[sdiy] SDIY MATH GOALS--need real help!

[cheater cheater]

Hi guys,
I'd suggest being aware that David's reply was at least a little bit misleading.
I asked David in private to correct his mistakes, cursorily pointing
them out, only to meet with an attack in return and lack of action by
David for several days.

Let's go through the post and spot the problems. There are some
excerpts of my private conversation with David in this post and a full
quote in the attachment.

On Sun, Mar 1, 2009 at 9:56 PM, David G. Dixon <dixon at interchange.ubc.ca> wrote:
> Dan,
>
> A triangle pointing up to the left of a variable (say, x) is to be read
> "Delta x" and it simply means "macroscopic change in x" or "x_2 minus x_1"
> (as opposed to "dx" from calculus, which means "infinitesimal change in x"
> and is the limit as Delta approaches zero).

Not really. Stochastic and other finite-differential equations are
very uncommon in analogue electronics, you're not very likely to come
across those. The most common use of the Delta symbol (notice the big
D which means it's a big Delta as opposed to the small delta as used
in e.g. continuity proofs) is with quadratic equations, where it is
used for the discriminant of the equation. Everyone knows this
formula:

Delta = b^2 - 4ac.

I write this in hope the readers of this list not advanced with
mathematics will have a chance to learn it, instead of (wrongly)
wondering why you're taking a square root of a differential in order
to find the root of a parabola.

>  If the triangle is pointing
> down, this is to be read "del x" or "nabla x" and is simply a stand-in for
> the differential expression "dx/d?".

This isn't true. The expression you used is valid, but not in the
context you meant. the right expression you are looking for might be
Del y = dy/dx, but that's still wrong (it's fixable in single-valued
single-variable functions where you can define the ortonormal base to
be {1}). However, as you can check for example on wikipedia, Del y is
an operator which gives you a vector of partial derivatives. A partial
derivative is written with a curly 'd', which can be signified by @ on
the internet. You can see this curly 'd' on the partial derivative
article on Wikipedia.

Suppose you have a function y of three variables x_1, x_2, x_3: y =
f(x_1, x_2, x_3). Suppose your vector space has the ortonormal base
{i, j, k}.
Then, Del y = @y/@x_1* i + @y/@x_2 * j + @y/@x_3 * k.

In the simplest form, i is the vector (1, 0, 0), j is (0, 1, 0), and k
is (0, 0, 1).
As you can see we get a vector.

Some more advanced people might point out that in dimensions higher
than 1 when calculating the (total) derivative of a function we get a
matrix. That is still not the vector signified by the Del operator.

> Generally, if x is differentiable by
> more than one independent variable (for example, say x is a 2D variable in
> the y-z plane, so it may be differentiated separately with respect to y or z
> (i.e., dx/dy and dx/dz), then the upside-down triangle (or nabla) signifies
> all possible differentials at once.

Those are partial derivatives. As you can see here,
http://en.wikipedia.org/wiki/Del#Definition
They're using that funny curly d. And they're not 'all at once',
they're components of a vector.

>  This is what we call "vector notation"
> as it doesn't differentiate between the various dimensions of the problem.
> There is a whole subfield of calculus called "vector calculus" which deals
> with this.

Vector calculus deals with vector-valued functions which the Del
operator does not apply to. The Del operator applies only to
vector-argument functions, which is a very big difference from the
point of view of what tools you can use in mathematics to analyze such
a function. The study of functions that take vectors as their
arguments is called multivariable calculus or, less often,
multivariate calculus.

