This table is a format for me to develop ideas for creating a small animation on the amplitude (size) of the harmonics on the vibrating strings of guitar chords. (wave equation/pde)

The shifting, interwoven patterns created and implied by these harmonics are an essential part of the music played on the instrument and their amplitude, size (or strength) is probably best described here with calculus (If it can be simplified beyond what I have done so far - "Calculus 2" animation)

I don't think the subject should be avoided on this website, the question is how to make it most presentable in animation format. I expect that much of the information here will not be explicit in the animation, but taken into consideration so as to avoid significant error.

There are somewhat contradictory Musemath objectives here:

1. First, whatever is done should fit the subject of (guitar and string) and be of some interest to a practicing musician.

2. Supplement (at least not interfere with) the education of those persuing the subject.

It seems to me, at this time, any small improvement in any of these things may be worth a great deal of effort. The internet may multiply the effects to the general advantage of all.

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.

http://www.maa.org/features
/faceofcalculus.html

http://www.maa.org/features
/092404bressoud.html

Treating calculus as the language of science;

1. What is basic fluency in it?

2. How is fluency obtained?

***

The subject takes some work. and significant preparation.

I am looking at the following 2 sites to bolster my agebra:

http://www.algebralab.org/

http://www.themathpage.com
/alg/algebra.htm

In general, the U.S. standards for "Advance Placement" in math for (I'm guessing) 14-17 year old students.

I'm treating the general comprehension of most of this material (regardless of when it is actually learned) as a sort of basic literacy for 'science specialists'.

It is more than should or will be covered in Musemath but may provide me with a sort of guideline to the understanding of people who know more than I do. Hopefully it will help me avoid blatant error.

In mathematics, partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and its (resp. their) partial derivatives with respect to those variables.

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.

Partial Differential Equation:.

For all the above discussion, the end result is likely to be simply a Partial Differential Equation:.

Vibrating string

If the string is stretched between two points where x=0 and x=L and u denotes the amplitude of the displacement of the string, then u satisfies the one-dimensional wave equation in the region where 0<x<L and t is unlimited. Since the string is tied down at the ends, u must also satisfy the boundary conditions

u(t, 0) = 0, u(t, L) = 0,

as well as the initial conditions

u(t, x) = f(x), u [sub t](0, x) = g(x),

The method of separation of variables for the wave equation

u[sub tt] = c^2 u[sub xx],

leads to solutions of the form

u(t, x) = T(t)X(x),

where

T^n + (k^2)(c^2)T = 0

where the constant k must be determined. The boundary conditions then imply that X is a multiple of sin kx, and k must have the form

k= n(pi)/L,

where n is an integer. Each term in the sum corresponds to a mode of vibration of the string. The mode with n=1 is called the fundamental mode, and the frequencies of the other modes are all multiples of this frequency. They form the overtone series of the string, and they are the basis for musical acoustics. The initial conditions may then be satisfied by representing f and g as infinite sums of these modes...

Specifically, the 'Wave Equation" for a... Vibrating string

Fourier Series, Fourier Integral, and Boundrry Conditions are important here and at least mentioned in the article.

... Newton claimed that his method of fluxions was conceived in 1665. Its fullest treatment is in De methodis fluxionum et serierum infinitorum, dated 1671... which was actually a letter that was never meant to be published.

http://www.math.rutgers.edu
/~cherlin/History/Papers1999
/kijewski.html

Isaac Newton

Kerry Kijewski

History of Mathematics Term Paper, Rutgers, Spring 1999

This gives some dates and titles, some helpful explanations of Newton's conception of calculus, and suggestions for further reading.

Some of the Newtonian explanations here might be animated to give some idea of his approach.

In general his work may be better suited to animated demonstrations but Leibniz' formal symbolic calculus is the accepted form today.

Where possible they might be compared in animation.

Scholium

"The foundation of the general method is contained in the preceding Lemma."

