Calculus
July 5, 2006-updated September 26, 2007
I am working on an animation, "Calculus 2", to demonstrate the math used to determine the relative amplitudes of the partials. I am first exploring some of the symbols I think will be needed to make up an interesting animation sufficient for an introductory and relatively superficial discussion.
The actual math is fairly simple for a string with fixed ends, only sine waves are required for the basic theory and that's all the Musemath animations have used so far. However, most applications, (and most music) need both sine and cosine to describe the wave phenomena and most discussions will put the equation in the more complex form so it might as well be addressed.
I want people to spend about a minute on it, gain some knowledge, then, if interested have some links to find more ideas on the subject by more qualified authors. A guitarist might say "aha! thats why I have to hold that difficult chord, I better sign off and start practicing." An engineering student might say "OK, but I'd better do a search on "boundry conditions" if I don't want my bridge to fall down.
Here are some of the links I am looking at and learning from:
http://www.math.mun.ca/~apics/calculus/welcome.php#Sec2
NOTE- offline 09/26/07
Preparing for University Calculus
What is calculus?
Calculus is a branch of mathematics that deals with rates of change. Its roots go back as far as Ancient Greece and China, but calculus as we know it today began with Newton and Leibnitz in the 17th century. Today it is used extensively in many areas of science.
Basic ideas of calculus include the idea of limit , derivative , and integral.
The derivative of a function is its instantaneous rate of change, with respect to something else. Thus, the derivative of height , (with respect to position) is slope ; the derivative of position , (with respect to time) is velocity ; and the derivative of velocity (with respect to time) is acceleration.
The integral of a function can be thought of as the area under its graph, or as a sort of total over time. Thus, the integral of slope is (up to a constant) height ; the integral of velocity is, up to a constant, position; and the integral of acceleration (with respect to time) is velocity.
As you may have guessed, integrals and derivatives are related, and are in a sense opposites.
Now, many functions (though not all) can be represented by algebraic expressions. For instance, the area of a circle is related to its radius by the formula A= r2; and the distance that a body falls in a time t, starting at rest, is given by x = 1/2 a t^2. Given such an expression, calculus allows us to find expressions for the integral and derivative of the function, when they exist.
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Why is calculus important?
...in cases where a deterministic model is at least a good approximation, calculus is a powerful tool to study the ways in which the variables interact. Situations involving rates of change over time, or rates of change from place to place, are particularly important examples.
(ror Note: -a deterministic system is a system in which no randomness is involved in the development of future states of the system. Deterministic models thus produce the same output for a given starting condition
a non deterministic model apparently is referred to as
stochastic
1) Generally, stochastic (pronounced stow-KAS-tik, from the Greek
stochastikos, or "skilled at aiming," since stochos is a target) describes an
approach to anything that is based on probability.
(I play darts and have a great intuitive feel for this definition)
It is my understanding that there is calculus here as
well:
From Wikipedia, the free encyclopedia
Stochastic calculus is a
branch of mathematics that operates on stochastic processes. It allows a
consistent theory of integration to be defined for integrals of stochastic
processes with respect to stochastic processes. It is used to model systems that
behave randomly.
__________
You cannot just memorize everything; you have to understand it. You have to do that yourself, and be an active participant...
...material consists of a rather small number of big ideas, and a moderate number of formulae you will need to know...
... try to understand the underlying patterns.
Here are the stages by which you will learn and master a new
idea in calculus.
1. Work on problems... It's better to work for an hour or so several times a week than in one killer session. This is the heart of the learning process...
2. If you don't understand something, decide what it is that you don't understand and go for help.
3. Make sure you understood each bit before going on.
Experience has shown that success in calculus requires a
fairly high level of preparation.
(ror - this is pretty much the definition of limit for me - my algebra is too rusty and probably never sufficient in the first place, I may keep a lookout for websites which invite me to learn but the reality is, most of the work in this area will have to be done by younger, better qualified people - especially those who can test their ideas and change them as reality dictates.)
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http://www.math.temple.edu/~cow/
Calculus on the Web
COW is an internet utility for learning and practicing calculus.
