Word Finder - calculus

definition

Calculation; computation.

definition

Any formal system in which symbolic expressions are manipulated according to fixed rules.

example

lambda calculus

definition

(often definite, the calculus) Differential calculus and integral calculus considered as a single subject; analysis.

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A stony concretion that forms in a bodily organ.

example

renal calculus ( = kidney stone)

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Deposits of calcium phosphate salts on teeth.

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A decision-making method, especially one appropriate for a specialised realm.

But no carefully devised calculus can take the place of insight, observation and experience.

It is not, however, necessary that the notation of the calculus should be employed throughout.

Hence, early empiricism makes ethics simply a calculus of pleasures ("hedonism").

The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work.

In 1747 he applied his new calculus to the problem of vibrating chords, the solution of which, as well as the theory of the oscillation of the air and the propagation of sound, had been given but incompletely by the geometricians who preceded him.

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.

Sidgwick holds that intuition must justify the claims of the general happiness upon the individual, though everything subsequent is hedonistic calculus.

The former was professor of mathematics at Bologna, and published, among other works, a treatise on the infinitesimal calculus.

He not only freed it from all trammels of geometrical construction, but by the introduction of the symbol b gave it the efficacy of a new calculus.

It was his just boast to have transformed mechanics (defined by him as a "geometry of four dimensions") into a branch of analysis, and to have exhibited the so-called mechanical "principles" as simple results of the calculus.

General aspects of the subject are considered under Mensuration; Vector Analysis; Infinitesimal Calculus.

If He Had Happened To Think Of Them As " Products," He Might Have Anticipated Grassmann'S Discovery Of The Extensive Calculus.

The discoveries of Johann Kepler and Bonaventura Cavalieri were the foundation upon which Sir Isaac Newton and Gottfried Wilhelm Leibnitz erected that wonderful edifice, the Infinitesimal Calculus.

Substituting for H its value from (3), and employing the notation of the calculus, we obtain the relation S - s =0 (dp /do) (dv/do),.

Jacques Bernoulli cannot be strictly called an independent discoverer; but, from his extensive and successful application of the calculus and other mathematical methods, he is deserving of a place by the side of Newton and Leibnitz.

With the marquis de l'Hopital he spent four months studying higher geometry and the resources of the new calculus.

Among these were the exponential calculus, and the curve called by him the linea brachistochrona, or line of swiftest descent, which he was the first to determine, pointing out at the same time the relation which this curve bears to the path described by a ray of light passing through strata of variable density.

Meanwhile the study of mathematics was not neglected, as appears not only from his giving instruction in geometry to his younger brother Daniel, but from his writings on the differential, integral, and exponential calculus, and from his father considering him, at the age of twenty-one, worthy of receiving the torch of science from his own hands.

He contributed two memoirs to the Philosophical Transactions, one, "Logometria," which discusses the calculation of logarithms and certain applications of the infinitesimal calculus, the other, a "Description of the great fiery meteor seen on March 6th, 1716."

He also showed that the roots of a cubic equation can be derived by means of the infinitesimal calculus.

Beyond this point, analytical methods must be adopted, and the student passes to trigonometry and the infinitesimal calculus.

The general method of constructing formulae of this kind involves the use of the integral calculus and of the calculus of finite differences.

If, as is usually the case, the ordinate throughout each strip of the trapezette can be expressed approximately as an algebraical function of the abscissa, the application of the integral calculus gives the area of the figure.

The establishment of these formulae involves the use of the integral calculus.

Discussions of the approximate calculation of definite integrals will be found in works on the infinitesimal calculus; see e.g.

For the methods involving finite differences, see references under DIFFERENCES, CALCULUS OF; and INTERPOLATION.

The determination of the shortest distance between two small circles on a sphere is given in the article Variations, Calculus Of.

The second volume (1817) relates to the Eulerian integrals, and to various integrals and series, developments, mechanical problems, &c., connected with the integral calculus; this volume contains also a numerical table of the values of the gamma function.

It will thus be seen that Legendre's works have placed him in the very foremost rank in the widely distinct subjects of elliptic functions, theory of numbers, attractions, and geodesy, and have given him a conspicuous position in connexion with the integral calculus and other branches of mathematics.

As a mathematician he occupied himself with many branches of his favourite science, more especially with higher algebra, including the theory of determinants, with the general calculus of symbols, and with the application of analysis to geometry and mechanics.

Papers published in 1776 were concerned with quartz, alum and clay and with the analysis of calculus vesicae from which for the first time he obtained uric acid.

In the pursuit of this inquiry he rashly invaded other departments of science, and much of the Common Place Book is occupied with a polemic, as vigorous as it is ignorant, against the fundamental conceptions of the infinitesimal calculus.

Wallis having meanwhile published other works and especially a comprehensive treatise on the general principles of calculus (Mathesis universalis, 1657), he might take this occasion of exposing afresh the new-fangled methods of mathematical analysis and reasserting his own earlier positions.

Pascal also distinguished himself by his skill in the infinitesimal calculus, then in the embryonic form of Cavalieri's method of indivisibles.

There has been some discussion as to the fairness of the treatment accorded by Pascal to his rivals, but no question of the fact that his initiative led to a great extension of our knowledge of the properties of the cycloid, and indirectly hastened the progress of the differential calculus.

He managed to make practical use of his calculus about his farms, and seems to have been remarkably apt in the practical application of mechanical principles.

Democritean physics without a calculus had necessarily proved sterile of determinate concrete results, and this was more than enough to ripen the naturalism of the utilitarian school into scepticism.

A mathematico-physical calculus that would work was in question.

With Hobbes logic is a calculus of marks and signs in the form of names.

The propositions which deal with actual existence are still of a unique type, with whatever limitation to the calculus.

There is a formal-symbolic logic engaged with the elaboration of a relational calculus.

Hence, and in this lies the main element of the symmetry and simplicity of the quaternion calculus, all systems of three mutually rectangular unit lines in space have the same properties as the fundamental system i, j, k.

Hamilton, in fact, remarks, 2 " regard it as an inelegance and imperfection in this calculus, or rather in the state to which it has hitherto been unfolded, whenever it becomes, or seems to become, necessary to have recourse.

Neither of these men professed to employ the calculus itself, but they recognized fully the extraordinary clearness of insight which is gained even by merely translating the unwieldy Cartesian expressions met with in hydrokinetics and in electrodynamics into the pregnant language of quaternions.

Examples will be found in textbooks of the calculus and of analytical statics.

In his desire to bring science home to the imperfectly educated he published anonymously Calculus made Easy by " F.R.S."

In the years 1815-1817 he contributed three papers on the "Calculus of Functions" to the Philosophical Transactions, and in 1816 was made a fellow of the Royal Society.

Along with Sir John Herschel and George Peacock he laboured to raise the standard of mathematical instruction in England, and especially endeavoured to supersede the Newtonian by the Leibnitzian notation in the infinitesimal calculus.

The length of any arc may be determined by geometrical considerations or by the methods of the integral calculus.

At the commencement of his new career he enriched the academical collection with many memoirs, which excited a noble emulation between him and the Bernoullis, though this did not in any way affect their friendship. It was at this time that he carried the integral calculus to a higher degree of perfection, invented the calculation of sines, reduced analytical operations to a greater simplicity, and threw new light on nearly all parts of pure mathematics.