> Your question is a good one, because it leads directly to the most
> fundamental idea in differential calculus; namely, the "central limit
> theorem".  This says that a function evaluated at x + Delta-x minus that
> function evaluated at x, all over Delta x, approaches the differential df/dx
> as Delta-x approaches zero.  Draw a random curve on an x-y plot, take any
> two points on the curve, and draw a straight line through them.  This is the
> algebraic linear slope of those two points, or "Delta-y/Delta-x".  Now
> slowly bring one point closer to the other, continuing to draw straight
> lines through the two points.  When the two points essentially become one,
> the last straight line you draw is the tangent slope of the function f(x) at
> point x, or df(x)/dx.  This, in a nutshell, is differential calculus.  Every
> single-valued function (i.e., one which has only one value at a given x) has
> a unique differential (or tangent slope) at every point (unless there is a
> discontinuity, or step-change, in which case there are two -- one on either
> side of the step).  See?  It's really not that hard.  You can prove that
> calculus is correct by taking two points an equal small distance in x from
> the point of interest, say x+ and x-, by solving (f(x+) - f(x-))/(x+ - x-).
> This will give you a very close approximation of df/dx at x (in the middle).
> We call this the "finite difference approximation method".

This is a very fuzzy reminiscence of something once seen on a
blackboard. This is actually the classical way of teaching children
about derivatives. Take a smooth curve, fix a point x, find a point x'
a bit away from it on that same curve, calculate the slope between the
points and you get an approximation of the derivative at x of the
function generating the curve. The closer x' is to x, the better the
approximation for such curves. Now let's see what can be fixed about
your explanation of this:

> Your question is a good one, because it leads directly to the most
> fundamental idea in differential calculus; namely, the "central limit
> theorem".

The central limit theorem is a theorem in statistics, and has nothing
to do with the topic.

> This says that a function evaluated at x + Delta-x minus that
> function evaluated at x, all over Delta x, approaches the differential df/dx
> as Delta-x approaches zero.

I am guessing (looking past the terrible notation) that you mean the
fact that for a function f,

lim_((Delta x )-> 0) ( f(x+ Delta x) - f(x) )/(Delta x) )

can exist, and if it exists then its numeric value at x is the
derivative of f at x, signified by f'(x) or df/dx(x). This is Newton's
definition of the derivative, and not a theorem. It's very important
to understand this, since it just makes you look funny in
conversations.

> Draw a random curve on an x-y plot, take any
> two points on the curve, and draw a straight line through them.  This is the
> algebraic linear slope of those two points, or "Delta-y/Delta-x".

To be strict, it's a line, and this line's slope is what we call Deltay/Deltax.

> Now
> slowly bring one point closer to the other

David, you forgot to mention that we can't move the x point, only the
'other point', otherwise we're not calculating the differential at x
anymore.

> continuing to draw straight
> lines through the two points.  When the two points essentially become one,
> the last straight line you draw is the tangent slope of the function f(x) at
> point x, or df(x)/dx.

Only if such a slope is uniquely defined! This is what is meant in
Newton's definition by the words 'if the limit exists'. Take for
example a function which looks a bit like the triangle wave, except it
only has one 'valley':

f(x) = |x|. This looks a little bit like this: \/
with the valley being in the point 0 and the left branch extending
towards x=-infinity in the same way as y=-x, while the right branch
extends towards x=+infinity in the same way as y=x.

Now if you look at that valley, the point (x, y) = (0, 0), then by
calculating differently you can get different results. The by taking
closer and closer approximations with the 'second point' being to the
right of the Y axis, you will get the value +1, with the points being
to the left of Y you get the value -1, and without such restrictions,
your term

f(x+ Delta x) - f(x) )/(Delta x)

will randomly alternate between +1 and -1 yielding no limit at all.

Hence the function is not differentiable, but by using your wrong
'test' one might be drawn to think that it is.

Also, to fully scrutinize, the statement 'last line you draw' is
pretty wrong, since you're talking about a process with an infinite
amount of possible iterations. Sure, you can stop somewhere and look,
but there's no 'last line'. There's always another line that can be
drawn which will (for some functions) approximate the function's
derivative better.

> This, in a nutshell, is differential calculus.

I cringe at the day when people start teaching this as differential calculus *

> Every
> single-valued function (i.e., one which has only one value at a given x) has
> a unique differential (or tangent slope) at every point

NO.

> (unless there is a
> discontinuity, or step-change, in which case there are two -- one on either
> side of the step).

NO.

Most functions don't even have a one-sided derivative at any point.
There are functions that are not differentiable at any point.