It was designed at Temple by two members of the Temple University Mathematics
Department, Gerardo Mendoza and Dan Reich.
The principal purpose of COW
is to provide you, the student or interested user, with the opportunity to learn
and practice problems in calculus (and in the future other topics in
mathematics) in a friendly environment via the internet. The most important
feature of the COW is that you get to know whether your answer is correct almost
immediately. It is as if you had a tutor looking over your shoulder and helping
you along as you work. This will be true no matter where you are or what
computer you use, as long as it is connected to the internet and has a web
browser.
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http://www.humboldt.edu/~mef2/calcsites.html
A list of calculus links, some a bit dated but highly recommended is:
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http://www.zweigmedia.com/RealWorld/
Very Good.
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(ror - The following paper was found in a googleSearch for
"Locus ad tres et quattuor lineas" which I understand to be something of a
precurser to the calculus. Here are some excerpts, modified, to see if you are
interested.)
http://www.math.rutgers.edu/~cherlin/History/Papers1999/schmarge.html
Conic Sections in Ancient Greece
Ken Schmarge
...a systematic introduction to the work of the Greek
geometers involved in the development of conic section theory...
After...(Menaechmus, Aristaeus, Euclid, Archimedes)...the culmination of the subject came at the hands of Apollonius...(his) Book V introduces the idea of "maximum" and "minimum" lines to refer to tangents and normals, respectively. The propositions and relationships it proves, which today are more easily shown using differential calculus, are rigorously explored in the classic Greek geometric fashion (Heath, 1961, pp. lxxv-lxxvi)...
...With the passing of Pappus and perhaps Proclus, conics disappeared for over 1000 years until being re-born in the 15th and 16th centuries. Though the work of scientists and mathematicians, like Kepler who was both, conics evolved from a novel intellectual exercise in Ancient Greece, to a powerful modeling tool for explaining the physical laws of the universe.
____________
Appendix B. Approximate Time line of Major Figures
350 B.C.
Menaechmus
310 B.C. Aristaeus the Elder
300 B.C. Euclid
287-212 B.C.
Archimedes
262-? B.C. Apollonius
290-350 A.D. Papus
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http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html
A history of the calculus
The main ideas which underpin the calculus developed over a very long period of time indeed. The first steps were taken by Greek mathematicians.
... Archimedes, around 225 BC, made one of the most significant of the Greek contributions. His first important advance was to show that the area of a segment of a parabola is 4/3 the area of a triangle with the same base and vertex and 2/3 of the area of the circumscribed parallelogram. Archimedes constructed an infinite sequence of triangles starting with one of area A and continually adding further triangles between the existing ones and the parabola to get areas
A, A + A/4 , A + A/4 + A/16 , A + A/4 + A/16 + A/64 , ...
The area of the segment of the parabola is therefore
A(1 + 1/4 + 1/42 + 1/43 + ....) = (4/3)A.
This is the first known example of the summation of an infinite series.
Archimedes used the method of exhaustion to find an approximation to the area of a circle. This, of course, is an early example of integration which led to approximate values of pi.
________
In the Method, Archimedes described the way in which he
discovered many of his geometrical results (see [7]):-
... certain things first became clear to me by a mechanical method, although they had to be proved by geometry afterwards because their investigation by the said method did not furnish an actual proof. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.
(ror - June 6, 2006 - Descartes also used mechanical
constructions for some of his discoveries, and of course Galileo...Newton was
always messing about with his hands...Einstein is thought of as a theorist only
but then he was a violinist...anyway, one of the ideas of Musemath is virtual
mechanical constructions...not so much the finished animation, but the process
of creating them...every frame in the animation is "instantaneous" motion in the
sense of Zeno's paradoxes and The Calculus - when you start playing around with
such constructions, your understanding changes, the compromises necessary to
"make it work" alter the approach to the subject in a beneficial way. With a
drawing/animation program, some of these benefits can be obtained at a very
young age - before the technicalities and intimidations of calculus and
programming languages raise their heads)
Perhaps the brilliance of Archimedes' geometrical results is best summed up by Plutarch, who writes:-
It is not possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanations. Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearances, easy and unlaboured results. No amount of investigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required.
