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1. First, whatever is done should fit the subject of guitar and string and be of interest to at least some practicing musicians.
2. Supplement (at least not interfere with) the education of those studying calculus and math in more detail.
3. The original idea from July 17, 2006 (summarized)
The guitar involves variations in the strengths of several frequencies sounding simultaneously. Fourier analysis if it can be made brief and to the point, seems the best way to understand it.
With a bit of work it may be possible to create an animation which demonstrates some of the key ideas and which would satisfy the natural curiosity expected in a casual visitor to the website.
I am thinking of a rather long term commitment to create a two minute animation for the general public.
4. There will be supplemental, optional animations.
http://betterexplained.com/articles
/a-gentle-introduction-to-learning-calculus/
Arithmetic is about manipulating numbers...
Algebra finds patterns between numbers:...entire sets of numbers...
Calculus finds patterns between equations:..
Algebra & calculus are a problem-solving duo: calculus finds new equations, and algebra solves them.
http://www.ericdigests.org/pre-9217/calculus.htm
...calculus will remain the principal point of entry to most mathematically based scientific careers...
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Euler's Introductio in analysis infinitorium of 1748 is a primer on infinite sets. (,,,a preparation for calculus)
"Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that though they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use..."
http://en.wikipedia.org/wiki/Calculus
...Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of equations...
More generally, calculus ... may refer to any method or system of calculation guided by the symbolic manipulation of expressions...
...In mathematics, foundations refers to the rigorous development of a subject from precise axioms and definitions.
...There is more than one rigorous approach to the foundation of calculus. The usual one today is via the concept of limits defined on the continuum of real numbers.
...An alternative is nonstandard analysis, in which the real number system is augmented with infinitesimal and infinite numbers, as in the original Newton-Leibniz conception. The foundations of calculus are included in the field of real analysis...
...In the 19th century, infinitesimals were replaced by limits. Limits describe the value of a function at a certain input in terms of its values at nearby input. They capture small-scale behavior, just like infinitesimals, but use the ordinary real number system. In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits are easy to put on rigorous foundations, and for this reason they are usually considered to be the standard approach to calculus.
http://www.math.unl.edu/~shermiller2/calc/
... it will be especially important to understand what is useful or applicable in a particular context.
... understanding the process for solving a particular type of problem is emphasized over memorizing formulas.
...memorizing becomes completely unnecessary...you construct the necessary tools when needed.
...to be good at it, you must DO a lot of Calculus in order to be successful.
http://www.macalester.edu/aratra/chapt2/chapt2_6.html
Calculus derives its name from its use as a tool of calculation. At its most basic level, it is a collection of algebraic techniques that yield exact numerical answers to geometric problems.
***
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.
Why is calculus important?... calculus is a powerful tool to study the ways in which the variables interact.
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.
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...
...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.
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
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.
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. 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
http://www.math.hawaii.edu/~lee/calculus/
The importance of calculus is that most of the laws of science do not provide direct information about the values of variables which can be directly measured.
This is why it's important to have a mathematical way of talking about change. That's why you see the concept of the derivative used throughout science
...in the real world, you usually don't have a formula. The formula, in fact, is what you would like to have: the formula is the unknown. What you do have is some information, given by the laws of science, about the way in which the function changes.
...it provides a language, a conceptual framework for describing relationships that would be difficult to discuss in any other language.
The classical approach to the integral starts by considering the problem of finding the area under the graph of a function between points x = a and x = b on the x-axis. One deals with this problem by dividing the area under the curve up into a large number of very narrow vertical strips. One then treats each vertical strip as if it were a rectangle and adds up the resulting areas. (The result is called a Riemann sum.) Taking the limit of these Riemann sums as the width of the vertical strips is made narrower and narrower, one finds the desired area.
Thus presented, the integral is a quite formidible concept. Furthermore, the indicated calculation seems almost impossible to actually carry out in practice.
However in practice, the evaluation of integrals has nothing to do with dividing areas into little vertical strips and taking Riemann sums.
