Theorems Sentence Examples
Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.
These theorems, which hold for the motion of a single rigid body, are true generally for a flexible system, such as considered here for a liquid, with one or more rigid bodies swimming in it; and they express the statement that the work done by an impulse is the product of the impulse and the arithmetic mean of the initial and final velocity; so that the kinetic energy is the work done by the impulse in starting the motion from rest.
If, however, we defined the strength of the source by the statement that the strength divided 1 The beginner is often puzzled by the constant appearance of the factor 47r in electrical theorems. It arises from the manner in which the unit quantity of electricity is defined.
Let us apply these theorems to a portion of a tube of electric force.
Mensuration involves the use of geometrical theorems, but it is not concerned with problems of geometrical construction.Advertisement
They are sometimes known as Guldinus's Theorems, but are more properly described as the Theorems of Pappus.
The theorems are of use, not only for finding the volumes or areas of solids or surfaces of revolution, but also, conversely, for finding centroids or centres of gravity.
These theorems may prove useful in preliminary calculations where the pressure-curve is nearly straight; but, in the absence of any observable law, the area of the pressure-curve must be read off by a planimeter, or calculated by Simpson's rule, as an indicator diagram.
In 1851 Mr Spottiswoode published in the form of a pamphlet an account of some elementary theorems on the subject.
The former of these theorems has been generally accepted and may be taken as proved, but the second has been closely criticized and probably must be modified.Advertisement
Euclid devotes his third book entirely to theorems and problems relating to the circle, and certain lines and angles, which he defines in introducing the propositions.
To the former belong the theorems (t), (2), and (3), and to the latter especially the theorem (4), and also, probably, his solution of the two practical problems. We infer, then, [t] that Thales must have known the theorem that the sum of the three angles of a triangle are equal to two right angles.
He published a number of these theorems without demonstration as a challenge to contemporary mathematicians.
The preceding theorems are purely kinematical.
The full working out is in general difficult, the comparatively simple problem of three bodies, for instance, in gravitational astronomy being still unsolved, but some general theorems can be formulated.Advertisement
The validity of the recilirocal theorems of Helmholtz and Lord Rayleigh, already referred to, is not affected by frictional forces of the kind here considered.
These theorems are too absolutely stated, and require much modification to adapt them to real life.
But the truth of Ricardo's theorems is now by his warmest admirers admitted to be hypothetical only.
Honore Fabri (Synopsis geometrica, 1669) treated of the curve and enumerated many theorems concerning it.
It occupies twenty-four octavo pages, and consists of four theorems and seven problems, some of which are identical with some of the most important propositions of the second and third sections of the first book of the Principia.Advertisement
And some years ago I lent out a manuscript containing such theorems; and having since met with some things copied out of it, I have on this occasion made it public, prefixing to it an introduction, and joining a Scholium concerning that method.
In this memoir by Gergonne, the theory of duality is very clearly and explicitly stated; for instance, we find " da p s la geometrie plane, a chaque theoreme ii en repond necessairement un autre qui s'en deduit en echangeant simplement entre eux les deux mots points et droites; tandis que dans la geometrie de l'espace ce sont les mots points et plans qu'il faut echanger entre eux pour passer d'un theoreme a son correlatif "; and the plan is introduced of printing correlative theorems, opposite to each other, in two columns.
In these papers the subject was recast and enriched by new and important theorems. through which the name of Jacobi is indissolubly associated with this branch of science.
The general equations expressing the motion of a planet considered as a material particle round a centre of attraction lead to theorems the more interesting of which will now be enunciated.
Some valuable but isolated facts and theorems had been previously discovered and proved, but it was he who first clearly grasped the idea of force as a mechanical agent, and extended to the external world the conception of the invariability of the relation between cause and effect.Advertisement
In his Discorso intorno alle cose the stanno su l'acqua, published in 1612, he used the principle of virtual velocities to demonstrate the more important theorems of hydrostatics, deducing from it the equilibrium of fluid in a siphon, and proved against the Aristotelians that the floating of solid bodies in a liquid depends not upon their form, but upon their specific gravities relative to such liquid.
Hesse, "they are, like P. Fermat's theorems, riddles to the present and future generations."
Other theorems were published in his Opera Varia, and in John Wallis's Commercium epistolicum (1658).
The position assigned to logic by Kant is not, in all probability, one which can be defended; indeed, it is hard to see how Kant himself, in consistency with the critical doctrine of knowledge, could have retained many of the older logical theorems, but the precision with which the position was stated, and the sharpness with which logic was marked off from cognate philosophic disciplines, prepared the way for the more thoughtful treatment of the whole question.
According to Mises, all of the categories, theorems, or laws of economics are implied in the action axiom.
Obviously the converse is true of Individual B. Two Theorems of welfare economics And now two theorems of welfare economics.
They commonly treat the historical method with a sort of patronizing toleration as affording useful exemplifications or illustrations of their theorems.
Their theorem prover is written in the functional programming language lisp which is also the language in which theorems are represented.
Given these, we can prove some useful theorems about the policy.
We also implement a proof checker for SPL which derives theorems in the HOL system from SPL proof scripts.