Heath adds his opinion of the quality of Archimedes' work [T L Heath, A history of Greek mathematics II (Oxford, 1931).]:-
The treatises are, without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.
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http://soucc.southern.cc.oh.us/home/jdavidso/MathAdvising/AboutCalculus.html
What is Calculus?
Jon Davidson
... This article attempts to explain just what calculus is about--where it came from and why it is important...
The word "calculus" comes from "rock", and also means a
stone formed in a body. People in ancient times did arithmetic with piles of
stones, so a particular method of computation in mathematics came to be known as
calculus...
(ror - Why is it often called "The" calculus?)
( ...after a very worthwhile history...)
...There are three main facets to being a successful calculus student:
--You must be good at algebra skills. It is not enough to have passed algebra, you must also remember what you learned! If you have to relearn algebra while learning calculus then the burden can overwhelm.
--Memorization of computational patterns is not enough. Some people can get by in algebra by memorization without understanding. In calculus it is quite necessary to pay attention and learn the concepts in order to apply them. This is learning at a mature level.
--You must be dedicated to study. Don't skip any classes except for the most dire reasons. Take notes. Above all, practice lots of problems, without which those concepts will not be reinforced and learned.
The author also has a music page
http://www.sscc.edu/home/jdavidso/Music/Music.html
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(ror - 060106 - I had seen this stuff about algebra before and have noted my insufficiency, The following web site is at or near the top of my list to rectify the problem (for today, July 5, 2006)
http://www.themathpage.com/index.html
The Math Page
Lawrence Spector
Bourough of Manhatten Community
College
City College of New York
(ror - You will note the last two site are community colleges designed to rectify shortcomings in the lower school levels. This is pretty much my situation although the 40 or so years since high school probably has something to do with it as well. I should think a good look at the way these topics are addressed would itself help to guide the K-12 curriculum towards improvement.)
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A collection of applets from International Education Software, they can be purchased.
http://www.ies.co.jp/math/products/calc/menu.html
http://www.ies.co.jp/math/products/trig/applets/graphFourier/graphFourier.html
http://www.ies.co.jp/math/products/trig/applets/graphSinX/graphSinX.html
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Java Components for Mathematics
Version 1.0
Hobart and
William Smith Colleges
August 2001
http://math.hws.edu/javamath/config_applets/index.html
A set of thirteen configurable JCM applets is available. Note that there are no restrictions on using these applets.
(Evaluator SimpleGraph FamiliesOfGraphs
MultiGraph
AnimatedGraph Parametric
EpsilonDelta Derivatives SecantTangent
RiemannSums IntegralCurves FunctionComposition
ScatterPlotApplet )
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Math as a Language in Its Own Right
book review
Alfred Scharff Goldhaber(reviewer)
Einstein's Heroes: Imagining the World Through the Language of Mathematics. Robyn Arianrhod. xii + 323 pp. Oxford University Press, 2005. $28.
[...Goldhaber(reviewer)
the current view of theoretical
physicists, and perhaps also mathematicians, is that integral calculus, although
mathematically equivalent to differential calculus, is more intuitively
accessible. That's why Faraday could express his concepts in words and pictures
rather than more abstruse symbols.
]
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http://www.understandingcalculus.com/
Calculus is the study of mathematically defined change...
...Change is a relative concept that can involve any pair of dimensions, time, force, mass, length, temperature etc...
...The aim is for you to understand how to apply your mind in a systematic manner towards understanding the world around you
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re: the "discovery" of calculus
http://www1.umn.edu/ships/9-1/calculus.htm
... in his History of the Calculus, published in 1949, (p. 187) Boyer wrote that "Few new branches of mathematics are the work of single individuals. The analytic geometry of Descartes and Fermat was the outgrowth of several mathematical trends which converged in the sixteenth and seventeenth centuries. . . . Far less is the development of the calculus to be ascribed to one or two men."
