Why do we need The Fourier Transforms in Physics?
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The Fourier transform is used throughout physics. All the Fourier representation is, is an expansion of some variable, function or data in terms of sines and cosines.\n\nThis is valuable because:\n1. Simple harmonic oscillators are a very valuable model in physics (in mechanics, quantum mechanics, electromagnetism, etc), and the solutions of the simple harmonic equation are sines and cosines. Therefore sines and cosines are a natural basis to use in physics. \n2. The eigenfunctions of the infinitesimal generator of translations are complex exponentials, which can be expanded easily in the sum of a sine and a cosine by Euler's formula. \n\nTherefore, an expansion in sines and cosines is very valuable, because they can be used as a parsimonius representation of many phenomena in physics. In otherwords, it is a compact way to describe many physical processes. \n\nAnother reason that the Fourier transform is important in physics is that it is very useful for solving equations in physics like the heat equation, Maxwell's equations, fluid mechanics equations, the wave equation, etc. \n\nAnother reason that the Fourier transform is useful in physics is that many important physical properties are captured in things like dispersion relations. This even lead to Schrodinger's equation. Fourier transform pair uncertainty principles are reflected in the Heisenberg uncertainty principle. \n\nHow does the Fourier transform show up in nature? All over the place, such as in the spectra of gases, stars, etc. Rainbows are a visual example of a Fourier transform in nature.
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FFT's are totally cool. Sometimes this mathematical method is run on a computer, and other times on a specialized signal-processing chip.\n\nIt is a high speed way to convert changing data, for example an audio signal (louder and softer), or a photo (pixels change as you look from point to point).\n\nThe FFT converts the data from the realm of  time (say for audio) or space (like for a still photo) into frequency. And it can also run the other way, and convert frequency data back.\n\nAll FFT techniques involve complex numbers. These numbers are represented as two 16 bit numbers. A 16 bit number has 65,536 possible combinations. Another way to describe this is that a 16 bit number has a "resolution" of 65,536 parts.\n\nDepending on what kind of data you are measuring, there are standard and also one-of-a-kind ways of converting analog information into a digital number.\n\nFor example, in the telephone industry, an international group called the CCITT sets the representation of audio signals into digital. A common standard is the CCITT recommendation G.711, "Pulse Code Modulation (PCM) of Voice Frequencies,".  This standard, voted on in Geneva, 1972 is still in widespread use in phones and audio.\n\nIn some cases, "proprietary" (private, one-of-a-kind) schemes are used to represent signal data in 16 bit binary numbers.\n\nUsually the FFT device could care less about WHAT your coding scheme is, it just chews up the incoming  16 bit pairs (16 bits for real component and a second 16 bits for the imaginary component) and spits out the answer. It is up to you to interpret the results based on your coding scheme.\n\nBest of luck!
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I'm not familiar with that particular book, but the differential equations of simple harmonic motion are interesting.  You can make the system as complex as you like.  I've studied SHM in both maths and physics.  I simulated SH oscillators with operational amplifiers; in other words an analogue computer.  First you simulate a frictionless pendulum, then you introduce friction so the oscillations die away, then you introduce forcing terms of various frequencies to see what happens; like pushing a child on a swing.  If you're really clever, you can introduce hysteresis (delay), and extra degrees of freedom (flexible joints in the pendulum), and vary the Q of the system (how sharply you can tune it to the resonant frequency).  It's about the most open-ended project I ever saw.  I did it in second-year physics.  At its simplest, it's just sixth form maths, but you can turn it into an impossibly complex exercise in chaotic systems if you're clever.
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Feynman diagrams are used to compute various quantities in an area called quantum field theory. Quantum field theory is an application of quantum mechanics to collections of particles and fields (like electromagnetic fields for example). For example, one might want to compute a "cross section", that is, the chance that a certain type of particle will be found at a certain place in an experiment where the particles are bouncing off a target. \n\nInstead of solving the probabilities using the partial differential equation known as Schrodinger's equation, Feynman formulated the problem as an integral equation (that is, instead of describing the probabilities in terms of an equation for how they change, an equation involving the sums of probabilities was produced instead). However, the integral equation cannot be solved directly, but instead is solved as a infinite series, or sum of terms. Each term in this series can be thought of as a certain kind of interaction by the particles in the system. Each of these terms in the series is represented by a Feynman diagram. A Feynman diagram is a pictorial characterization of this interaction. Feyman diagrams give an intuitive way to think about the ways these particles associate with each other and effect each other, and also make extremely challenging computations much more tractable. Each term is produced, then evaluated and summed to some desired level of accuracy. \n\nThis is only gives a very vague sketch of the basics of Feynman diagrams, but hopefully they will give you some idea about what they are good for and how they are used.
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Math is used all the time in the engineering sciences.  Many physical phenomena, including fluid flow, heat transport, and diffusion can be described by appropriate partial differential equations.  In order to solve these equations and make predictions, you need to be good at calculus.\n\nIf you limit yourself to addition, subtraction, multiplication, and division, you would be stuck with linear models for everything.  This means that any more complicated relationship between two variables could not be modeled continuously, and you would never be able to extrapolate with any confidence.  Thus development of new technology would be strictly trial and error (and you wouldn't even have a good trial and error system without statistics).\n\nHang in there!  Get through that math.  You don't have to like it, but you will inevitably appreciate the developments that math has enabled.
