colleague David Wilkins, who is our resident expert on Hamilton,
>and has probably spent more hours poring over his works,
>both hand-written and printed, than anyone else -
>but who unfortunately is not a devotee or sci.math -
>writes as follows:

>For my transcription of relevant Hamilton papers, see
>
>http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/SecEssay.pdf
>http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/BARep34A.pdf
>
>Hamilton uses \varpi in conjunction with \eta, see in particular pp. 4-5
>of the first of these papers.

Great! I like equation 12 in the first paper.

>Hamilton probably didn't think of his quantities \varpi_1, varpi_2 etc.
>as 'momenta' but as partial derivatives of his characteristic and
>principal functions; analogous to the quantities \sigma, \tau and
>\upsilon that appear, in the analogous optical context, in his
>famous Third Supplement (the paper containing the prediction of
>conical refraction)."

But of course the modern, sophisticated concept of "momentum" is
precisely the thing whose components are the partial derivative of the
Lagrangian, or equivalently of Hamilton's principal function,
with respect to velocity. So, Hamilton's \varpi is indeed the
ancestor of the modern p.

I looked at the first paper to try to guess why Hamilton chose the
letter \varpi, but I didn't see any obvious reason - it just seemed
to be one among many Greek letters he used for various quantities
(as mentioned above). He may not have considered it to be particularly
important.

So, the answer "why is momentum called p" may be "because Hamilton
chose \varpi for this variable" - no better reason.

tdp...@gmail.com

Nov 29, 2005, 7:41:35 AM11/29/05
to

John Baez wrote:
> Thanks to Dirk Schlimm, I've finally found an online map showing
> the bridge where Hamilton carved his famous formula defining the
> quaternions:
>
>
> You can see where Broombridge Road crosses the Royal Canal -
> that's the place. For pictures and bus directions, try this:
>
> http://math.ucr.edu/home/baez/dublin/index.html#brougham_bridge

Open the "Placemark"
at the URL below to see a closeup of the bridge on Google Earth.


text:
# derive using the chain rule

Given the polinomyal $f(x)=\frac{x^3}{(4-x^2)^3}$ find $f'(x)$

So, If I try to derive this, first I must to apply the chain rule in the denominator and then derive of the division (...)

$$f'(x)=\frac{x^3}{3(4-x^2)^2(-2)} = \frac{x^3}{-6(4-x^2)^2}$$

(...)?

Or there is another way to do this?

• Does your problem requires you to use only the chain rule? – Ataraxia May 19 '13 at 23:10
• the exercise says find f'(x) but I don't know how to solve this – Tomi May 19 '13 at 23:19

You're doing wrong. A good way to do this when still not expert in derivatives, is to write $$f(x)=\frac{g(x)}{h(x)}$$ where $$g(x)=x^3,\qquad h(x)=(4-x^2)^3.$$

Then, first of all, you have to differentiate the quotient: $$f'(x)=\frac{g'(x)h(x)-g(x)h'(x)}{(h(x))^2}$$

So you need to compute $g'(x)$ and $h'(x)$. The first one is easy: $$g'(x)=3x^2.$$ The second one needs the chain rule: $$h'(x)=3(4-x^2)^2(-2x)=-6x(4-x^2)^2.$$ Now plug in the formula above: \begin{align} f'(x)&= \frac{3x^2\cdot(4-x^2)^3 - x^3\cdot(-6x(4-x^2)^2)}{(4-x^2)^6}\\[2ex] &=\frac{3x^2(4-x^2)^3+6x^4(4-x^2)^2}{(4-x^2)^6}\\[2ex] &=\frac{3x^2(4-x^2)+6x^4}{(4-x^2)^4}\\[2ex] &=\frac{12x^2+3x^4}{(4-x^2)^4}=\frac{3x^2(4+x^2)}{(4-x^2)^4} \end{align}

A different strategy may be noting that $$f(x)=\left(\frac{x}{4-x^2}\right)^3$$ so, letting $$g(x)=\frac{x}{4-x^2}$$ you have, by the chain rule, $$f'(x)=3g(x)^2g'(x)$$ The derivative of $g(x)$ can be computed again with the quotient rule: $$g'(x)=\frac{1\cdot(4-x^2)-x\cdot(-2x)}{(4-x^2)^2}= \frac{4-x^2+2x^2}{(4-x^2)^2}=\frac{4+x^2}{(4-x^2)^2}$$ and so $$f'(x)=3\left(\frac{x}{(4-x^2)}\right)^2\frac{4+x^2}{(4-x^2)^2}$$ which gives the same result as before.