Newton's `Principia':
Book Two, Lemma II

http://scanserver.ulib.org
/is/scanserver/newton/xml
/doc.scn?rs=0&br=0.0&rt
=0&rp=http%3A%2F%2F
scanserver.ulib.org%2Fis
%2Fscanserver%2FSearch
.asp%3FgroupId%3Dnewton
%26bookId%3Dnewton%26
query%3Dbook&pg=260&
auth=c2b421b91241a34ea8

http://www.maths.tcd.ie/pub
/HistMath/People/Newton
/Principia/Bk2Lem2/PrBk2Lm2.html

These are I believe Newton's own final words on the subject, from his own revised edition of the Principa written with half a century of reflection.

The book is generally considered the most important work in science, from within its framework. an image of our place in the material world has been drawn.

By 1676, Leibniz had invented the notation that made calculus easy to learn and apply: dy/dx , INT y and INT y dx . The first account of this appears in Nova methodus... published in his own scientific journal Acta Eruditorum in 1684.

By 1690, Leibniz had discovered most of the ideas of elementary calculus, including differential equations, but he did not write up a complete treatment of this material, which was first done by L'Hospital (1651--1704) and Jean Bernoulli (1667--1748).

There is some excellent historical treatment of the subject on this site,

This apparently translates some of Leibniz's original discussions. Here is an animation (currently little more than a placeholder - eventually it may explore some ways to make the explanations a bit more appropriate for the NetAge.

Leibniz

Mathematical for Descartes was what could be grasped in a single intuition or could be reduced in clearly understood steps to such immediate apprehension.

http://www.princeton.edu
/~mike/articles/canons
/canons.htm

INFINITESIMALS AND TRANSCENDENT
RELATIONS:
THE MATHEMATICS OF
MOTION IN THE LATE
SEVENTEENTH CENTURY

Michael S. Mahoney
Princeton University

04/17/08

Descarte's ideas of visual mechanical linkages as logical steps is discussed a bit in Origins 12TET 2, page 29 and Origins 12TET 3, page 14 .

Apparently, in 1637 it was most important for Descarte to demonstrate the validity of the new algebraic operations by picturing them in terms of well known, existing geometrical methods.

"A quantitative relation," wrote Leibniz, (11) "is a way of finding one quantity by means of another", and, as will become clear below, an equation of whatever degree constituted a modus inveniendi. (modus inveniendi - a way of discovery)

...Taking algebraic equations as themselves basically intelligible, he moved to widen the concept of the equation by shifting focus from the notion of a compound relation to that of a modus inveniendi, a way of finding (one quantity by means of others), thereby encompassing the new, transcendent relations that were so interesting both mathematically and physically.

citing J.E. Hofmann

"... Leibniz strove for a technique of representation which is simplified and formalized down to the [detail] by means of appropriate symbols, yet which cannot be immediately grasped but must be learned. Whoever can acquire this has an unimaginable advantage over the uninititated, even when he has no particularly deep insights into the connections: the formalism thinks for him.

My thinking is that the animations, and probably to a greater extent their actual construction by students in some software program provides some of the useful "insights and connections."

That's pretty much the way I think I'm learning this stuff - along with this historical approach. (and I still need to try making up for some formal education lapses by going to the basic problem oriented math courses).

That the straight and the curved coalesced at the level of the infinitesimal was a premiss of the calculus, indeed its raison d'être.

,,, Newton and Leibniz knew how to correctly give the derivatives of most common functions, but they did not have a precise definition of "derivative"; they could not actually prove the theorems that they were using.

.... They explained a derivative as a quotient of two infinitesimals (i.e., infinitely small but nonzero numbers).

Ultimately, the biggest difference between the infinitesimal approach and the epsilon-delta approach is in what kind of language you use to hide the quantifiers:

The numbers epsilon and delta are "ordinary-sized", in the sense that they are not infinitely small. They are moderately small, e.g., numbers like one billionth. We look at what happens when we vary these numbers and make them smaller. In effect, these numbers are changing, so there is motion or action in our description. We can make these numbers smaller than any ordinary positive number that has been chosen in advance.

The approach of Newton, Leibniz, and Robinson involves numbers that do not need to change, because the numbers are infinitesimals -- i.e., they are already smaller than any ordinary positive number.

