Who Is Fourier A Mathematical Adventure: How Fourier Transformed Mathematics, Physics, Engineering,

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_OC_InitNavbar("child_node":["title":"My library","url":" =114584440181414684107\u0026source=gbs_lp_bookshelf_list","id":"my_library","collapsed":true,"title":"My History","url":"","id":"my_history","collapsed":true,"title":"Books on Google Play","url":" ","id":"ebookstore","collapsed":true],"highlighted_node_id":"");Who is Fourier?: A Mathematical AdventureTransnational College of LEX.Language Research Foundation, 1995 - Mathematics - 435 pages 1 ReviewReviews aren't verified, but Google checks for and removes fake content when it's identifiedMany people give up on math in high school - they do not feel comfortable with it, or they do not see the need for it in everyday life. These "mathematically-challenged" people may have had little recourse available in the past. Now, however, there is LRF's Who is Fourier?, which takes readers gently by the hand and helps them with both simple and intimidating concepts alike. By using everyday examples it enables the reader to develop an understanding of the language of Fourier's wave analysis. For instance, Fourier Series is explained with a comparison to the contents of 'Veggie-veggie' juice! The student authors take the reader along on their adventure of discovery, creating an interactive work that gradually moves from the very basics ("What is a right triangle?") to the more complicated mathematics of trigonometry, exponentiation, differentiation, and integration. This is done in a way that is not only easy to understand, but actually enjoyable. From inside the book if (window['_OC_autoDir']) _OC_autoDir('search_form_input'); What people are saying - Write a reviewReviews aren't verified, but Google checks for and removes fake content when it's identifiedUser Review - Flag as inappropriateLearning Fourier Series, Fourier Coefficients and Periodic Wave Fourier was the most elegant thing. I have never seen something more elegantly simply put on paper. Very powerfully presented, compelling reading once you get started. Origins and logic deriving e to the x power, i is similarly so derived , Pi, and the unit radial rotating vector around the Origin that is e to the i Pi power. Differentiation, integration, and more Changed my life. I can now venture into higher mathematics able to understand McLauren Series, Taylor Series, the various Laplace transforms etc all because thie book opened a door to their understanding that never existed for me before.John Soper

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Who Is Fourier A Mathematical Adventure

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Like most real masterpieces, the paper broke rules. Fourier's intuition led him where his logic couldn't always follow. The work offended many great mathematicians, and for 15 years he fought to get it published. It didn't come out until 1822. By then it was a full book and the most important mathematical work of his age.

The Egypt adventure touched Fourier's entire life. It began a lifelong obsession with heat and with the healing powers of heat. In his later years he swathed himself, mummy-like, in his overheated Paris apartment. At the end, he died of a chronic illness he'd contracted in Egypt. But by then Fourier's work had permanently expanded the very character of engineering.

Last Friday, Dr. No&eacute and I were finally able to come up with thecorrect (very close) mathematical model of the patterns seen. Itinvolved taking the Fourier Transform of the circular apertureilluminated by a Gaussian beam of light. We used Mathematica tointegrate over the radius of the pinhole to get a graph of theintensity of the field formed at an angular position, w, on the screenwith a fiber-pinhole separation L. The integrand included a Gaussianpart, a phase shift part, and a zeroeth order Bessel Function part.We discovered the general solution to a circular aperture in the book"Principles of Optics" by Max Born and Emil Wolf. Dr. No&eacute callsthis book the Bible of optics, and I can see why-it's packed with somedeep stuff. The reason I said the model is close is because it givesaccurate patterns, but they are not in the exact separations measuredexperimentally. This may be an artifact of the way I found the zeroposition of the separation not being very good. I would gently touchthe pinhole and gently move the fiber close until I just barely feltit make contact. There is some systematic error associated with thisthat could throw off the measurements. I will have to figure out justwhat this error is (Dr. No&eacute had the idea that it may be due tothe thickness of the aperture itself), and attempt to account for it.But anyway, I got something that fits PRETTY well, so we went out forbeers that afternoon :)

