Read e-book online 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 may be accurately dated. it'd be acknowledged to start for the time being that Euler coined the identify calculus of diversifications yet this can be, after all, no longer the real second of inception of the topic. it can no longer were unreasonable if I had long gone again to the set of isoperimetric difficulties thought of by way of Greek mathemati­ cians reminiscent of Zenodorus (c. two hundred B. C. ) and preserved via Pappus (c. three hundred A. D. ). i haven't performed this considering the fact that those difficulties have been solved through geometric capability. as an alternative i've got arbitrarily selected first of all Fermat's stylish precept of least time. He used this precept in 1662 to teach how a mild ray was once refracted on the interface among optical media of alternative densities. This research of Fermat turns out to me specially applicable as a place to begin: He used the equipment of the calculus to reduce the time of passage cif a gentle ray during the media, and his process used to be tailored via John Bernoulli to unravel the brachystochrone challenge. there were a number of different histories of the topic, yet they're now hopelessly archaic. One by way of Robert Woodhouse seemed in 1810 and one other by means of Isaac Todhunter in 1861.

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He proceeds to say that he has sent his solutions to Leibniz and asked him to serve as their judge. In August 1697 he then himself issued a list of six problems to be solved which widened yet further the class of special cases that had been considered. 56 On 15 October of the same year, probably to fulfill the conditions of the challenge by his brother James, he sent off a letter to a M. Varignon which appeared in print in December. 25 when an arbitrary ordinate PZ is a power n of the corresponding ordinate PF, but he does not consider when it depends on the corresponding arc BF, which is really the key point.

He proceeds to say that he has sent his solutions to Leibniz and asked him to serve as their judge. In August 1697 he then himself issued a list of six problems to be solved which widened yet further the class of special cases that had been considered. 56 On 15 October of the same year, probably to fulfill the conditions of the challenge by his brother James, he sent off a letter to a M. Varignon which appeared in print in December. 25 when an arbitrary ordinate PZ is a power n of the corresponding ordinate PF, but he does not consider when it depends on the corresponding arc BF, which is really the key point.

12"). 5 by rotating FGB 18 1. Fermat, Newton, Leibniz, and the Bernoullis cannot be part of a surface of least resistance, as we mentioned above. 12'}-have y > 0 and, in fact, y ~ + 00 as x ~ + 00. 8 picks the points D and G, through which the minimizing curve passes, above the axis of revolution. It is not hard to find a parametric representation for a minimizing arc in terms of the quantity q = - dx / dy. 13) a( - ~ + 2q + 3q3) if we regard x, y as functions of the parameter q. Then integrating as to that variable, we find x = -a(q2 + ~q4 -Iogq) + b == -a~(q) + b.

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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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