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

[David G. Dixon]

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).  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?".  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.  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, 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".  

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
 
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> -----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"
> 
> 
> 
> 
> 
> --------------------------------------------
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> 
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> 
> (updated monthly)
> 
> http://www.soundclick.com/lossnyc.htm
> 
> 
> 
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> 
> (or for techno) http://www.myspace.com/snazelle
> 
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> 
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> 
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> 
> 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
> >