To a large extent, mathematics -- or any kind of abstract reasoning -- works by selectively suppressing information. We choose a notation or terminology that hides the information we're not currently concerned with, and focuses our attention on the aspects that we currently want to vary and study.

http://www.maths.tcd.ie/pub
/HistMath /People/Leibniz
/RouseBall /RB_Leibnitz.html

D.R. Wilkins

...Euler published his differential calculus book, Institutiones calculi differentialis... in 1755.

...Euler is stuck with the paradox that the quantity dx is, in some sense, both zero and
not zero. ....Essentially, he introduces some rules for the use of infinite and infinitesimal quantities,
roughly equivalent to our techniques for manipulating limits...

http://www.maa.org/editorial
/euler/how%20euler%20did
%20it%2035%20foundations
%20of%20calculus.pdf

How Euler Did It
by Ed Sandifer
Foundations of Calculus
September 2006

I have not gone very deeply into this material. (at all)

Jerome Keisler's book on Non-Standard Calculus is now available on-line (free, April 18, 2008)

http://www.math.wisc.edu
/~keisler/calc.html

http://mrfronius.com
/download4.php?file=limits.pdf

Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus

Judith V. Grabiner,

http://www.maths.uwa.edu
.au/~schultz/3M3/L27Cauchy.html

Lecture 27 The Calculus of Cauchy.

Phill Schultz
11 September, 2000

http://askville.amazon.com/simple-explanation-calculus-concept-limit
-delta-epsilon/AnswerViewer
.do?requestId=2465032

"Need a simple explanation for the calculus concept of limit, especially delta and epsilon"

Karl Theodor Wilhelm Weierstrass 1815 – 1897

At the time, there were ambiguous definitions regarding the fundamentals of calculus, hence theorems could not be properly proven. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and other eminent mathematicians such as Cauchy had only vague definitions of limits and continuity of functions. ...

Weierstrass also formulated similar definitions of limit and derivative still taught today.

http://hyperphysics.phy-astr.gsu.edu/Hbase/calc.html#calch

http://hyperphysics.phy-astr.gsu.edu/Hbase/hmat.html#hmath

Ron Nave

http://www.mc.maricopa.edu
/~dschultz/LimitStudent.html
#MapleAutoBookmark1

David Schultz,
Mesa Community College
Mesa, Arizona

http://www.mc.maricopa.edu/
~dschultz/

Limit Animations were immediately helpful to me.

Caveat, links to class websies sometimes dissapear, either because they were never intended for the non-paying non-student or possibly just because the class ended.

http://archives.math.utk.edu
/visual.calculus/0/functions.11
/index.html

http://www.calculus-help.com

Preface
"The goal is genuine comprehension of mathematics as an organic whole and as a basis for scientific thinking and acting."

P273...The idea of a functional relationship was more or less identified with the existence of a simple mathematical formula expressing the exact relationship. This concept proved too narrow for the requirements of mathematical physics...

What is Mathematics?: An Elementary Approach To Ideas and Numbers

By Richard Courant, Herbert Robbins - Contributor Ian Stewart
Published 1996 (The original 1941 edition I use is reproduced exactly and an update chapter added at the end by Stewart,)
Oxford University Press
566 pages
ISBN:0195105192

.

A principle source for some time. On re-reading a number of ideas seem to have been implanted, even if not understood on first reading. (Or in many cases on second and probably future readings for that matter - though at this point I can pretty much just look for sections that are insufficiently annotated.)

Eutocius/Eratosthenes description of mesolabium. There are 2nd hand descriptions but the original may be needed for the linkage (if it exists). Many original works are not yet on the internet.

This is the location of most complete second hand description I have found so far- it is possible some of the unavailable sections have the actual letter

The Ancient Tradition of Geometric Problems
by Wilbur Richard Knorr

(google.books p.17 Re: account of Mesolabium by Eutocius)

I'm looking to verify some concepts of "Linkage" such as used by Descartes and possibly somewhat differently in the Mesolabium animation, Origins 12TET 3

I'm satisfied with an approximate solution and the idea of a steam locomotive piston was exactly what came to mind when I thought of the linkage used in the animation.

