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Future Plans:

March 8 , 2010

I've done as much as I can do at the moment on the Fourier the latest version is Fourier 3 and much of it is "placeholders". More preliminary math will be studied before I proceed further with it. No date.

As to the future of the website itself, this depends on the future of the Flash Player. Apple has dismissed the tool in no uncertain terms, there have been real difficulties with the processing power of mobile devices and the problems with the last version of the authoring tool were very, very bad. Adobe promises better. It may be a no holds barred battle for market share, no prisoners taken. For what I am doing, there is nothing better than Flash though most of it could've been done in Flash MX (Player version 6), my interest in working with it depends on continued communication to a wide enough audience to make it worthwhile. I expect to buy the CS5 version in presumably a few months and keep my eyes open on the possibilities. As usual, everything changes.

July 23, 2009

The immediate goal, has been an accurate, popularized Fourier animation by September 6, 2009, the start of the school year here and a season when the greatest interest in the website can be generated.

Apparently, this goal will not be reached. My mathematics is still inadequate,

To avoid the pressures of an unrealistic deadline I have put up what seems the most presentable preliminary animated version I can, Fourier 2 based on an earlier animation (Calculus 2, now obsolete) which has links to WWW sites seemingly most useful to learning the math thus enabling those both interested and capable to take the matter further if desired. The objective is that such information can be found on the Web, how and by whom is immaterial.

Thereafter, I will improve upon the animation with a new deadline of Februay 2010 for completion, historically another good season for generating interest.

The material I am now looking at is outlined as follows:

General Theme

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http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.0020439

Joel E. Cohen
Cohen JE (2004) Mathematics Is Biology's Next Microscope, Only Better

... analysis, including the calculus of Newton and Leibniz and probability theory, is the line between ancient thought and modern thought. Without an understanding of the concepts of analysis, especially the concept of a limit, it is not possible to grasp much of modern science, technology, or economic theory.

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Fourier

A search of links from

http://www.intute.ac.uk/sciences/cgi-bin/search.pl?term1=fourier&gateway=mathematics&limit=0&submit.x=9&submit.y=11

http://www.intute.ac.uk/cgi-bin/redir.pl?url=http://www.dwsimpson.com/fourieranalysis.html&handle=1001934612-18346

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http://hyperphysics.phy-astr.gsu.edu/hbase/audio/Fourier.html

Rod Nave's Hyperphysics puts everything in small static, relatively easy to handle, modules with good navigation.

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/powstr.html#c2

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/powstr.html#c1
How much power is transported by a string wave?
Energy in a String Wave

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http://www.astro-med.com/knowledge/fourier.html

verbal description - Fourier analysis

Astro-Med is a world leader in the data acquisition and recording market and has been developing and manufacturing recording systems since 1969.
(the description is from their knowledge base)

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http://ccrma-www.stanford.edu/~lantz/fourier.html

What I learned in Music 320
Very rarely (at Stanford, at least) do you see an explanation of Fourier Transforms which makes sense!

The Fourier Transform
In a nutshell, this is it: the Fourier Transform is a projection of any function onto complex exponentials of the form exp(jwx), where w is the frequency. Mathematically, the integral of the product of two functions is an inner product and the complex exponentials are a convenient set of orthogonal basis functions for an arbitrary function space. That's Fourier's theorem.

...the DFT (Discrete Fourier Transform) is a convenient way of breaking down a signal into its frequency components...
...efficient algorithms (called Fast Fourier Transforms) exist to compute the DFT...

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(ror - I am quite sure that at present I do not make sense of the first paragraph above... however, due to recent studies there is some danger that I might do so in the distant, but not too distant future.

As an acoustic instrumentalist I think of this material as stagecraft in the sense that performance is most often heard through earplugs these days. Those musicians using electronics as their instrument likely approach this differently... I would.)

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http://www.phy.ntnu.edu.tw/oldjava/OTHERS/fourier2/index.html

Fourier Series Approximation

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The Open University provides high-quality university education to all. - The Open University is incorporated by Royal Charter (RC 000391), an exempt charity in England & Wales and a charity registered in Scotland (SC 038302).

Broadening access to education is taken to another level with OpenLearn, our major new open content initiative. OpenLearn makes a range of OU educational resources freely available on the internet, with state-of-the-art learning support and collaboration tools to connect learners and educators.

http://openlearn.open.ac.uk/course/view.php?id=2790

Introduction
This unit is concerned with the technique of expressing a periodic function as a sum of terms, where each term is a constant, a sine function or a cosine function. There is a strong analogy with the technique of expressing a (non-periodic) function as a Taylor series, which is a sum of terms that are powers of the independent variable(s); in both cases, working with just the first few terms generally gives a useful approximation. This unit assumes the following background knowledge: the definition of the period; forced oscillations and resonance; integration by parts.

