# Calculus Sentence Examples

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**.AdvertisementGeneral 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.AdvertisementWith 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**.AdvertisementBeyond 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.AdvertisementFor 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.He soon commenced to read the Principia, and at sixteen he had mastered a great part of that work, besides some more modern works on analytical geometry and the differential

**calculus**.The Principia gives no information on the subject of the notation adopted in the new

**calculus**, and it was not until 1693 that it was com municated to the scientific world in the second volume of Dr Wallis's works.Newton's admirers in Holland had informed Dr Wallis that Newton's method of fluxions passed there under the name of Leibnitz's

**Calculus**Di fferentialis.The work on [[Trigonometry]] and Double Algebra (1849) contains in the latter part a most luminous and philosophical view of existing and possible systems of symbolic

**calculus**.Late in 1847 De Morgan published his principal logical treatise, called Formal Logic, or the

**Calculus**of Inference, Necessary and Probable.A notion related to that of infinitesimals is presented in the Greek " method of exhaustion "; the more perfect conception, however, only dates from the 17th century, when it led to the infinitesimal

**calculus**.At the age of twenty-five he published a treatise on the integral

**calculus**, as a supplement to De l'Hopital's treatise, Des infiniment petits.A certain common agreement has been reached concerning the impossibility of regarding pleasure as the sole motive criterion and end of moral action, though different opinions still prevail as to the place occupied by pleasure in the summum bonum, and the possibility of a hedonistic

**calculus**.It must be admitted that any intelligent comprehension of the subject requires at least a grasp of the fundamental conceptions of analytical geometry and the infinitesimal

**calculus**, such as only one with some training in these subjects can be expected to have.They afford yet another great advantage in that the derivation of the results requires only the analytic operations of the infinitesimal

**calculus**.Cavalieri's " indivisibles " into the infinitesimal

**calculus**, all accomplished during the 17th century, immeasurably widened the scope of exact astronomy.The area of the ellipse is 7rab, where a, b are the semi-axes; this result may be deduced by regarding the ellipse as the orthogonal projection of a circle, or by means of the

**calculus**.He studied the properties of the cycloid, and attempted the problem of its quadrature; and in the "infinitesimals," which he was one of the first to introduce into geometrical demonstrations, was contained the fruitful germ of the differential

**calculus**.He clarified the principles of the

**calculus**by developing them with the aid of limits and continuity, and was the first to prove Taylor's theorem rigorously, establishing his well-known form of the remainder.Starting from simple elementary propositions, Steiner advances to the solution of problems which analytically require the

**calculus**of variation, but which at the time altogether surpassed the powers of that**calculus**.He discovered a simpler method of quadrating parabolas than that of Archimedes, and a method of finding the greatest and the smallest ordinates of curved lines analogous to that of the then unknown differential

**calculus**.Combinatorics was introduced by Ramsey to solve a special case of the decision problem for the first-order predicate

**calculus**.Propensity theories obey the probability of

**calculus**.He sent his results on making the differential

**calculus**into a purely algebraic theory to the Royal Society.To calculate accurately the motion of the Earth around the Sun, Isaac Newton was inspired to invent

**calculus**.In 1702 he applied the

**calculus**to clocks driven by a spring.For example the controversy over whether Newton or Leibniz discovered the

**calculus**first can easily be answered.It enables us to extend matrix algebra

**calculus**in an easy way.Have you suggestions for the best way to introduce

**calculus**?Recently, I started working on X, a

**calculus**based on the sequent**calculus**for Classical Logic.Such a question is irrelevant at the level of predicate

**calculus**.The term " derivative, taken from infinitesimal

**calculus**, refers to an isolated aspect, or " function " of a real quantity.Overview of the thesis Chapter 2 is an introduction to stochastic

**calculus**.John has worked on a range of topics related to Hamiltonian mechanics and variational

**calculus**, and in particular their application to water waves.Adams

**calculus**is aimed at the upper end of the three semester**calculus**course.The theory is a strictly smaller theory than the lambda