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re: function
http://id.mind.net/~zona/mmts/functionInstitute/functionInstitute.html
the following notes are to remind me what's on the page, it's rather friendly appearance is what makes it interesting to me.
...a function tells you how to match up one group of numbers with another group of numbers...
________
can be expressed in at least three ways.
(1) As a data table:
(2) As a graph:
(3) Or, as an
equation
________
The group, or set, of numbers that "goes into" a function is called the DOMAIN of the function.
The set of numbers that "comes out of" a function is called the
RANGE of the function.
________
...not everything with an equal sign in it is a function.
Technically speaking, not every equation is a function.
(there are:)
ONE-TO-ONE FUNCTIONS
When exactly one unique input number
yields exactly one unique output number, the equation is a function.
MANY-TO-ONE FUNCTIONS
It is still a function if two or
more different inputs yield the same output.
ONE-TO-MANY EQUATIONS (which are not functions)
The
equation is not a function if one uiique input cn get you to two or more
different outputs
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From an old textbook I bought at a yard sale
for $1. Its relevence here is to the history of calculus.
Physics
Principles and Applications
Margenau, Watson,
Montgomery
1949
(...in modern terminology...the force between any two mass particles...is an attraction that acts along the line joining the particles and has the magnitude
F = (Gm1m2)/r^2
The constant G is called the constant of universal gravitation...
...if the force between extended bodies is wanted...each of them must be regarded as decomposed into particles and the interaction between all particles must then be computed by integration. It is not correct to (...take for r) the distance between the centers of (the two masses).
Newton was well aware of this and there is some evidence to indicate that one of the motives that led Newton to invent the calculus was the desire to be able to perform this integration!
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http://www.fordham.edu/halsall/mod/newton-optics.html
[Induction]
As in mathematics, so in natural philosophy, the investigation of difficult things by the method of analysis, ought ever to precede the method of composition. This analysis consists in making experiments and observations, and in drawing general conclusions from them by induction and admitting of no objections against the conclusions, but such as are taken from experiments, or other certain truths. For hypotheses are not to be regarded in experimental philosophy. And although the arguing from experiments and observations by induction be no demonstration of general conclusions; yet it is the best way of arguing which the nature of things admits of, and may be looked upon as so much the stronger, by how much the induction is more general. And if` no exception occur from phenomena, the conclusion may be pronounced generally. But if` at any time afterwards any exception shall occur from experiments, it may then begin to be pronounced with such exceptions as occur. By this way of analysis we may proceed from compounds to ingredients, and from motions to the forces producing them; and in general, from effects to their causes, and from particular causes to more general ones, till the argument ends in the most general. This is the method of analysis: And the synthesis consists in assuming the causes discovered, and established as principles, and by them explaining the phenomena proceeding from them, and proving the explanations.
Isaac Newton, Optics, or, a Treatise of the Reflections,
Refractions, IrMlections and Colours of Light, 4th ed. (London, 1730).
[Capitalization and spelling modernized. ]
This text is part of the Internet Modern History Sourcebook.
http://www.fordham.edu/halsall/mod/modsbook.html
The Sourcebook is a collection of public domain and copy-permitted texts for introductory level classes in modern European and World history.
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http://aleph0.clarku.edu/~djoyce/java/elements/elements.html
Euclid's Elements
The text of all 13 Books is complete, and all of the figures are
illustrated using the Geometry Applet, even those in the last three books on
solid geometry that are three-dimensional.
guide sections are included in
more modern language
by
D.E.Joyce
Clark University
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The idea behind the site is to offer mathematics as well as
some fun bits, and to combine the two wherever possible.
developed and
maintained by Rod Pierce
General
The main content of the site is aimed
at basic math skills. However you will find some more complex stuff, and some
easier bits. Hopefully there should be something for everybody.
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http://www.wmueller.com/precalculus/
William Mueller
The MathWorks, Inc.
...Some learners... benefit from a big-picture-first presentation when embarking on a new educational endeavour. Such a mental framework, however broadly sketched and indistinct at first, can provide a comfortable storage place for arranging new discoveries.
Am I Ready for Calculus?
