Bernoulli, Johann

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)... 

By 1730 Bernoulli longed to return to Basle, but despite numerous attempts, he lost out in ballots for academic positions until 1732. However, in 1734 the French Academy of Sciences awarded a joint prize to Daniel and his father in recognition of their work. Johann found it difficult to admit that his son was at least his equal, and once again the house of Bernoulli was divided. 

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. 

In Johann Bernoulli s problem, the brachistochrone, it is required to find the shape of a wire such that a bead slides from point 0, 0 to xi, yi in the shortest time T under the force of gravity. The energy equation mv2 = —mgy implies v = V—2gy, so that... 

This rejection occurred in the context of a dispute carried out in 1700-01 between Homberg and Johann Bernoulli on the luminescence of mercury enclosed in barometers. 

Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands... 

During the time of Newton, Bernoulli family contributed a lot to engineering. Jacob Bernoulli (1655-1705) along with his brother Johann Bernoulli (1667-1748) founded the calculus of variations. With the help of calculus of variations, one can find an integrand function that minimize or maximize an integral. For example, consider the following integral expression (called functional, i.e., function of functions) ... 

Calculus of variations seems to have started by solving brachistochrone (shortest time) problem. In 1696, Johann Bernoulli posed the following problem. Find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction) from one point to the other in the least time. It is said... 

The Bernoulli brothers, Jakob and Johann, proved that a chain hanging from two points has the shape of a catenary, not a parabola, and that of aU possible shapes, the catenary has the lowest center of gravity and thus the minimal potential energy. 

Faculty of Mathematical and Natural Science Institute Johann Bernoulli University of Groningen The Netherlands... 

Carmady, T. and Kobus, H. 1968, Hydrodynamics by Daniel Bernoulli and Hydraulics by Johann Bernoulli, Dover Publications. Inc., New York. 

The rule is named after the 17th-century French mathematician Guillaume de I Hopital, but the rule was likely discovered by the Swiss mathematician Johann Bernoulli. 

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