With the waist size and the wavelength of the light used, one cancalculate the divergence of the beam, which is proportional to thewavelength over the waist size. A waist of 4.8 microns produces adivergence of 0.1 radians. When I graphed the hyperbola with the 3 pointsI was able to measure, if looked like this.July 16, 2008I've decided I probably need to start working towards an end result,as with 3 weeks left, I think it's a good idea to have a good goal inmind. So I've been doing some brainstorming about what exactly I wantto have/say in my talk on August 8. Its preliminary title is"Explaining an Accident," in reference to how we accidently discoveredthe diffraction phenomena I've been studying. In my talk, I'lldeffinitely want to include something about the basics of diffraction,while not including TOO much math, because I've learned that equationstend to make the audience lose interest. I'll discuss the wholepinhole phenomenon, and attempt to explain it, using some of theMathematica graphs that Nityan and I created, as well as actual photos(that I still need to take!). I learned yesterday that a lot of theother REU students deal with Fourier transforms in their projects, sothat kind of put me off a little bit about discussing that subject.However, they mostly deal with computer algorithms, while we uselenses and the Beam 2 program. So, I probably will mention it some,and the picture I included in my last post will be in the talk forsure. So with that, a tentative list of things to complete: 1. Finish scanning the beam a few more times and fitting the data to a(hopefully Gaussian) curve.2. View the Fourier plane of my setup, and attempt to recreate the imageby using the 4f setup mentioned earlier.3. Explain mathematically the diffraction patterns observed, usingFresnel/Fraunhofer ideas.4. Maybe think of something else related to diffraction toexplore/explain.5. Write up a paper about everything I've done.6. Write a web paper to post on this site.7. Make a PowerPoint presentation of "Explaining an Accident."

We've been talking a lot about far field approximations and how itdeals with Fraunhofer Diffraction, because that seems to be thesituation of the pinhole experiment we've been doing. Also,Dr. No&eacute, Dr. Cohen, and I have spoken about Fourier transformsof the pinhole and how we should use that to develop a mathematicalexpression for the diffraction pattern. However, I was basically inthe dark about what a Fourier transform actually was. There was areally cool website found that explained the basics of FT's withoutall the difficult math behind it. The link can be found on my linkspage. Fourier analysis is when you add up all the spatial frequenciesof light coming from the source(s). What you get is a Fourier plane,where rays with common angles come together at a spot in the plane.What's really cool though is that if you use this new pattern as asource, you should get the original image out! This means the Fouriertransform operation is its own inverse! We read that lenses performFourier transforms by themselves (if the source is at the front focalplane, the Fourier plane will be at the back focal plane).I learnedhow to use the program "Beam 2," which lets you set up optical systemsand send in rays at different angles and positions to see whathappens. Using Beam2, I made a 2 lens system, with 3 point sourcesof light, each with rays emitted at 5 different angles. As you can see in this BEAM2 diagram,after the first lens there are 5 spots where the light comes together.These 5 spots correspond to the 5 angles emitted by each source (thecommon angles converge together), and form the Fourier plane. Afterthis, the second lens essentially "undoes" the Fourier transform, andwhat results is the image of the original object. This setup iscalled a "4f system," because the length of the setup is 4 times thefocal length of the lenses.July 7, 2008So, it's been a while since my last post on here, sorry about thatone! It has been a really busy week. Dr. Hal Metcalf has begun togive lectures on quantum mechanics, which I'm actually really excitedabout. The first couple of talks have concentrated mostly on matrixmathematics. We talked about the easy things like multiplyingmatrices and also operators such as energy and momentum. We alsodiscussed what eigenvalues/vectors are, and we were given the homeworkassignment of learning how to find them. Good thing I took a linearalgebra class, so I have a little understanding of "eigenstuff." Itdid take a little time, however, for me to remember certain methods.He wants to try and explain the "elegant" parts before the disgustingmath parts, which is the opposite of how most classes are taught. Ihaven't had much quantum yet, so it'll be cool to learn something newand possibly get a jump on next year's courses. We also hadDr. Dominic Shneble give a talk on his research with Bose-EinsteinCondensates, which started out really cool, but got a little bit overmy head towards the end! When he went in to how individual atoms canget caught in energy wells (or something like that), he pretty muchlost me. Afterwards was nice, because we got to go into his lab andget a tour from some of the students who work there. There were lotsof mirrors, lasers and vacuum pumps, and a CCTV camera for seeing theatoms condensed together. The weird part is that it's all done withRubidium, and I've had experience with experiments with Rubidiumspectroscopy and diode lasers. So I've been working on a mini-projectlooking at the diffraction patterns of light passing through a singlemode fiber optic and a pinhole. We discovered some strange phenomenonof a dark spot occurring at the center when the pinhole gets close tothe end of the fiber, then a dark ring moves outward as it getscloser. It took a while to set it up properly, because I had to makea 3D translational stage out of 3 1D stages, but afterwards it was wayeasier to get the beam into the pinhole and line it up correctly. Thetough part now is figuring out how to quantify or organize the data toget some kind of final coherent results. I'm still working on thatone... Oh! and last weekend we went on an adventurous walk to themovie theatre. I say adventure because it was like a 2 mile walkalong the highway, about 4 feet from zooming cars, after taking awrong turn (of course) getting off campus. I no longer trust GoogleMaps with my life, as I doubt they planned on people WALKING along theroutes they give. But we saw The Hulk, with Edward Norton, and it waspretty good I must say. Always an adventure around here though. 2ff7e9595c

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Who Is Fourier A Mathematical Adventure: How Fourier Transformed Mathematics, Physics, Engineering,

Who Is Fourier A Mathematical Adventure

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