...The most important precalculus concept is the notion of a functional relationship between two variable quantities. This relationship may take many forms: linear functions, power functions, exponential functions, logarithmic functions, trig functions, polynomial functions, rational functions... Functions from these basic families may be combined, transformed, and inverted to produce still more functional forms. Functions also appear in various representations: formulas, graphs, data sets... You will have to be familiar with the basic families of functions, and all of their representations, in order to succeed in your study of calculus. The concept of function underlies everything that calculus considers.
(...algebra...)
Finally, it is helpful to be acquainted with the main ideas of calculus, even before you get there. These are the ideas of rate of change and accumulation...
...The strategies that may have worked for you in previous math courses may not have worked as well for you in this course. Precalculus and calculus are more conceptual than any of the mathematics you have studied before. To succeed in calculus you must know more than "how" to do things, you must also understand "why". You must also see how the different things that you do fit into a bigger picture: understanding functions and how they behave...
Don't hurry to study calculus. (ror - good) Calculus is the lingua franca of mathematics, engineering, and all of the sciences. You want to speak it well, with genuine understanding. You want to carry out calculations involving realistic problems with confidence...
...What, then, might the derivative be a model of? It's not the model of a thing ...but rather of an idea.
...If you can provide a clear description of a changing quantity — that is, if you can provide a function — the integral will describe the amount that is accumulating. The integral models accumulations in the same way that the derivative models rates of change
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http://www.math.gatech.edu/~bourbaki/JavaProjects.html
...a collection of applets designed for use in conjuction with the calculus courses at Georgia Tech.
(ror - I went to the level IV first, to check them out - it looks like they would be interesting and even fun if I knew some of the earlier stuff - so this may be a worthwhile collection to spend some time on.)
Eric A. Carlen
(they take a while to download - but)... These applets have many features to facilitate experimentation -- both to provide feedback, and to premit the user to make changes in the set-up. For example, the badge up above the graph gives the coordinates of the cursor. The textfield at the bottom can be used to enter new functions, and one can "zoom in" for a closer view, and change the region shown with simple mouse clicks and drags...
All code is copyrighted by the authors. It is however placed here for public non-comercial use -- you are free to use it as long as you do not include it in or bundle it with (in any way) any product being sold at any price. So academic use and freeware is fine.
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http://www.calculus-help.com/index.html
W. Michael
Kelley
A different approach from the author of "The Complete Idiot's Guide to Calculus".
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http://archives.math.utk.edu/topics/calculus.html
A selection of links that has proven useful, including some of the above.
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http://www.understandingcalculus.com/chapters/01/index.php
I'll have to read this a bit further
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April 4. 2005
Calculus (for Fourier Analysis) :
Still studying "Who is Fourier? book (slow going here - p 129/409). 01/10/05 - Found one problem but discovered another one - progress has been made in its usual form. 03/21/05 I have actually done a bit of work here but scripting and programming have higher priority at the moment. (July 5, 2006 - I returned to the book and finally got past page 129 - but not much past it)
The general idea here is that if you start with the Complex wave, the amplitude and phase of the individual component waves seem to be determined by integral calculus (...areas and such). (in Circular Functions 5 - the amplitude of each wave is the same as the radius of its circle - this is generally sufficient for derivation of equal temperament scales from a stretched string which needs only sine waves but will not suffice for analysis of vocal and other sounds which use combinations of sine and cosine waves. I do not think that elipses or any other geometric animation that might illustrate it can readily be verified or drawn to any accuracy without this kind of math. And such an animation might not be as easy to understand as the math itself.
However, a lot has been forgotten, not enough was ever learned, other priorities arise, this may take a bit of time. Some basics will need reviewing and then perhaps:
The following sites explore a bit of the Descartes/Locus ad tres et quatro lineas - intermediary step per Turnbull.
Adolph Karger has a fairly good exposition of the locus problem
- see Figure 1 and surrounding paragraphs of::
http://www.heldermann-verlag.de/jgg/jgg01_05/jgg0202.pdf
http://www-personal.umich.edu/~pberman/renmath.html