You can probably use quotient rule,

$$h(x)=\dfrac{f(x)}{g(x)}$$

$$h'(x)=\dfrac{f'(x)g(x)-g'(x)f(x)}{(g(x))^2}$$

You want to use the quotient rule then the chain rule.

$$\frac{d}{dx}\frac{f(x)}{g(h(x))}=\frac{g(h(x))f'(x)-f(x)g'(h(x))h'(x)}{g(h(x))^2}$$

so:

$$\frac{d}{dx}\frac{x^3}{(4-x^2)^3} = \frac{(4-x^2)^3(3x^2)-x^3(3(4-x^2)^3)(-2x)}{(4-x^2)^6}$$

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## One-Sketch-for-All: Non-linear Random Features from Compressed Linear Measurements

### Xiaoyun Li · Ping Li

Keywords: [ Algorithms, Optimization and Computation Methods ] [ Data Compression ]

[ Abstract ]
Wed 14 Apr 12:45 p.m. PDT — 2:45 p.m. PDT

Abstract: The commonly used Gaussian kernel has a tuning parameter $\gamma$. This makes the design of quantization schemes for random Fourier features (RFF) challenging, which is a popular technique to approximate the Gaussian kernel. Intuitively one would expect that a different quantizer is needed for a different $\gamma$ value (and we need to store a different set of quantized data for each $\gamma$). Fortunately, the recent work~\citep{Report:Li_2021_RFF} showed that only one Lloyd-Max (LM) quantizer is needed as the marginal distribution of RFF is free of the tuning parameter $\gamma$. On the other hand, \citet{Report:Li_2021_RFF} still required to store a different set of quantized data for each $\gamma$ value. In this paper, we adopt the one-sketch-for-all'' strategy for quantizing RFFs. Basically, we only store one set of quantized data after applying random projections on the original data. From the same set of quantized data, we can construct approximate RFFs to approximate Gaussian kernels for any tuning parameter $\gamma$. Compared with \citet{Report:Li_2021_RFF}, our proposed scheme would lose some accuracy as one would expect. Nevertheless, the proposed method still perform noticeably better than the quantization scheme based on random rounding. We provide statistical analysis on the properties of the proposed method and experiments are conducted to empirically illustrate its effectiveness.

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Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

# Derivation of yield stress in the honeycomb

Here we estimate the stress at which the honeycomb starts to deform irreversibly where some of the faces are aligned parallel to the loading axis and the predominant contribution to the strain comes from the bending of the diagonal faces. (see here) The thickness of the cell wall is t, its through-thickness width is w and the length of each of the hexagonal faces is l.

From beam bending theory, the maximum stress, σmax, in a cantilevered beam of length L due to a force, W, applied normal to the beam is

${\sigma _{{\rm{max}}}} = \frac{{6{\rm{ }}W{\rm{ }}L}}{{w{\rm{ }}{t^2}}}$

Now in our situation W = F cos θ and L = l/2, which gives smax as

${\sigma _{{\rm{max}}}} = \frac{{3{\rm{ }}F{\rm{ }}l}}{{w{\rm{ }}{t^2}}} \cdot \cos \theta$

We assume that yielding will start when σmax = σY, giving the applied force, F, at which yielding starts as

$F = \frac{{w{\rm{ }}{t^2}}}{{3{\rm{ }}l{\rm{ }}\cos \theta }} \cdot {\sigma _{\rm{Y}}}$

Now the applied force F acts over an area wl cos θ, so that the stress, σ, can be expressed as

$\sigma = \frac{F}{{w{\rm{ }}l\cos \theta }}$

Substituting for F gives the applied stress at which the honeycomb starts to yield in terms of the yield stress of the cell wall material, σY

$\sigma = \frac{1}{3} \cdot {\left( {\frac{t}{l}} \right)^2} \cdot \frac{1}{{{{\cos }^2}\theta }} \cdot {\sigma _{\rm{Y}}}$

For a regular hexagon, where θ = 30°, this becomes

$\sigma = \frac{4}{9} \cdot {\left( {\frac{t}{l}} \right)^2} \cdot {\sigma _{\rm{Y}}}$

At this value of σ, yielding begins at the surfaces of the cell wall, at the cantilevered end of the beam where the stresses will be greatest. For plastic yielding to spread through the thickness of the cell wall requires that the stress is increased by a factor of 1.5, giving a macroscopic flow stress of

$\sigma = \frac{2}{3} \cdot {\left( {\frac{t}{l}} \right)^2} \cdot {\sigma _{\rm{Y}}}$