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http://en.wikipedia.org/wiki/Fourier_optics\nhttp://www.supercrawler.com/Facts/fourier_optics.html\n\nYou can download a optics program to learn fournier opticas in\nhttp://www.aptrio.com/Education/Science/wave-optics-on-the-computer-download-12206.html\nGood luck
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D'un point de vue mathématique, calculer la vitesse de la balle revient à résoudre un système de n équations différentielles avec conditions initiales. C'est un peu laborieux, donc dans la pratique on utilise effectivement un radar à effet Doppler, le plus souvent sous forme de canon.\n\n Ce radar est pourvu d'un émetteur à radio-fréquences (dans la gamme des micro-ondes) dont les ondes sont réfléchies par la balle en déplacement. Le délai de retour de l'onde est proportionnel à la vitesse de la balle. Ce délai est amplifié, filtré et analysé par une puce DSP qui pratique une correction d'erreurs et envoie les données analogiques à l'afficheur. \n\nA noter que les joueurs amateurs peuvent eux aussi chronométrer leur coup droit, grâce à un module de type Speed Trac qui s'accroche sur le filet. Ludique et pas trop cher.
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I don't see how components are necessary.  \n\n"r" and "v" are just shorthand for r(t) and v(t), assuming we are looking at position (r) and velocity (v) as time varying "vectors".  And v(t) is just the time derivative of r(t) (by DEFINITION), that is, v(t) is DEFINED as r'(t).\n\nSo there isn't any actual CONTENT in expanding them into components. We are just choosing different ways to symbolize the quantities involved and there isn't anything to prove.
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Yes, Richard Feynman suggested this way back when. \n\nThe original paper is available here: http://prola.aps.org/abstract/PR/v76/i6/p749_1\n\nWithout a subscription you'll only be able to read the abstract.
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Think of it as attempting to approach an impulse in the frequency domain.  The fourier transform of an impulse is a constant.  Try this:\n\nb = fir1(100,0.000001);\nstem(b)\n\nAll of the filter coefficients are positive.  Your result will look almost like a Gaussian distribution. That's the result of convolution.  Your filter coefficients are trying to "knock down" anything that is periodic.
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For recent physics and maths preprints, try here\n\nhttp://lanl.arxiv.org/\n\nor one of it's mirror sites
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see\nhttp://hyperphysics.phy-astr.gsu.edu/hbase/surten2.html#c1\n\n here you can find the answer for your question with MATHEMATICAL EXPLANATIONS.
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CIH is spot on, but you should also realise from what he said that electromagnetic waves or particles are ways of describing and calculating what a photon will do, they are not the actual photon itself (whatever that is). \n\nSometimes it easy to think that the description of an object IS the object, especially if it fits it well. If it was possible to write an equation or model which fully described a motor vehicle, you still wouldn't be able to drive it to the shops, but you would be able to accurately descibe how it drove there.\n\nIsn't physics wonderful! Philosopy and science all neatly wrapped up in one unifying theory of lots of theories!
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I've done a good bit with linear transforms in general (LaPlace,Fourier, Z), so hopefully I can help you. The most important thing for your class is to know whether or not they will allow the use of tables or not. Even the simplest function will usually require integration by parts to compute a transform directly. Usually I use tables of transforms coupled with properties of transforms (like the differentiation property, which lets you use algebra instead of calculus to solve an ordinary linear DE). I really can't go into any depth without pencil and paper. Check out the link below (it contains an example of using LaPlace transforms to solve a linear ordinary DE).
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Differential equations is very useful to electronics inductries. This is commonly used in designing electrical circuit and their responses to the signals. Input current response, voltage responses and even time decay is critical to the computation of integrated circuit responses. Without is certain electrical functions such as response time, triggering mode, active and inactive function regions will not be possible.
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In electronics, which is my specialty, determining the dynamic response of a circuit (with respect to time or with respect to frequency) often requires the use of differential and integral calculus.  In structural engineering, designing buildings or bridges to make them strong and earthquake resistant also requires calculus.  Certainly physics and mechanics rely heavily upon calculus.  I'm not sure about chemistry or biology.
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Complex numbers are used to make things simpler which may otherwise become very very complex without complex numbers. \n\napplications of complex numbers:\n\nControl Theory: In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The systems poles and zeros are then analysed in the complex plane.\n\nSignal analysis: Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals. The absolute value i.e mod z is interpreted as the amplitude and the argument arg(z) as the phase of a sine wave of given frequency.\n\nImproper integrals: In applied fields, complex analysis is often used to compute certain real-valued improper integrals by means of complex valued functions.\n\nQuantum mechanics: The complex number field is also of utmost importance in quantum mechanics since the underlying theory is based on (infinite dimensional) Hilbert spaces.\n\nRelativity: In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary.\n\nApplied mathematics: In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system.\n\nFluid dynamics: In fluid dynamics complex functions are used to describe potential flow in 2d.\n\nFractals: Certain fractals are plotted in the complex plane e.g. Mandelbrot set and Julia set.