This is because the Fundamental Theorem of Calculus says that differentiation and integration are reverse operations. Using this, one computes integrals by finding anti-derivatives. In fact, if asked what an integral is, I believe that almost all students would give an answer in terms of anti-derivatives.
However when it comes to applications of integration, the Riemann sums re-appear -- ``with a vengeance'' one might almost say.
In order to derive the integral formula for each new application -- a volume of revolution, the force on a dam, the work done by a moving force -- one essentially re-invents the integral by going back to Riemann sums.
http://en.wikipedia.org/wiki/Partial_differential_equation
Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, elasticity.
Interestingly, seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic
http://www.oswego.edu/multi-campus-nsf/descartes1.htm
David Dennis
René Descartes' Curve-Drawing Devices:
[Summarized Points.]
Experiments in the Relations Between Mechanical Motion and Symbolic Language... Descartes' demonstrated the compatibility of geometrical and algebraic representation...the ability of algebraic language to represent geometry accurately...the results of symbolic algebraic manipulations are consistent with independently established geometrical results. Curves were constructed by the geometrical actions of mechanical apparatuses. After curves had been drawn Descartes introduced coordinates and then analyzed the curve-drawing actions in order to arrive at an equation that represented the curve. Equations did not create curves; curves gave rise to equations...nowhere in the Geometry did he ever graph an equation....he emphasized the importance of making strong connections between physical actions and their possible representations in diagrams and language.
http://mathforum.org/cgraph/history/descartes.html
Descartes' appendix on mathematics was called La Géometrie. Although its title means geometry, it focussed on the connections between geometry and algebra.
Descartes used reference lines to analyze the curves he studied. He also used algebra to investigate curves. ...Like Apollonius, he usually drew his curves first, then drew reference lines to analyze them with. Descartes often used reference lines that were tilted, not at right angles. Also, Descartes thought that negative numbers did not represent "real" physical quantities, so he ignored negative roots of equations, and he avoided measuring in more than one direction on a line whenever possible.
Descartes had gathered all the tools for coordinate graphing. Because of this accomplishment, he is often given credit for inventing the coordinate plane, even though he never graphed an equation.
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Math Through the Ages By William P. Berlinghoff, Fernando Quadros Gouvêa
p 172
In the geometry appendix, simply called La Geometrie, were the main ingredients for anlalytic geometry.
His main graphical device was essentially the same as the one devised by Fermat: the independent variable, now called "x", marked off along a horizontal reference line, and the dependent variable, now "y", represented by a line segment making a fixed angle with the "x" segment.
Descartes, perhaps even more than Fermat, emnphasized that the angle choice was a matter of convenience, and need not always be a right angle.
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Unifying the Universe By Hasan Padamsee
p530
"Descartes essential insight was that numbers can describe spatial location and introduce a powerful, quantitative element into geometry."..."Creating a new order for mathematical space, Descartes proceeded to fuse algebra with geometry, turning lines and curves into numbers."
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Descartes Embodied: Reading Cartesian Philosophy Through Cartesian Science
By Daniel Garber
Edition: 2, Cambridge University Press, 2001
ISBN 0521789737, 9780521789738
p309 All knowledge properly speaking "scientia", must come from intuition and deduction: completed science will have the structure of conclusions deduced from initially intuited premises. His method is a proceedure for constructing such a science
http://www.math.rutgers.edu/courses/436/Honors02/descartes.html
Although the association of algebra and geometry was proposed even by the Greeks ... no satisfying procedure had been found to merge the two disciplines into one until the development of the Cartesian plane.
http://mathforum.org/cgraph/history/notation.html
By the middle of the 1600s, all of the concepts involved in coordinate graphing had been discovered. But they hadn't just been discovered: they had also been published.
http://mathforum.org/cgraph/history/newton.html
Newton - "Enumeration of Curves of Third Degree":written 1676, published 1704, - (one of) the first collections of graphs that consistently used two perpendicular axes...included both positive and negative numbers
http://www.fordham.edu/halsall/mod/newton-optics.html
Isaac Newton, Optics, or, a Treatise of the Reflections, Refractions, IrMlections and Colours of Light, 4th ed. (London, 1730). [Capitalization and spelling modernized. ]
[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.