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Home / Basic Electrical (page 2)

# Basic Electrical

## Capacitor Types | Fixed and Variable Capacitors

Variable capacitors Variable capacitors are distinguished by the fact that their capacitance can be changed. Basically, there are two most common types of such capacitors named as a trimmer and rotor-stator capacitors. Rotor-Stator Capacitor The rotor-stator type of capacitor comprises two metallic plate sets. The moving plates are attached conjointly …

## Ohm’s Law | Ohm’s Law Formula

As the voltage increases in a circuit (resistance remaining constant), the current increases by the same amount. Hence, if the voltage is doubled, the current will double. Also, the amount of current in a circuit is inversely proportional to its resistance when the voltage remains unchanged. Stated another way, if …

## Capacitive Reactance | Reactance of Capacitor

If a sinusoidal voltage is applied to a pure capacitance ( no series or parallel resistance), the current is maximum when the voltage begins to rise from zero.one-quarter a cycle later, the current is zero when the voltage across the capacitor is maximum. This condition, illustrated in figure 1, shows …

## Inductive Reactance | Reactance of Inductor

Inductive Reactance Opposition to the flow of an alternating current by the inductance of the circuit; equal to 2πfL and measured in ohms. When a changing current flows through an inductor, a self-induced voltage is developed. Its polarity is such that it opposes the change. The emf varies directly with …

## Electrical Formulas | Electrical Formulas Sheet

A Absolute Permittivity ${{\varepsilon }_{o}}=8.84*{{10}^{-12}}$ Active Power $\text{P=VICos(}\theta \text{) Watt}$ Apparent Power $\text{S=VI volt-amp}$ B C Capacitance $\text{C=}\frac{\text{ }\varepsilon {{\text{ }}_{\text{o}}}\text{ }\varepsilon {{\text{ }}_{\text{r}}}\text{A}}{\text{d}}$ Where, εo= Absolute Permittivity εr= Relative Permittivity A=Plates Area d= distance between plates Conductance $\text{Conductance}=\frac{1}{\text{Resistance}}=\frac{1}{\text{R}}$ Capacitive Reactance ${{\text{X}}_{\text{C}}}\text{=}\frac{1}{2\pi fC}$ Capacitive Susceptance ${{\text{B}}_{\text{C}}}\text{=}\frac{1}{{{\text{X}}_{\text{C}}}}$ Current in Series Circuit …

## Factors Affecting Capacitance | Dielectric Constant

There are three main factors affecting the capacitance of the capacitors that will be discussed in this tutorial in detail. The SI unit of capacitance is farad, named in honor of the English physicist and chemist Michael Faraday. The unit symbol for the farad is F. capacitance is the ability …

## Types of Resistors | Power Resistor

Resistor An element used to reduce supply voltages to some desired value or to limit current. A resistor is a small component with two leads. A wide variety of resistors is used in the electronics industry today. Carbon composition resistor The carbon composition resistor is the basic mass-produced resistor of …

## Nonlinear Resistors | Characteristics Curves of Different Nonlinear Devices

In most circuits, we can assume that resistance is constant with relation to current and voltage. This linear relation can be graphically shown in figure 1. Fig.1: Plot of Linear Relation between Current and Voltage For example, if 3V is applied to a certain resistor and 1A flows, then 6V …

## Resistor Color Code | Resistor Color Bands

Carbon resistors are color coded- that is, they have several color bands painted around the body near one end- to identify their ohmic values. Other types of resistors are not color-coded; instead, they have their ohmic values and, sometimes, identifying part numbers printed on them. The code has been established by …

## Resistor Power Rating | Power resistor

The physical size of a resistor is not determined by its resistance but by how much power, or heat, it can dissipate. It electric circuits, the unit of power is the watt (W), named in honor of James Watt. One watt is the power dissipated when one ampere flows under …


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## Mathematical Programming: Turing completeness and applications to software analysis

Mathematical Programming is Turing complete, and can be used as a general-purpose declarative language. We present a new constructive proof of this fact, and showcase its usefulness by discussing an application to finding the hardest input of any given program running on a Minsky Register Machine. We also discuss an application of Mathematical Programming to … Read more

## Code verification by static analysis: a mathematical programming approach

Automatic verification of computer code is of paramount importance in embedded systems supplying essential services. One of the most important verification techniques is static code analysis by abstract interpretation: the concrete semantics of a programming language (i.e.the values $\chi$ that variable symbols {\tt x} appearing in a program can take during its execution) are replaced … Read more