What you mean is that for a function differentiable everywhere in the
neighborhood of a point, but not at the point itself, there are two
values you could get (like the -1 taking the 'left derivative' and +1
taking the 'right derivative' from our function |x| earlier). This is
misleading to say that this happens in every, or even the *majority*
of situations.

If you consider all functions that can be defined on the real numbers
and which have values that are real numbers, so your usual, normal,
pleasant functions f(x) : R->R, then if you try to randomly choose a
function, the probability can be calculated that you find one that's
differentiable at even a signle point. This probability is 0,
demonstrating how rare those functions actually are, and demonstrating
how important it is to add certain warnings to any statement about
mathematical objects.

> See?  It's really not that hard.

It actually is. It is hard and you did not understand it.

> You can prove that
> calculus is correct by taking two points an equal small distance in x from
> the point of interest, say x+ and x-, by solving (f(x+) - f(x-))/(x+ - x-).

...again in some practical situations only. Not in all, not even in
most. Only for continuous functions that are everywhere differentiable
can you allow yourself such mental slack.

> This will give you a very close approximation of df/dx at x (in the middle).
> We call this the "finite difference approximation method".

> Sometimes it almost seems like magic that this calculus stuff works, but it
> always does!
>
> David G. Dixon
> Professor
> Department of Materials Engineering
> University of British Columbia
> 309-6350 Stores Road
> Vancouver, B.C. V6T 1Z4
> Canada

David, I am surprised that you undersign your email with the title of
Professor, but display a lack of understanding of calculus that most
children in my country have mastered by the time they're freshmen in
high school. This is truly appalling that you give your posts some
mark of quality with this title, yet spew total and utter
disinformation. The things I pointed out are not 'a mathematician's
mockings'. They're not even nit picking. Those are not concepts 'only
mathematicians need to care about' - those are very basic concepts
that literally *have* to be understood by every engineer who wants to
be successful in what he does. Many of the mistakes were not in the
details, but in the gross meaning of what you were saying.

I post this mostly because almost all of those concepts you have taken
on are important parts of a base for further knowledge to be built
upon. Those concepts are used even in teaching people who will not use
them later on, just to create a pattern of thought that can be later
related to. Like Newton's definition of a derivative, which you
completely butchered and, defiling Newton's name, removed it from the
name of the definition showing a complete disregard for one of the
greatest thinkers in written history.

The act of not recognizing something as basic as the definition of the
derivative, or multiple instances of mistaking the derivative for a
differential all add up to my judgment that you are in no way fit to
teach people about calculus. Please don't do so.

I certainly hope you do not teach where ever you are called Professor.
If you do, please *at least* take the time to revise. This is your job
and you are doing it wrong. As a Professor you need to have a PhD
which stands for Doctor of Philosophy. It is certainly required for a
Doctor of Philosophy to know the most important theorems conceived by
some of the most important philosophers to exist to date, like
Leibniz, Newton, or Cauchy.

It's never the wrong time to go back to books, and I know it might
look funny for a Professor to sit down with an undergrad or even high
school level book, but I think it's better to do that than to find
yourself in a situation like this one where you start talking and
don't even realize you might be wrong. Or judging by your response
off-list, don't even admit the possibility.

When looking at your signature of 'Perfecta fingamus serviat natura',
I wonder if it really means you are a servant in taking nature apart
to see how it works, or if the 'fingere' in your slogan actually uses
the other meaning, which would turn out to 'servant in feigning
nature'. I can't say you had me tricked with your version of nature,
though.