**calculus**.They constitute the first step toward the conception of tensor

**calculus**.We extend their result to more general structures of first order predicate

**calculus**.The combinatorics was introduced by Ramsey to solve a special case of the decision problem for the first-order predicate

**calculus**.In Morgan's refinement

**calculus**it appears with respect to initial variable values.Renal cell carcinomas can present with ' clot ' colic similar in nature to ureteric colic due to renal

**calculus**.In this approach we use differential

**calculus**and differential geometry both to filter and analyze multi-dimensional images.Students are only required to have some training in basic

**calculus**and some familiarity with first-order difference equations.Language goes no further in this direction, but the mathematical symbolism of the differential

**calculus**allows of an indefinite extension of this procedure.Systematic analysis of dental remains focuses on caries, ante-mortem tooth loss, abcesses,

**calculus**, and enamel hypoplasias.In Chapter 3, MJ is extended to a

**calculus**for an intuitionistic modal logic (due to Curry) called Lax Logic.These include complete sets of curriculum material for various college and university level courses, including

**calculus**, linear algebra and engineering mathematics.When you are doing

**calculus**you always use radians.**Calculus**, sometimes called tartar, is hardened calcified plaque.As plaque ages, it hardens and calcifies to form tartar (called dental

**calculus**).Course Description This module will extend the vector algebra of the first year to the

**calculus**of three dimensional vector algebra of the first year to the**calculus**of three dimensional vectors.Speculations concerning a

**calculus**of reasoning had at different times occupied Boole's thoughts, but it was not till the spring of 1847 that he put his ideas into the pamphlet called Mathematical Analysis of Logic. Boole afterwards regarded this as a hasty and imperfect exposition of his logical system, and he desired that his much larger work, An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities (1854), should alone be considered as containing a mature statement of his views.Peacock powerfully aided the movement by publishing in 1820 A Collection of Examples of the Application of the Differential and Integral

**Calculus**.Hence, early empiricism makes ethics simply a

**calculus**of pleasures (" hedonism ").Arithmetic, algebra, and the infinitesimal

**calculus**, are sciences directly concerned with integral numbers, rational (or fractional) numbers, and real numbers generally, which include incommensurable numbers.By means of this "

**calculus**of derived functions" Lagrange hoped to give to the solution of all analytical problems the utmost "rigour of the demonstrations of the ancients"; 6 but it cannot be said that the attempt was successful.Some writers have attempted unification by treating algebra as concerned with functions, and Comte accordingly defined algebra as the

**calculus**of functions, arithmetic being regarded as the**calculus**of values.This is a particular case of Taylor's theorem (see Infinitesimal

**Calculus**).The quest of a solvent for

**calculus**in the bladder and kidneys was pursued by him as by others at the period, and he devised a form of forceps which, on the testimony of John Ranby (1703-1773), sergeant-surgeon to George II., extracted stones with "great ease and readiness."Furthermore it can be shown by the application of the

**calculus**of variations that the condition for a minimum value of the function W, is that vV = o.Starting from any ordinate ue,o, the result of integrating with regard to x through a distance 2h is (in the example considered in § 61) the same as the result of the operation 3h(I + 4E + E 2), where E r denotes the operation of changing x into x+h (see Differences,

**Calculus**oF).In mensuration, "cubature" is sometimes used to denote the volume of a solid; the word is parallel with "quadrature," to determine the area of a surface (see Mensuration; Infinitesimal

**Calculus**) .Formulae of the

**calculus**of finite differences enable us from the chronograph records to infer the velocity and retardation of the shot, and thence the resistance of the air.The Arithmetica infinitorum relates chiefly to the quadrature of curves by the so-called method of indivisibles established by Bonaventura Cavalieri in 1629 (see Infinitesimal

**Calculus**).Leibnitz (see Infinitesimal

**Calculus**).The equations to the tangent and normal at the point x' y are yy' = 2a(x+x) and aa(y - y')+y'(x - x')=o, and may be obtained by general methods (see Geometry, Analytical, and Infinitesimal