Perhaps something else is needed for greater precision:

http://kmoddl.library.
cornell.edu/tutorials/04/

How to Draw a Straight Line
by Daina Taimina

(see also http://www.math.ru
/teacher/kempe/index.htm

On the other hand, with virtual mechanisms, the simplest linkage to transmit the idea quickly seems to me the best demonstration technique. (followed perhaps by a "virtual footnote" on the problem) - If a good description by Eratoshenes exists I may be able to avoid the complication.

Apparently the "mechanistic" approach was supplanted by Leibniz' "modus inveniendi", a way of finding (one quantity by means of others) as noted above, and perhaps later by a more refined notion of function but it would be nice to get this linkage idea correctly explained..

Calculus
>
http://www.humboldt.edu
/~mef2/calcsites.html

ListOfCalcSites

First impression of these sites - the concern is whether they can help me, i.e. outside the classroom.
__________

http://www.math.hmc.edu
/calculus/tutorials/

Online Mathematics Tutorials at Harvey Mudd College!
first impression - VG
http://www.math.hmc.edu
/calculus/tutorials/transformations/
much of the rest lost me fairly quickly
___________

http://math.smith.edu/Local/cicintro/cicintro.html

The story of the Five College Calculus Project began almost forty years ago, when the Five Colleges were only Four: Amherst, Mount Holyoke, Smith, and the large Amherst campus of the University of Massachusetts.

PDF format - not reviewed - what I'm lookoing for is quick transmission of good, organized information - There are some PDF links below but they sounded and turned out to be more appropriate for me.
_______________

http://web01.shu.edu
/projects/reals/reals.html

Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in one real variable.

biographies good - Map navigation good (v similar to "Hyperphysics" > 4.1 Series and Convergence > trouble getting to the punchline of 4.1.1 Zeno - Achilles/tortoise - on second look clicked on small ? Icon got:

"The problem with Zeno's paradox is that Zeno was uncomfortable with adding infinitely many numbers. In fact, his basic argument was: if you add infinitely many numbers, then - no matter what those numbers are - you must get infinity. If that was true, it would take Achilles infinitely long to reach the tortoise, and he would loose the race. However, reducing the infinite addition to the limit of a sequence, we have seen that this argument is false."

good for animation of "Limit" ??

___________________

http://www-cm.math.uiuc.edu/what

not reviewed - need to buy Mathematica software

____________

http://www.math.harvard.edu
/~calculus/

gone

The Calculus Consortium (at Harvard)
Harvard Mathematics Department Missing Page

a reform movement, some criticisms, books and material for sale - no online course found

_____________

Calculus, Concepts, Computers and Cooperative Learning (C4L)

gone

http://www.math.purdue.edu/~ccc/
found this summary but not the actual course:

http://www.pnc.edu/Faculty
/kschwing/C4L.html
According to this emerging theory, students need to construct their own understanding of each mathematical concept. Hence, we believe that the primary role of teaching is not to lecture, explain, or otherwise attempt to "transfer" mathematical knowledge, but to create situations for students that will foster their making the necessary mental constructions. A critical aspect of our approach is a decomposition of each mathematical concept into developmental steps following a Piagetian theory of knowledge based on observation of, and interviews with, students as they attempt to learn a concept.

_____________

8. Project CALC (Duke)

http://www.math.duke.edu
/education/proj_calc/index.html

Not reviewed due to:
We have designed this book to be viewed in the Mozilla Firefox browser with installed MathML fonts...Many of our student interactions use prepared computer algebra files. In the current version, we provide files for Maple (version 9.5 or higher) and Mathcad (version 13 or higher).

In a number of places we use animations, video clips, sound files, and Flash applets.

try at school

Material starts here
http://www.math.duke.edu
/education/calculustext/index.html

______________

9. Calculus from Graphical, Numerical, and Symbolic Points of View (Ostebee and Zorn, St. Olaf College)

http://www.stolaf.edu
/people/zorn/ozcalc

textbook, nothing found online

________________

10. JPCalculus The Java Powered Calculus Project

gone

http://www.usm.maine.edu
/~flagg/jpc/

The page that you requested is not available at this address.

_______________

11. Why Slopes and More Math, A Calculus Primer & Companion by A.Selby

gone
prob just a textbook anyway...