On Mon, Mar 2, 2009 at 7:58 AM, David G. Dixon <dixon at interchange.ubc.ca> wrote:
> Dear Cheater,
>
>> David,
>> I've found your post to be highly offensive to any mathematician
>
> Well, I'm an engineer, so I officially don't care!
>
>> and very disinformative. I am posting this off-list to give you a chance
>> to correct this yourself. Please do so, something has to be said
>> on-list by someone. If nothing happens I will post an errata.
>
> If your "errata" are as incoherent as your rebuttal, then heaven help us
> all!
>
>>
>> Best regards
>> D.
>>
>> On Sun, Mar 1, 2009 at 9:56 PM, David G. Dixon <dixon at interchange.ubc.ca>
>> wrote:
>
>
>> > Dan,
>> >
>> > A triangle pointing up to the left of a variable (say, x) is to be read
>> > "Delta x" and it simply means "macroscopic change in x" or "x_2 minus
>> x_1"
>> > (as opposed to "dx" from calculus, which means "infinitesimal change in
>> x"
>> > and is the limit as Delta approaches zero).
>
>
>>
>> Unclear. Someone learning calculus will just get confused by this.
>>
>
> Perhaps, but accurate nonetheless.  I don't know how else to say it.  Delta
> is a triangle pointing up, and it means "difference".  Nabla is an
> upside-down Delta, and it means "differential (as a vector, i.e., in 3D).
>
>> > If the triangle is pointing
>> > down, this is to be read "del x" or "nabla x" and is simply a stand-in
>> for
>> > the differential expression "dx/d?".
>>
>
>> Wrong.
>
> How so?  I tried to clarify below...
>
>> Well, I wonder if the water in your pipes flows against gravity with
>> this definition.
>
> I think you are confusing the definition of nabla with the constitutive laws
> which govern most physical processes, in which the flux of any conservable
> quantity is proportional to the negative gradient of some intensive property
> related to that quantity (e.g., Fourier's law of heat conduction, in which
> heat flows in the direction of the negative temperature gradient, and which
> is generally written "q = -k [nabla] T" or, in one direction, say x, "q = -k
> dT/dx", where q is heat flux, k is thermal conductivity, T is temperature,
> and x is distance along some direction)
>
>> > Generally, if x is differentiable by
>> > more than one independent variable (for example, say x is a 2D variable
>> in
>> > the y-z plane, so it may be differentiated separately with respect to y
>> or z
>> > (i.e., dx/dy and dx/dz), then the upside-down triangle (or nabla)
>> signifies
>> > all possible differentials at once.
>
>> Do you mean it signifies a sum of all partial derivatives? Big difference.
>
> No, I most emphatically do not mean that, and yes, it is a big difference.
> It signifies exactly what I said it signifies: all the differentials at
> once, as components of a 3D vector.
>
>> > This is what we call "vector notation"
>> > as it doesn't differentiate between the various dimensions of the
>> problem.
>> > There is a whole subfield of calculus called "vector calculus" which
>> deals
>> > with this.  In fact, in my professional work, I tend to use vector
>> calculus
>> > quite often, as I am dealing with flow in 3D, and vector calculus is
>> more
>> > convenient for me in this regard (as I only have to do it once, rather
>> than
>> > 3 times).
>> >
>> > Your question is a good one, because it leads directly to the most
>> > fundamental idea in differential calculus; namely, the "central limit
>> > theorem".  This says that a function evaluated at x + Delta-x minus that
>> > function evaluated at x, all over Delta x, approaches the differential
>> df/dx
>> > as Delta-x approaches zero.  Draw a random curve on an x-y plot,
>
>> ...which is as unprobable a function as anything else, since curves
>> are a set of measure zero in the set of functions wherefore this
>> example does not define the differential without explaining this quite
>> important bit of information..
>
> You lost me there, I'm afraid...
>