**Calculus**).The length of a parabolic arc can be obtained by the methods of the infinitesimal

**calculus**; the curve is directly quadrable, the area of any portion between two ordinates being two thirds of the circumscribing parallelogram.But from the new point of view its method was inadequate too, its contentment with an induction that merely leaves an opponent silent, when experiment and the application of a

**calculus**were within the possibilities.Leibnitz, fresh from the battle of the

**calculus**in the mathematical field, and with his conception of logic, at least in some of its aspects, as a generalized mathematic,' found a fruitful inspiration, harmonizing well with his own metaphysic, in Bacon's alphabet of nature.A curve came to be treated as a sequence of infinitesimal straight lines; a tangent as the extension of an infinitesimal chord; a surface or area as a sequence of infinitesimally narrow strips, and a solid as a collection of infinitesimally small cubes (See Infinitesimal

**Calculus**).His great work De maxims et minimis brought him into conflict with Rene Descartes, but the dispute was chiefly due to a want of explicitness in the statement of Fermat (see Infinitesimal

**Calculus**).Ultrasonic scalers are used to remove

**calculus**rapidly from the tooth surface.Sonic scalers remove

**calculus**from the surface of the tooth.Dental scaler Removing the hard

**calculus**from teeth helps eliminate the places where bacteria can lurk in the mouth.As plaque ages, it hardens and calcifies to form tartar (called dental

**calculus**).Of course, the two are linked through the fundamental theorem of

**calculus**.If one used a utilitarian

**calculus**in this your child or your dog?The course is also a vehicle for the introduction of theorems in vector

**calculus**that have wide application in physics.Course Description This module will extend the vector algebra of the first year to the

**calculus**of three dimensional vectors.In

**calculus**class, I learned how to integrate a function.Whether you want to learn all about macrame or understand how

**calculus**can change your everyday life, there are online courses for you.They offer help in Math, Algebra,

**Calculus**, Statistics, Economics, Finance, Accounting, Physics, Programming, Biology, Chemistry, and Psychology.No, sweater vests are the tacit signature look of the chess team captain, the

**calculus**expert, the valedictorian and a young lad who prefers Stephen Hawking to Curious George.Among the specific courses offered are options in botany, finite math,

**calculus**, astronomy, geology, classics, and political science.Dental

**calculus**is a very hard substance and, as such, is difficult to remove.In dentistry,

**calculus**refers to a hardened yellow or brown mineral deposit from unremoved plaque, also called tartar.In fact, for some students, the field of poetry is the equivalent of a verbal

**calculus**.Both these methods, differing from that now employed, are interesting as preliminary steps towards the method of fluxions and the differential

**calculus**.While still an undergraduate he formed a league with John Herschel and Charles Babbage, to conduct the famous struggle of "d-ism versus dot-age," which ended in the introduction into Cambridge of the continental notation in the infinitesimal

**calculus**to the exclusion of the fluxional notation of Sir Isaac Newton.He at once took a leading position in the mathematical teaching of the university, and published treatises on the Di f ferential

**calculus**(in 1848) and the Infinitesimal**calculus**(4 vols., 1852-1860), which for long were the recognized textbooks there.This latter work included the differential and integral

**calculus**, the**calculus**of variations, the theory of attractions, and analytical mechanics.For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL

**CALCULUS**; NUMBER; QUATERNIONS; VECTOR ANALYSIS.During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential

**calculus**by Newton and Leibnitz and the discovery of gravitation.At the age of nineteen he communicated to Leonhard Euler his idea of a general method of dealing with "isoperimetrical" problems, known later as the

**Calculus**of Variations.In expounding the principles of the differential

**calculus**, he started, as it were, from the level of his pupils, and ascended with them by almost insensible gradations from elementary to abstruse conceptions.The leading idea of this work was contained in a paper published in the Berlin Memoirs for 1772.5 Its object was the elimination of the, to some minds, unsatisfactory conception of the infinite from the metaphysics of the higher mathematics, and the substitution for the differential and integral