___________

12. The Interactive Textbook for PFP 98

http://dept.physics.upenn.edu
/courses/gladney/mathphys/java
/Contents.html

Some decent introductory language - physics oriented - written material ok but requires Maple software for problems

example:
The first application of any new theory of physics is usually to explain previously unexplained experimental results. Newton did this be deriving Kepler's Laws of planetary motion. He immediately moved on to the next step, still followed today, predict measurements which have not or can not be made directly, but which can be verified by consistency checks with things that can be or have been measured. For Newton, one of the first checks along these lines was to determine the ratio of Jupiter's mass to the earth's mass. Even without the value of the gravitational constant, G, he could do this by application of his version of Kepler's Third Law.

________________

13. Tutorial - Java Powered EXCELLENT on-line interactive tutorial - Java and Javascript resources. Many excellent real-world explanations and examples. (originally associated w/ Hofstra University)

new URL
http://www.zweigmedia.com
/RealWorld/

Finite Mathematics & Applied Calculus Resource Page

_________________

14. From Projects to Themes: The Evolution of Calculus Classes at New Mexico State University

http://www.math.nmsu.edu
/evolution.html
The webpage cannot be found

________________

15. Calculus@Internet

http://calculus.sjdccd.cc.ca.us/
The webpage cannot be found

________________

16. calculus@internet (a service of WebPrimitives Cambridge, Mass)

http://www.distancecalculus.com/

selling COURSE OFFERINGS, no material

_____________________

17. Calculus on the Web (Temple University- web problem sets/tutorials)

http://cow.math.temple.edu/

COW

An internet tutoring utility for learning and practicing calculus. COW gives the student or interested user the opportunity to learn and practice problems.

includes some preCalc

COW is a project of
Gerardo Mendoza and Dan Reich
Temple University

____________________

18. Interactive Learning in Calculus and Differential Equations (IUP)

http://www.ma.iup.edu/projects
/CalcDEMma/Summary.html

Requires Mathematica software - I require Flash and Java Applets, I want to avoid anything more than that.

The Mathematics Department at Indiana University of Pennsylvania (IUP) established a computerized learning environment, consisting of a classroom with 31 Macintosh Centris 650s and a laboratory with 12 Macintosh LCs, all equipped with Mathematica.

____________________

19. Calculus Reform at Cornell

http://math.cornell.edu
/~harelb/calc-reform.html

gone - Internet Explorer cannot display the webpage

_______________

20.The Connected Curriculum Project: Frank Wattenberg et al.'s grand vision of learning (including calculus) using the www. These three sites are co-linked as well.

Connected Curriculum Project -- Duke
http://www.math.duke.edu
/education/ccp/index.html
problems only??

Connected Curriculum Project -- Montana State
http://www.math.montana.edu
/frankw//ccp/home. htm
seems to require maple or mathmaticaof T92 graphing calc software

21. Connected Curriculum Project -- Cal Poly
http://grandmac.calpoly.edu/
Internet Explorer cannot display the webpage

_________________

22. Demonstration of materials that go with Edwards/Penney : Calculus, 5th Edition [Some nice animations and interactive features... much included of value in these demos using MathView (Mathworks - commercial software??).]

http://www.edwardspenney.com
/demo/

seems pretty commercial w/following interesting link to HS math pg which includes student outline of the couse and some of their annonymous letters to future students
http://www.kent.k12.wa.us
/staff/DavidWright/calculus
/book/index.html#P

The following links will send you to a section in the book which has been outlined by a group of three students from the 1999 and 2000 Kentridge AP Calculus Lab classes. What you will find is their perception of the section, along with ideas which may help you understand each topic. Each section also comes with a practice problem which will (hopefully) challenge you. The problem and it's solution were also developed by each group of students. The work has not yet been checked so readers should be warned that some problems or solutions may be inaccurate!

Student outline of the KR Calculus Book
http://www.kent.k12.wa.us
/staff/DavidWright/calculus
/book/index.html

(I never did this stuff myself and there are no students about so this is about as close as I get to what's going on in high schools.)

