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KEWPIE VARIATIONS (36) / 2001
(  Satoshi Kinoshita )
| Series: | Prints on canvas: Still Lifes / Landscapes | | Medium: | print on canvas | | Size (inches): | 26 x 20 (image size) | | Size (mm): | 660 x 508 (image size) | | Catalog #: | PC_052 | | Description: | From an edition of 25.Signed, titled, date, copyright, edition in magic ink on the reverse /Aside from the numbered edition of 5 artist's proofs and 2 printer's proofs.
VARIATIONS, CALCULUS OF, in mathematics. The calculus of variations arose from the attempts that were made by ~j,, mathematicians in the 17th century to solve problems olthe of which the following are typical examples. (i) It Calculus, is required to determine the form of a chain of given length, hanging frem two fixed points, by the condition that its centre of gravity must be as low as possible. This problem of the catenary was attempted without success by Galileo Galilei (1638). (ii) The resistance of a medium to the motion of a body being assumed to be a normal pressure, proportional to the square of the cosine of the angle between the normal to the surface and the direction of motion, it is required to determine the meridian curve of a surface of revolution, about an axis in the direction of motion, so that the resistance shall be the least possible. This problem of the solid of least resistance was solved by Sir Isaac Newton (1687). (iii) It is required to find a curve joining two fixed points, so that the time of descent along this curve from the higher point to the lower may be less than the time along any other curve. This problem of the brachistochrone was proposed by John (Johann) Bernoulli (1696).
The contributions of the Greek geometry to the subject consist of a few theorems discovered by one Zenodorus, of whom little Earl is known. Extracts from his writings have been prehisto served in the writings of Pappus of Alexandria and Theon Y. of Smyrna. He proved that of all curves of given perimeter the circle is that which encloses the largest area. The problems from which the subject grew up have in common the character of being concerned with the maxima and minima of quantities which can be expressed by integrals of the form (X1F(x y, y)dx, J xs in which y is an unknown function of x, and F is an assigned function of three variables, viz. x, y, and the differential coefficient of y with respect to x, here denoted by y; in special cases x or y may not be explicitly present in F, but y must be. In any such problem it is required to determine y as a function of x, so that the integral may be a maximum or a minimum, either absolutely or subject to the condition that another integral or like form may have a prescribed value. For example, in the problem of the catenary, the integral (XIy(1 +ylldx must be a minimum, while the integral (Xl(1 +y)f dx has a given value. When, as in this example, the length of the sought curve is given, the problem is described as isoperilnetrlc. At the end of the first memoir by James (Jakob) Bernoulli on the infinitesimal calculus (1690), the problem of determining the form of a flexible chain was proposed. Gottfried Wilhelm Leibnitz gave the solution in 1691, and stated that the centre of gravity is lower for this curve than for any other of the same length joinin the same two points. The first step towards a theory of suc problems was taken by James Bernoulli (1697) in his solution of the problem of the brachistochrone. He pointed out that if a curve, as a whole, possesses the maximal or minimal property, every part of the curve must itself possess the same property. Beyond the discussion of special problems, nothing was attempted for many years.
The first general theory of such problems was sketched by Leonhard Euler in 1736, and was more fully developed by him in his Euler treatise Methodus inveniendi - . - published in 1744.
He generalized the problems proposed by his predecessors by admitting under the sign of integration differential coefficients of order higher than the first. To express the condition that an integral of the form (XIF(x y, y, y, - - - y())dx may be a maximum or minimum, he required that, when y is changed into y+u, where u is a function of x, but is everywhere infinitely small, the integral should be unchanged. Resolving the integral into a sum of elements, he transformed this condition into an equation of the form I3F daF di3F d aFi ~ . +(i)~~j =0,
and he concluded that the differential equation obtained by equating to zero the expression in the square brackets must be satisfied. This equation is in general of the 2nth order, and the 2fl arbitrary constants which are contained in the complete ,primitive must be adjusted to satisfy the conditions that y, y, y, . . . y(~i) have given values at the limits of integration. If the function y is required also to satisfy the condition that another integral of the same form as the above, but containing a function 4, instead of F, may have a prescribed value, Euler achieved his purpose by replacing F in the differential equation by ~ and adjusting the constantA so that the condition may be satisfied. This artifice is known as the isoperimetric rule or rule of the undetermined multiplier. Euler illustrated his methods by a large number of examples.
*To be continued.
-1911 edition of the Encyclopedia Britannica
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