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http://www.youtube.com/watch?v=gZNm7L96pfY

Lecture 1 | The Fourier Transforms and its Applications

Stanford University
Video Lecture by Professor Brad Osgood for the Electrical Engineering course. July 3, 2008

(first 9 min are for the class only, actually can skip to 16:30)
Fourier Series intro
math analysis of periodic phenomena
FourierTransform a limiting case of Fourier Series...

(This is for people more advanced than I am, so I'm restricted to introductory material and even there will have to be cautious about what I can learn at this level.)

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Derivatives

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mathtv.com

http://www.youtube.com/watch?v=cRfNOg9Q22U&feature=channel

Calculus: Derivatives 0
The definition of a derivative
September 26, 2007

(ror - there are several other very short derivative videos here which I found an excellent introductory overview. Presumably he is "selling his product" here - for my purposes these are quite useful though each individual may approach the subject differently and have different requirements. Here, it will not take long to find out if the material is suitable for you.)

http://www.youtube.com/watch?v=JwY4lpejaWw&NR=1
Calculus: Derivatives 1

et. seq.

Charles Mckeague
Home Page
http://www.mathtv.com/#

multiple student vids <= 1 min each
multiple perspectives - gives you a choice of approach

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http://www.youtube.com/watch?v=jbIQW0gkgxo&feature=SeriesPlayList&p=590CCC2BC5AF3BC1

Lec 1 | MIT 18.01 Single Variable Calculus, Fall 2007
Derivatives, slope, velocity, rate of change
vid: 51:32
Prof. David Jerison

(ror - Fast paced - for talented MIT types.
However:
The only prerequisites are high school algebra and trigonometry.)

Syllabus:
"The basic objective of Calculus is to relate small-scale (differential) quantities to large-scale (integrated) quantities. This is accomplished by means of the Fundamental Theorem of Calculus. Students should demonstrate an understanding of the integral as a cumulative sum, of the derivative as a rate of change, and of the inverse relationship between integration and differentiation."

Vid 25:40 - 26:40
Perception of Difficulty
paraphrased
'Problems in calculus are generally asked "in context" and there are many other things going on. And so the little piece of the problem which is calculus is actually fairly routine and has to be isolated and gotten through. All the rest of it relies on everything else you've learned in mathematics from grade school through high school.'

Vid 37:05 - 40:20
Hardest part of Calculus
paraphrased
'We call it single variable calculus but actually we deal with any number of variables with various interrelationships between them. That's what makes things complicated. The manipulations we do are are algebraic. When doing derivatives we just consider one variable at a time, hence the phrase "single variable".

More subtly, we are deliberately sloppy about the way we deal with variables. [for instance, the symbol "y" means different things in different parts of the explanation - here it represented both an intercept on a curve and the y axis in different formulas ] - there are different things that get slipped by under the same name. An easy mistake to make if you're not paying attention to what is represented on the diagram. - We do this constantly because it is much more complicated not to do it. It is much more convenient to allow ourselves this flexibility to change the roll that the letter represents in the middle of the computation.'

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khanacadamy complete playlist
http://www.khanacademy.org/#
http://www.khanacademy.org/#Differential%20Equations

http://www.youtube.com/watch?v=1CMDS4-PKKQ
Introduction to partial derivatives.

http://www.youtube.com/watch?v=-u0mqFqpMNY&feature=SeriesPlayList&p=19E79A0638C8D449
Partial Derivatives 2

There is also a 6 vid introductory series on "Derivatives" I did not find as helpful as many of the other videos at Khan Acadamy I will review them again later.

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http://www.mathtutor.ac.uk/Differentiation/Main.html

So far:

Differentiating Logs and Exponents

The Quotient Rule

 

 

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So near as I can tell, Real Analysis is an undergraduate Sophmore subject where you draw all the previous material together into a working tool. This book perhaps is not as deep as some science majors may need but clarifies things a bit for the rest of us. Nothing is going to be a cakewalk at this level.

http://www.macalester.edu/aratra/chapt1/chapt1_0.html

A Radical Approach to Real Analysis 2nd edition
2006 David M. Bressoud

Modify "Contents" - This is something I've had in mind for several years now, to organize the site for legibility and viewability. Its priority is high but lower than having correct and sufficient mathematical content. It would not do to organize the site around faulty or inadequate mathematics.

Some animation titles are 'spur of the moment' filenames thought up very late at night so I could save the file and get some sleep - they might mean very little to first time viewers. It may be possible to rename and reorganize them so as to make the "Contents" page a better guide for viewers

Some lists of keywords from various possible categories of organization. They are just some commonly understood topics in the disciplines listed and covered to some extent in the animations. The rows are not related.

Caveats:

a. Structuring the site into a 'Formal Contents Page' means less creative freedom and the math and programming necessary to advance the site beyond its present configuration is going to take some time.

b. There is a lot of work involved in changing all the names and buttons and then testing the site. Do I really want to do this right now? So far, no.

First we take the math, then...the path will have some foundation, take us where it will.

 

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