_________________

23. Cengage Learning - commercial

http://academic.cengage.com/

Thomson Learning Announces New Name – CENGAGE Learning

selling books - no access to free material

mathematics page
http://academic.cengage.com/
cengage/discipline.do?
disciplinenumber=1

______________________

24. Calculus Modules OnLine (PWS OnLine Series) - Brooks/Cole Publishing Company

http://www.brookscole.com
/engineering/math/modules
/modules.htm

I made no notes here so it will be looked at again.
____________

25. Thomson Learning Mathematics Resources

gone > see 23 above

____________

26. S.O.S. Math - Calculus

http://www.sosmath.com
/calculus/calculus.html

http://www.sosmath.com/calculus/series/intro/intro.html
good intro to sum of a series

derivative intro not too bad - can simpler formulae be used??

sequence, differentiation, limit, intros confusing

http://www.sosmath.com
/fourier/fourier2/fourier2.html
Fourier Sine and Cosine Series - no easy read for me - animated gifs show what's discussed and can relate to some of the musemath stuff - loses me at several points

http://www.sosmath.com
/fourier/fourier6/fourier6.html
Application of Fourier Series to Differential Equations

Note that we will need the complex form of Fourier series of a periodic function. Let us define this object first:...
In order to apply the Fourier technique to differential equations, we will need to have a result linking the complex coefficients of a function with its derivative.

VERYgoodLookingLinksPage
http://www.sosmath.com
/wwwsites.html

_________________________________________

MoreLinks

________________

http://www.ima.umn.edu
/~arnold/graphics.html

These are excerpts from a collection of graphical demonstrations I developed for first year calculus. Those interested in higher math may also want to visit my page of graphics for complex analysis. This page is on the list of the most frequently linked math pages according to MathSearch.

"...e is the only number for which the tangent to the graph of y=ex through the point (0,1) has slope exactly 1. The important result that the function f(x)=ex is its own derivative follows easily from this fact and the elementary laws of exponents..."

java version (not working for me 5/2/08)
http://www.ima.umn.edu/~arnold/graphics-j.html#differential
________________

http://www.hoye.net/index.htm
Charley Hoye presents a rather unconventional (and non-technical) introduction to the underlying themes of Calculus.

more like tai chi than weight training - I didn't go for it at first try
__________________

http://www.karlscalculus.org/

Karl's Calculus Tutor

There are dozens of problems worked out for you step-by-step... There is also remedial coverage of algebra topics, number systems, exponentials, logs, trig functions and trigonometry,

http://www.karlscalculus.org/why.html
...It is hard. So is music...But the view from up there is breathtaking... It gives you a vantage point on the world that you cannot have any other way. It teaches you the language you must know to understand how the wind blows, how the waters flow, how the sun shines, how music reaches your ear, how the planets cycle through the heavens, and much more. Even the ebb and flow of such human activities as population dynamics and economics are better viewed from calculus' highlands.

http://www.karlscalculus.org/calc3_0.html
"continuity" - excelent verbal description (word picture) but I was a bit exhausted already by the half way point at deltaEpsilon
__________________

more links

http://homepages.roadrunner.com
/askmrcalculus/visuallinks.html

***
Graphics for the Calculus Classroom
Java applets demonstrate numerous calculus topics

______________________

http://www.sci.wsu.edu
/idea/

IDEA is Internet Differential Equations Activities, an interdisciplinary effort to provide students and teachers around the world with computer based activities for differential equations in a wide variety of disciplines.

NO: models of fish populations etc. - and outdated
______________

http://mathforum.org
/mathed/calculus.reform.html

The Forum's Internet Mathematics Library provides a page of links to Calculus Reform resources. A few selections from these pages are offered below.

______________

http://www.ies.co.jp
/math/products/trig
/menu.html

Trig Java AppletsExcellent Java applets that dynamically illustrate trigonometry.(International Education Software)

(check out the Fourier applet)

_______________

Math 1002 calculus at University Of Aberdeen: Calculus Notes from Ian Craw.

http://www.maths.abdn.ac.uk
/~igc/tch/ma1002/index/index.html

VeryGoodNotes

(I downloaded pdf version but suggest use html for ease of navigation)

GoodQuote

"Experience shows that very few people are able to use lecture notes as a substitute for
lectures; if it were otherwise, lecturing, as a profession would have died out by now. To put
it another way, "any teacher who can be replaced by a teaching machine, deserves to be".
***

revision pg 4

...The weakness of the triangle definition of the trig functions is that it only makes sense
for positive angles up to 90 degrees. We want a much more general definition than that, because we want to be able to take trig functions of almost any angle.
A better definition goes as follows.