**calculus**of an analogous method depending wholly on the serial development of algebraical functions.The

**calculus**of variations lay undeveloped in Euler's mode of treating isoperimetrical problems. The fruitful method, again, of the variation of elements was introduced by Euler, but adopted and perfected by Lagrange, who first recognized its supreme importance to the analytical investigation of the planetary movements.The

**calculus**of variations is indissolubly associated with his name.To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the

**calculus**of finite differences he contributed the beautiful formula of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler.A direct and an inverse

**calculus**is thus created, the object of the former being to determine the coefficients from the generating function, of the latter to discover the generating function from the coefficients.Reference may also be made to the special articles mentioned at the commencement of the present article, as well as to the articles on Differences,

**Calculus**Of; Infinitesimal**Calculus**; Interpolation; Vector Analysis.In the applications of the

**calculus**the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion.A characteristic feature of the

**calculus**is that a meaning can be attached to a symbol of this kind by adopting a new rule, called that of regressive multiplication, as distinguished from the foregoing, which is progressive.As in quaternions, so in the extensive

**calculus**, there are numerous formulae of transformation which enable us to deal with extensive quantities without expressing them in terms of the primary units.If, in the extensive

**calculus**of the nth category, all the units (including i and the derived units E) are taken to be homologous instead of being distributed into species, we may regard it as a (2'-I)-tuple linear algebra, which, however, is not wholly associative.For instance, there are the symbols A, D, E used in the

**calculus**of finite differences; Aronhold's symbolical method in the**calculus**of invariants; and the like.By applying the method of the differential

**calculus**, we obtain cos i= { (µ 2 - 1)/(n24-2n)} as the required value; it may be readily shown either geometrically or analytically that this is a minimum.He made use of the same suppositions as Daniel Bernoulli, though his

**calculus**was established in a very different manner.This

**calculus**was first applied to the motion of water by d'Alembert, and enabled both him and Euler to represent the theory of fluids in formulae restricted by no particular hypothesis.In the notation of the

**calculus**the relations become - dH/dp (0 const) = odv /do (p const) (4) dH/dv (0 const) =odp/do (v const) The negative sign is prefixed to dH/dp because absorption of heat +dH corresponds to diminution of pressure - dp. The utility of these relations results from the circumstance that the pressure and expansion co efficients are familiar and easily measured, whereas the latent heat of expansion is difficult to determine.In the notation of the integral

**calculus**, this area is equal to f x o udx; but the notation is inconvenient, since it implies a division into infinitesimal elements, which is not essential to the idea of an area.This discovery was followed by that of the

**calculus**of partial differences, the first trials of which were published in his Reflexion sur la cause generale des vents (1747).Roberval was one of those mathematicians who, just before the invention of the infinitesimal

**calculus**, occupied their attention with problems which are only soluble, or can be most easily solved, by some method involving limits or infinitesimals, and in the solution of which accordingly the**calculus**is always now employed.For the subjects of this heading see the articles DIFFERENTIAL EQUATIONS; FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION; FUNCTION OF REAL VARIABLES; FUNCTION COMPLEX; GROUPS, THEORY OF; INFINITESIMAL

**CALCULUS**; MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS,**CALCULUS**OF.In analytical invention, and mastery over the

**calculus**, the Turin mathematician was admittedly unrivalled.From A Merely Formal Point Of View, We Have In The Barycentric

**Calculus**A Set Of " Special Symbols Of Quantity " Or " Extraordinaries " A, B, C, &C., Which Combine With Each Other By Means Of Operations And Which Obey The Ordinary Rules, And With Ordinary Algebraic Quantities By Operations X And =, Also According To The Ordinary Rules, Except That Division By An Extraordinary Is Not Used.Taylor's Methodus Incrementorum Directa et Inversa (London, 1715) added a new branch to the higher mathematics, now designated the "

**calculus**of finite differences."