Calculus, Bernoulli’ development

Derivation of a ray path for the geometrical optics of an inhomogeneous medium, given v(r) as a function of position, requires a development of mathematics beyond the calculus of Newton and Leibniz. The elapsed time becomes a functional T [x(f)] of the path x(r), which is to be determined so that ST = 0 for variations Sx(t) with fixed end-points Sxp = Sxq = 0. Problems of this kind are considered in the calculus of variations [5, 322], proposed originally by Johann Bernoulli (1696), and extended to a full mathematical theory by Euler (1744). In its simplest form, the concept of the variation Sx(t) reduces to consideration of a modified function xf (t) = x(f) + rw(f) in the limit e — 0. The function w(f) must satisfy conditions of continuity that are compatible with those of x(r). Then Sx(i) = w(l)dc and the variation of the derivative function is Sx (l) = w (f) de. 

Upstream water level (in which case we could calculate the jet velocity by Bernoulli s equation) or if it were a jet of steam from a boiler with constant steam temperature and pressure (in which case we could calculate the jet velocity by the methods developed in Chap. 8). For a fixed jet velocity, at what velocity should we run the blade to get the maximum amount of useful work from it Here we want to maximize dWldm from calculus we know that this will be a maximum when d dWldm)ldV is zero. First we replace F, using Eq. 7,43 ... 

What have been the significant historical developments in the mathematics of minimum area surfaces John Bernoulli and his student Leonhard Euler were amongst the earliest workers to apply the methods of the calculus to the solution of these problems, thus laying the foundations for a new branch of the calculus, the Calculus of Variations. In a comprehensive work published in 1744 Euler derived his well known equation for the determination of minimum area surfaces and other variational problems that require the examination of a sequence of varied surfaces. The equation, in its simplest one dimensional form, is... 

Bernoulli Johann (1667—1748) Swiss math., developer of differential, integral and exponential calculus, law of quantity of conservation - mv ( vis viva ) Bertalanffy (von) Ludwig 9Q — 9T2) Austr. born, Can. biol., research in ordaining conception in biology, inventor of general organismic system theory and comparative physiology (book Modem Theories of Development 1933)... 

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