Draw the circle

x2 + y2 = 1

of radius 1 with centre at the origin. Let OX be the radius along the positive x-axis.
Let theta be any angle (positive or negative). Turn OX through angle theta so that it ends up at OP. Then cos theta is the x-coordinate of P and sin theta is the y-coordinate of P. i.e.

P = (cos theta; sin theta)

_____________

http://lc.brooklyn.cuny.edu/smarttutor/precalc/index.html

Copyright © Brooklyn
College Learning Center, 2006

Didn't make any notes - will look again

______________

http://mathforum.org
/library/topics/precalc/

The Best resources

The Math Forum - Math Library - Pre-Calculus
Blog posts dating back March, 2005, include "Wikis As Assessment For Learning
Tools" and "Great ...more>> · Dissemination of the Integration of Pre-calculus ...

____________

Animated Math Glossary - Math Advantage - Harcourt Brace School Publishers

http://www.harcourtschool.com
/glossary/math_advantage
/glossary8.html

A K-8 glossary of common mathematical terms, categorized by grade. Each entry has an example, most of which are animated.

An "other" section at the end of the alphabet gives tables of metric and customary measurements, time, formulas, and symbols

__________

http://www.webmath.com/

This site is composed of many math "fill-in-forms" into which you can type the math problem you're working on. Linked to these forms is a powerful set of math-solvers, that can instantly analyze your problem, and when possible, provide you with a step-by-step solution, instantly!

________________

http://www.brunnermath.com
/calculus.htm#32742593

excellent links by topic
____________

http://www.mathsnet.net/asa2/2004/c1.html

vGood but:
This site is no longer being developed. Visit the brand new version of this site

http://www.mathsnetalevel.com/
but
From 1st May 2008 access will be by paid subscription only. - (I'm 2 daysTooLate)

Need to take another look at the old site

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http://www.mathsnet.net/asa2/2004/c1.html

No notes on this one, will take another look.

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http://mathmistakes.info/index.html

Seems helpful to me - took a bit of playing around to get the idea - explanation immediately helpful - Since I make all kinds of mistakes the URL is to the homepage rather than to just the calculus mistakes page...

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http://www.ifigure.com/math/calculus/calculus.htm

calculator links - some math tutorials?? Needs another look.

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http://www.netn.net/27111.htm#MATH

National Education Telecommunications Network

MATHEMATICS

more links

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http://www.understandingcalculus.com/index.php

I've linked here before then deleted -throughout I find ideas I can use surrounded by diiscussions I don't want - will take another look - I think some of the ideas can be usefully abbreviated for my purposes...???

Understanding Calculus is a complete online introductory book that focuses on concepts.

Preface
The purpose of this book is to present Mathematics as the Science of Pure Reasoning and not as the Art of Manipulation.

A student must be able to understand how to set up an integral in a practical situation be it from electromagnetic forces to dynamic response of a skyscraper during an earthquake. Calculus is an utterly useless tool without this fundamental understanding of what integration is all about as the student will be able to play with Calculus but he or she will never know how to use it...the entire book is unique and has almost no connection to the modern textbook.

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http://eqworld.ipmnet.ru/

The international scientific-educational website EqWorld...seems somewhat above me...
links page looks good:

http://eqworld.ipmnet.ru
/en/info/mathwebs.htm

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http://www.math-atlas.org
/welcome.html

The Mathematical Atlas is a collection of articles about aspects of mathematics at and above the university level, but (usually) not at the level of current research. The goal of this collection is to introduce the subject areas of modern mathematics, to describe a few of the milestone results and topics, and to give pointers to some of the key resources where further information is to be found. Like any good atlas, we try to present several ways to look at each area and to show its relationship with neighboring areas and sub-areas.

Last modified 2002/03/14 by Dave Rusin