>> > take any
>> > two points on the curve, and draw a straight line through them.  This is
>> the
>> > algebraic linear slope of those two points,
>
>> Is not.
>
> Perhaps my English was lacking here a little, but as Dan is American like me
> (and therefore illiterate by definition), he'll probably understand what I
> was trying to say just fine!  (Do you, Dan? ;->)  Of course it helps if the
> curve is singly-valued, and not, say, an S shape, although the trick will
> still work for the latter.
>
>> > or "Delta-y/Delta-x".  Now
>> > slowly bring one point closer to the other, continuing to draw straight
>> > lines through the two points.  When the two points essentially become
>> one,
>> > the last straight line you draw
>
>> Which would never happen, unless we're considering a function that is
>> a straight segment curve..
>
> It happens all the time, D.!  That's why Newton and Leibniz were so fond of
> it!  Give it a try sometime!
>
>> > is the tangent slope of the function f(x) at
>> > point x, or df(x)/dx.  This, in a nutshell, is differential calculus.
>>  Every
>> > single-valued function (i.e., one which has only one value at a given x)
>> has
>> > a unique differential (or tangent slope) at every point (unless there is
>> a
>> > discontinuity, or step-change, in which case there are two -- one on
>> either
>> > side of the step).
>>
>> This is just awesomely wrong and misinformative
>
> Whatever.  If you can explain the central limit theorem any better without
> the use of diagrams, be my guest!
>
>> > See?  It's really not that hard.  You can prove that
>> > calculus is correct by taking two points an equal small distance in x
>> from
>> > the point of interest, say x+ and x-, by solving (f(x+) - f(x-))/(x+ -
>> x-).
>> > This will give you a very close approximation of df/dx at x (in the
>> middle).
>> > We call this the "finite difference approximation method".
>> >
>> > Sometimes it almost seems like magic that this calculus stuff works, but
>> it
>> > always does!
>> >
>> > David G. Dixon
>> > Professor
>> > Department of Materials Engineering
>> > University of British Columbia
>> > 309-6350 Stores Road
>> > Vancouver, B.C. V6T 1Z4
>> > Canada
>> >
>> > Tel 1-604-822-3679
>> > Fax 1-604-822-3619
>> >
>> > "PERFECTA FINGAMUS SERVIAT NATURA"
>> >
>> > The information in this email and in any attachments is confidential and
>> > intended solely for the attention and use of the named addressee(s).  It
>> > must not be disclosed to any person without the writer's authority.  If
>> you
>> > are not the intended recipient, or a person responsible for delivering
>> it to
>> > the intended recipient, you are not authorized to and must not disclose,
>> > copy, distribute, or retain this message or any part of it.
>> >> -----Original Message-----
>> >> From: Dan Snazelle [mailto:subjectivity at hotmail.com]
>> >> Sent: Saturday, February 28, 2009 10:39 PM
>> >> To: dixon at interchange.ubc.ca; sdiy
>> >> Subject: RE: [sdiy] SDIY MATH GOALS--need real help!
>> >>
>> >>
>> >> ok
>> >>
>> >> what about if it has little triangles? (are these delta? what school of
>> >> math is this? algebra?)
>> >>
>> >> also...i keep reading about transform functions...is that calculus?
>> >>
>> >> i am also seeing a few other weird symbols pop up.
>> >> thanks so much
>> >>
>> >>
>> >> spending my saturday night reading "mastering technical mathematics"
>> >>
>> >>
>> >>
>> >>
>> >>
>> >> --------------------------------------------
>> >> check out various dan music at:
>> >>
>> >>  http://www.myspace.com/lossnyc
>> >>
>> >> (updated monthly)
>> >>
>> >> http://www.soundclick.com/lossnyc.htm
>> >>
>> >>
>> >>
>> >> http://www.indie911.com/dan-snazelle
>> >>
>> >> (or for techno) http://www.myspace.com/snazelle
>> >>
>> >> ALSO check out Dan synth/Fx projects:
>> >>
>> >> AUDIO ARK:
>> >>
>> >> www.youtube.com/watch?v=TJRpvaOcUic
>> >>
>> >> www.youtube.com/watch?v=BqIa_lXQNTA&feature=channel_page