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.
http://www.maa.org/features /faceofcalculus.html
http://www.maa.org/features /092404bressoud.html
In spite of the pressures to take calculus while still in high school, students should never short-change their mathematical preparation in subjects such as algebra, geometry, or trigonometry. Solid mathematical preparation is far more important than exposure to calculus.
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http://ocw.mit.edu/OcwWeb/hs/calculus/calculus/
We have selected relevant material from MIT's introductory courses to support students as they study and educators as they teach the AP® Calculus curriculum.
(ror - In general, the U.S. standards for "Advance Placement" in math for (I'm guessing) 14-17 year old students. )
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http://vels.vcaa.vic.edu.au/downloads/vels_standards/velsrevisedmathematics.pdf
math standards
Victorian Essential Learning Standards
East Melbourne, Victoria, Australia
The Victorian Essential Learning Standards (VELS) replaced the Curriculum and Standards Framework (CSF) as the basis for curriculum and assessment in Victorian schools from 2006.
The VELS are based on the best practice in Victorian schools, national and international research and widespread consultation with school communities, educators, professional associations and community groups.
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http://www.utdanacenter.org/k12mathbenchmarks/downloads
/secondary_strand_may08.pdf
Apparently derived from:
http://www.achieve.org/node/337
Achieve
Created in 1996 by the nation’s governors and corporate leaders, Achieve is an independent, bipartisan, non-profit education reform organization based in Washington, D.C. that helps states raise academic standards and graduation requirements, improve assessments and strengthen accountability.
http://www.math.utoronto.ca/mathnet/questionCorner/impossconstruct.html
...if you start with some initial points whose coordinates are all rational numbers, then apply any sequence of compass-and-straightedge construction techniques, the coordinates of the points you end up with will be a very special kind of number: they will be obtainable from the rational numbers by a sequence of operations involving only addition, subtraction, multiplication, division, and the extraction of square roots.
The reason the three classical constructions (ror - 2x cube + 3x angle [+sqCircle]) are impossible is that they are asking you to be able to construct points whose coordinates are not numbers of this type.
Christos Obretenov
February 28, 2009 - link is broken, the page apparently removed but an archive version exists at:
http://web.archive.org/web/20040221170511/
http://www.math.sfu.ca/histmath/math380notes/math380.html
The Greeks were the 1st civilization concerned with proving results, instead of merely calculating them, as previous mathematicians.
“When you have mastered numbers, you will in fact no longer be reading numbers, any more than you read words when reading books You will be reading meanings ..." Harold S. Geneen - CEO of ITT (International Telephone and Telegraph) (1959-77)
http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley.art
http://mto.societymusictheory.org/issues/mto.93.0.3/toc.0.3.html
Lindley, Mark and Turner-Smith, Ronald
An Algebraic Approach to Mathematical Models of Scales
More and more in the course of the 19th century, the significance of enharmonic modulations lay not so much in their momentary effect as in the way they enabled composers to exploit the same physical scale in terms of two systems at once: harmonic and equal-division.
...
[4] According to Max Weber, there are two rational ways to construct a
system of tones: by means of harmonic relations or else by dividing the
octave into equal parts...
Max Weber, *Die rationalen und soziologischen Grundlagen der Musik*, ed. Theodor Kroyer (Munich 1921). (The English translation published in 1958 is, alas, so inadequate that it quite misrepresents Weber's thinking.)
http://www.bikexprt.com/tunings/tunings0.htm
John S. Allen:
A caution is in order: it is useful to describe musical structures through mathematics, but musical flexibility and creativity surpasses what can be described by simple mathematics. For example, the scale of a wind instrument established by the fingering may be represented through mathematics: but in the hands of a skillful player, the predefined scale serves only as the basis for a more flexible intonation controlled through the embouchure and breath control.