Herman. H. Goldstine's A History of the Calculus of Variations from the 17th PDF


By Herman. H. Goldstine

ISBN-10: 1461381061

ISBN-13: 9781461381068

ISBN-10: 1461381088

ISBN-13: 9781461381082

The calculus of diversifications is a topic whose starting could be accurately dated. it would be acknowledged to start in the mean time that Euler coined the identify calculus of adaptations yet this can be, after all, now not the real second of inception of the topic. it can now not were unreasonable if I had long gone again to the set of isoperimetric difficulties thought of via Greek mathemati­ cians reminiscent of Zenodorus (c. 2 hundred B. C. ) and preserved via Pappus (c. three hundred A. D. ). i have never performed this when you consider that those difficulties have been solved by way of geometric capability. in its place i've got arbitrarily selected firstly Fermat's stylish precept of least time. He used this precept in 1662 to teach how a gentle ray used to be refracted on the interface among optical media of other densities. This research of Fermat turns out to me specifically acceptable as a kick off point: He used the tools of the calculus to reduce the time of passage cif a gentle ray in the course of the media, and his procedure was once tailored through John Bernoulli to unravel the brachystochrone challenge. there were a number of different histories of the topic, yet they're now hopelessly archaic. One through Robert Woodhouse seemed in 1810 and one other through Isaac Todhunter in 1861.

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Additional info for A History of the Calculus of Variations from the 17th through the 19th Century

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11 Whiteside has chosen the point d so that BA = Aa = Da = a. Then let us set ad =~, ae = '11. The resistance on the surface generated by the rotation of the small arc BDd must be nearly the same as that on the surface generated by the rotation of BCd. , 2xy Since we could iterate this process as often as we wish, we ultimately can conclude that the value of the right-hand member of this last relation can be evaluated at any point d on the curve we wish, let us say at G, which we may view as a fixed point.

22. At the arbitrary point C on the minimizing arc he has drawn HF through C perpendicular to AH, the ordinate of C. The point D is "near to" C and CE = EF; EJ is parallel to AH and FD; Lon EJ is such that LG is the differential of EG; and EFDJ is a parallelogram. Since the arc CGD is the minimizing arc through C and D, then as we saw earlier (on p. 21), tCG + tGD = tCL + tLD, where, for instance, tCG means the time to descend from C to G along the arc CG (this equality is good through terms that are to be retained in the limit), and so tCG - tCL = tLD - tGD.

7. 1) for light and in an elegant manner saw how to transfer this principle to his own problem-perhaps this is why he was so pleased with the name "brachystochrone" since this notion of least time was basic to his approach. He certainly had at least an inkling of the importance that would attach to a careful pursuit of the analogy between optics and mechanics. 7. John Bernoulli's First Published Solution and Some Related Work 39 action, as we shall see later. 47 It is also clear that Bernoulli's method bears no resemblance to Newton's for the least resistance problem and is certainly not derived from Newton's.

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A History of the Calculus of Variations from the 17th through the 19th Century by Herman. H. Goldstine

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