>> >>
>> >> www.youtube.com/watch?v=V4nJPjGgOcU&feature=channel_page
>> >>
>> >>
>> >>
>> >>
>> >>
>> >> > Date: Sat, 28 Feb 2009 16:47:50 -0800
>> >> > From: dixon at interchange.ubc.ca
>> >> > Subject: RE: [sdiy] SDIY MATH GOALS--need real help!
>> >> > To: subjectivity at hotmail.com; synth-diy at dropmix.xs4all.nl
>> >> >
>> >> > Dan,
>> >> >
>> >> > If it's got "d something" over "d something" in it, its differential
>> >> > calculus (the d's on top and the somethings on the bottom can also
>> have
>> >> > exponents).  If it has a big S-shaped thingy to the left of
>> everything
>> >> > (often with little numbers or symbols above and below it), with a "d
>> >> > something" at the far right, its integral calculus.  If its got "e"
>> or
>> >> "exp"
>> >> > or "ln" or "log", then its an exponential or logarithmic function.
>>  If
>> >> it's
>> >> > got "sin" or "cos" or "tan" or "cot" or "sec" or "csc", with or
>> without
>> >> an
>> >> > "arc" in front or an "h" behind, then its trigonometry.  If
>> everything
>> >> is a
>> >> > function of "s" then it's a Laplace transform.  Alternatively, if
>> >> everything
>> >> > is a function of "jw" (where the "w" is really an undercase omega),
>> then
>> >> > it's a Fourier transform, which is really a Laplace transform where s
>> =
>> >> jw.
>> >> > If it's got lots of "j"s all over the place, then its complex math.
>> >> > Otherwise, it's just algebra!  (See how easy it all is?!?  ;->)
>> >> >
>> >> > David G. Dixon
>> >> > Professor
>> >> > Department of Materials Engineering
>> >> > University of British Columbia
>> >> > 309-6350 Stores Road
>> >> > Vancouver, B.C. V6T 1Z4
>> >> > Canada
>> >> >
>> >> > Tel 1-604-822-3679
>> >> > Fax 1-604-822-3619
>> >> >
>> >> > "PERFECTA FINGAMUS SERVIAT NATURA"
>> >> >
>> >> > The information in this email and in any attachments is confidential
>> and
>> >> > intended solely for the attention and use of the named addressee(s).
>>  It
>> >> > must not be disclosed to any person without the writer's authority.
>>  If
>> >> you
>> >> > are not the intended recipient, or a person responsible for
>> delivering
>> >> it to
>> >> > the intended recipient, you are not authorized to and must not
>> disclose,
>> >> > copy, distribute, or retain this message or any part of it.
>> >> >
>> >> >> -----Original Message-----
>> >> >> From: Dan Snazelle [mailto:subjectivity at hotmail.com]
>> >> >> Sent: Saturday, February 28, 2009 3:49 AM
>> >> >> To: dixon at interchange.ubc.ca; sdiy
>> >> >> Subject: RE: [sdiy] SDIY MATH GOALS--need real help!
>> >> >>
>> >> >>
>> >> >>
>> >> >>
>> >> >>> My advice to Dan and others would be not to set out to "learn
>> math",
>> >> but
>> >> >>> only electronics.  When something mathematical arises that you
>> don't
>> >> >>> understand, then learn the specific bit of math required to get
>> over
>> >> the
>> >> >>> hump -- read a wiki, ask a friend, crack a textbook, whatever --
>> but
>> >> >> always
>> >> >>> only in the service of electronics.  This way, the math will slip
>> in
>> >> by
>> >> >>> osmosis.
>> >> >>
>> >> >>
>> >> >> one question i have about this good advice is how to TELL what i AM
>> >> >> LOOKING AT. (so i can understand a specific bit)
>> >> >>
>> >> >> for example I just got the book on non linear electronics from
>> analog
>> >> >> devices. It seems full of  great stuff BUT is full of equations.
>> >> >> however how can i tell just from looking at some weird equation what
>> i
>> >> >> will need to learn to understand it
>> >> >> ? it's not as if they say "this is from algebra 2 or this is from
>> calc"
>> >> >>
>> >> >> so  that right there makes the piecemeal approach hard at times.
>> >> >>
>> >> >>
>> >> >> thanks
>> >> >
>> >
>> > _______________________________________________
>> > Synth-diy mailing list
>> > Synth-diy at dropmix.xs4all.nl
>> > http://dropmix.xs4all.nl/mailman/listinfo/synth-diy
>> >
>
>