Over all geometrical theorems they would be in complete agreement, only interpreting the words in terms of their respective intuitions.
We make use of the convolution theorems for the Mellin transforms to produce analytic structures for part of the calculation.
Fourteen years later the Academie Frangaise, in ignorance of Smith's work, set the demonstration and completion of Eisenstein's theorems for five squares as the subject of their "Grand Prix des Sciences Mathematiques."
The importance of this algebra arises from the fact that in terms of such complex numbers with this definition of multiplication the utmost generality of expression, to the exclusion of exceptional cases, can be obtained for theorems which occur in analogous forms, but complicated with exceptional cases, in the algebras of real numbers and of signed real numbers.
The theorems on the composition of forces in circular motion with which it concluded formed the true prelude to Newton's Principia, and would alone suffice to establish the claim of Huygens to the highest rank among mechanical inventors.
General Arithmetical Theorems. (i.) The fundamental laws of arithmetic should be constantly borne in mind, though not necessarily stated.
In his famous Geometria (1637), which is really a treatise on the algebraic representation of geometric theorems, he founded the modern theory of analytical geometry (see Geometry), and at the same time he rendered signal service to algebra, more especially in the theory of equations.
The theorems of hydrostatics are thus true for all stationary fluids, however, viscous they may be; it is only when we come to hydrodynamics, the science of the motion of a fluid, that viscosity will make itself felt and modify the theory; unless we begin by postulating the perfect fluid, devoid of viscosity, so that the principle of the normality of fluid pressure is taken to hold when the fluid is in movement.
The proof of these theorems proceeds as before, employing the normality principle; they are required, for instance, in the determination of the liquid thrust on any portion of the bottom of a ship.
If, however, the unit point charge were defined to be that which produces a unit of electric flux through a circumscribing spherical surface or the electric force at distance r defined to be 1/47rr2, many theorems would be enunciated in simpler forms.
The general theorems which enabled him to do this, after a start had been made, are A2n = 11A„A ' n (Snell's Cyclom.), P 2A„A' n - 2A' „AZ, Gre o A 2 ” - A n +A2n or A' n +A2„ (g r1') where A „, A'„ are the areas of the inscribed and the circumscribed regular n-gons respectively.
Reciprocity theorems relate different states occurring within the same domain.
The singularity theorems imply the universe had an infinite density at some time in the past.
The course is also a vehicle for the introduction of theorems in vector calculus that have wide application in physics.
Smith, at the request of a member of the commission by which the prize was proposed, undertook in 1882 to write out the demonstration of his general theorems so far as was required to prove the results for the special case of five squares.
As a geometer he is classed by Eudemus, the greatest ancient authority, among those who "have enriched the science with original theorems, and given it a really sound arrangement."
In 1709 he entered the university of Glasgow, where he exhibited a decided genius for mathematics, more especially for geometry; it is said that before the end of his sixteenth year he had discovered many of the theorems afterwards published in his Geometria organica.
In it Maclaurin developed several theorems due to Newton, and introduced the method of generating conics which bears his name, and showed that many curves of the third and fourth degrees can be described by the intersection of two movable angles.
But the desire to obtain general enunciations of theorems without exceptional cases has led mathematicians to employ entities of ever-ascending types of elaboration.
All the world, including savages who cannot count beyond five, daily "apply" theorems of number.
Given that such observations at the surface of the sea, at intermediate levels and at the bottom are sufficiently numerous and are of a high degree of precision, general conclusions as to the movements of the ocean may be deduced from established theorems in hydrodynamics.
His mathematical bent, however, soon diverted him from legal studies, and the perusal of some of his earliest theorems enabled Descartes to predict his future greatness.
In the theory of numbers he furnished solutions of many of P. Fermat's theorems, and added some of his own.
Besides this most important contribution to the general fabric of dynamical science, we owe to Lagrange several minor theorems of great elegance, - among which may be mentioned his theorem that the kinetic energy imparted by given impulses to a material system under given constraints is a maximum.
Astronomy was also enriched by his investigations, and he was led to several remarkable theorems on conics which bear his name.
The long-sought cause of the "great inequality" of Jupiter and Saturn was found in the near approach to commensurability of their mean motions; it was demonstrated in two elegant theorems, independently of any except the most general considerations as to mass, that the mutual action of the planets could never largely affect the eccentricities and inclinations of their orbits; and the singular peculiarities detected by him in the Jovian system were expressed in the so-called "laws of Laplace."
Theorems and formulae are appropriated wholesale without acknowledgment, and a production which may be described as the organized result of a century of patient toil presents itself to the world as the offspring of a single brain.
Notwithstanding the prolixity of writers and the number of the writings, all attempts at extracting an algebraic analysis from their geometrical theorems and problems have been fruitless, and it is generally conceded that their analysis was geometrical and had little or no affinity to algebra.
Fundamental theorems in the theory of equations are to be found in the same work.
At the same time, it delights the pure theorist by the simplicity of the logic with which the fundamental theorems may be established, and by the elegance of its mathematical operations, insomuch that hydrostatics may be considered as the Euclidean pure geometry of mechanical science.