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By Paul J. Nahin
What does quilting need to do with electrical circuit idea? the answer's only one of the attention-grabbing ways in which best-selling well known math author Paul Nahin illustrates the deep interaction of math and physics on the planet round us in his most modern publication of tough mathematical puzzles, Mrs. Perkins's electrical duvet. along with his trademark blend of interesting mathematical difficulties and the historic anecdotes surrounding them, Nahin invitations readers on a thrilling and informative exploration of a few of the various methods math and physics mix to create anything greatly extra strong, priceless, and fascinating than both is on its own. In a chain of short and mostly self-contained chapters, Nahin discusses a variety of themes during which math and physics are together established and together illuminating, from Newtonian gravity and Newton's legislation of mechanics to ballistics, air drag, and electrical energy. The mathematical topics diversity from algebra, trigonometry, geometry, and calculus to differential equations, Fourier sequence, and theoretical and Monte Carlo chance. every one bankruptcy comprises problems--some 3 dozen in all--that problem readers to attempt their hand at utilising what they've got realized. simply as in his different books of mathematical puzzles, Nahin discusses the historic history of every challenge, offers many examples, contains MATLAB codes, and offers whole and particular suggestions on the finish. Mrs. Perkins's electrical duvet will attract scholars drawn to new math and physics functions, lecturers searching for strange examples to take advantage of in class--and a person who enjoys renowned math books.
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Example text
Indeed, mathematicians do not count infinity as a number but rather recognize it as a (potentially rather slippery) concept. And it is not true that only mathematicians need to use great care when manipulating infinities; physicists need to be careful, as well. Let me give you a concrete example of this, from electric circuit theory. 1. The ladder is constructed from an infinite number, each, of two generally different-valued resistors, R1 and R2 . ) The question here is, what is the net input resistance R “seen’’ when “looking into’’ the two terminals marked a and b at the left?
That is, our recursive formulation of the pure reactive ladder (remember, we have set r = 0) does not have a positive real part to the input impedance, no matter how long we make the ladder. 4) for ω < 2/ LC . 6), for Feynman’s case of r = 0, we get an even more dramatic indication of that. 6) reduces to In = ωL + In−1 , 1 − ωC In−1 I1 = L h2 − 1 . C h Since ω = hω0 , we then have In = hω0 L + In−1 In−1 1 L+ = h√ √ 1 − hω0 C In−1 1 − h 1 C In−1 LC LC L C h L In−1 + =h = C 1−h C I L n−1 − h 2 In−1 + In−1 1−h C I L n−1 , or In = h L C − (h 2 − 1)In−1 1−h C I L n−1 , I1 = L h2 − 1 .
It all comes about so smoothly, in fact, that it is easy to 2 _ _ _ _ DISCUSSION 1 convince yourself that the job is done. But it is not, because that solution makes no physical sense, and it will be elementary mathematical arguments that show us that. The second example illustrates the converse: a physical problem that is easy to state but that clearly has a nonsensical mathematical solution. And it will now be physics that shows us the way out. The final example mirrors my earlier story of C, M, and P, that is, it will show you yet another example of physics “deriving’’ a mathematical theorem—probably the most famous theorem in all of mathematics, in fact.