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Mathematical analysis or just analysis is a branch of mathematics that includes the theories of differentiation, integration, measure, limits, infinite series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions.

CONTENT : A - F , G - L , M - R , S - Z , See also , External links

Quotes[edit]

Quotes are arranged alphabetically by author

A - F[edit]

• The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible.
  • W. W. Rouse Ball, History of Mathematics, (London, 1901), p. 463

• In the entire history of Greek mathematics, all but the incomparable Archimedes and a few of the more heterodox sophists appear to have hated or feared the mathematical infinite. Analysis was thwarted when it might have prospered.
  • Eric Temple Bell, The Development of Mathematics (1940)

• I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in certain order, or that that order, has been to great extent necessary – being determined by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for the evolution had arrived.
  • George Boole in a “Treatise on Differential Equations”, 1859, pp.485 quoted in: The Mathematical Monthly, Volume 2,1860 P.70-71

• THEY who are acquainted with the present state of the theory of Symbolical Algebra, are aware, that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. This principle is indeed of fundamental importance ; and it may with safety be affirmed, that the recent advances of pure analysis have been much assisted by the influence which it has exerted in directing the current of investigation.
  • George Boole, The Mathematical Analysis of Logic, Philosophical Library, (1847), p. ii: Lead paragraph of the Introduction

• The terms synthesis and analysis are used in mathematics in a more special sense than in logic. In ancient mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in Euclid, XIII. 5, which in all probability was framed by Eudoxus: 'Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it.'
  • Florian Cajori, A History of Mathematics (1893). p. 30

• The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery of synthetic proofs or solutions.
  • Florian Cajori, A History of Mathematics (1893). p. 31

Principles Of Mathematical Analysis Review

• Mathematical Analysis is... the true rational basis of the whole system of our positive knowledge.
  • Auguste Comte, The Positive Philosophy, 1853, Bk. 1, chap. 1, as cited in: in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book, (1914), p. 224

• Every attempt to refer chemical questions to mathematical doctrines must be considered, now and always, profoundly irrational, as being contrary to the nature of the phenomena. . . . but if the employment of mathematical analysis should ever become so preponderant in chemistry (an aberration which is happily almost impossible) it would occasion vast and rapid retrogradation....
  • Auguste Comte, The Positive Philosophy, 1853

• Those skilled in mathematical analysis know that its object is not simply to calculate numbers, but that it is also employed to find the relations between magnitudes which cannot be expressed in numbers and between functions whose law is not capable of algebraic expression.
  • Antoine Augustin Cournot. Mathematical Theory of the Principles of Wealth, [Bacon, N. T.], (New York, 1897), p. 3.

• So far we have studies how, for each commodity by itself, the law of demand in connection with the conditions of production of that commodity, determines the price of it and regulates the incomes of its producers. We considered as given and invariable the prices of other commodities and the incomes of other producers; but, in reality the economic system is a whole of which the parts are connected and react on each other. An increase in the incomes of the producers of commodity A will affect the demand for commodities Band C, etc., and the incomes of their producers, and, by its reaction will involve a change in the demand for A. It seems, therefore, as if, for a complete and rigorous solution of the problems relative to some parts of the economic system, it were indispensable to take the entire system into consideration. But this would surpass the powers of mathematical analysis and of our practical methods of calculation, even if the values of all the constants could be assigned to them numerically.
  • Antoine Augustin Cournot. Researches into the Mathematical Principles of the Theory of Wealth, 1897, English transl. by Nathaniel T. Bacon. p. 137

• Machine-held strings of binary digits can simulate a great many kinds of things, of which numbers are just one kind. For example, they can simulate automobiles on a freeway, chess pieces, electrons in a box, musical notes, Russian words, patterns on a paper, human cells, colors, electrical circuits, and so on. To think of a computer as made up essentially of numbers is simply a carryover from the successful use of mathematical analysis in studying models. Most of this series of lectures has been devoted to applications of computers, and this is not the time to give details about their usefulness. I merely wish to point out certain types of things being done with computers today that could not have been done in 1945. Some of these are technological, some are intellectual.
  • George ForsytheEducational implications of the computer revolution, 1963

• Mathematical analysis is co-extensive with nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures ; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind.
  Its chief attribute is clearness; it has no marks to express confused notations. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them. If matter escapes us, as that of air and light because of its extreme tenuity, if bodies are placed far from us in the immensity of space, if man wishes to know the aspect of the heavens at successive periods separated by many centuries, if gravity and heat act in the interior of the solid earth at depths which will forever be inaccessible, mathematical analysis is still able to trace the laws of these phenomena. It renders them present and measurable, and appears to be the faculty of the human mind destined to supplement the brevity of life and the imperfection of the senses, and what is even more remarkable, it follows the same course in the study of all phenomena; it explains them in the same language, as if in witness to the unity and simplicity of the plan of the universe, and to make more manifest the unchangeable order which presides over all natural causes.
  • Joseph FourierThe Analytical Theory of Heat (1878)

• The effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory is to demonstrate these laws; it reduces all physical researches on the propagation of heat, to problems of the integral calculus, whose elements are given by experiment. No subject has more extensive relations with the progress of industry and the natural sciences; for the action of heat is always present, it influences the processes of the arts, and occurs in all the phenomena of the universe.
  • Joseph Fourier, Theory of Heat, [Freeman], (Cambridge, 1878), Chap. 1, p. 12.

• Mathematical Analysis is as extensive as nature herself.
  • Jean Baptiste Joseph Fourier in 'Fourier Analysis' by T.W.Korner quoted in Mathematical Analysis, Universities Press, 2001, p. 120

G - L[edit]

• Perhaps the least inadequate description of the general scope of modern Pure Mathematics I will not call it a definition would be to say that it deals with form, in a very general sense of the term; this would include algebraic form, functional relationship, the relations of order in any ordered set of entities such as numbers, and the analysis of the peculiarities of form of groups of operations.
  • E. W. Hobson 'Presidential Address British Association for the Advancement of Science, Section A, (1910}' in: Nature, Vol. 84, p. 287

• [D]uring the last half-century, number and measurable quantity have been separated... the idea of number alone has been recognized as the foundation upon which Mathematical Analysis rests, and the theory of extensive magnitude is now regarded as a separate department in which the methods of Analysis are applicable, but as no longer forming part of the foundation upon which Analysis itself rests.
  • E. W. Hobson, Mathematics, from the points of view of the Mathematician and of the Physicist (1912) p. 8.

• I discovered that a whole range of problems of the most diverse character relating to the scientific organization of production (questions of the optimum distribution of the work of machines and mechanisms, the minimization of scrap, the best utilization of raw materials and local materials, fuel, transportation, and so on) lead to the formulation of a single group of mathematical problems (extremal problems). These problems are not directly comparable to problems considered in mathematical analysis. It is more correct to say that they are formally similar, and even turn out to be formally very simple, but the process of solving them with which one is faced [i.e., by mathematical analysis] is practically completely unusable, since it requires the solution of tens of thousands or even millions of systems of equations for completion.
  I have succeeded in finding a comparatively simple general method of solving this group of problems which is applicable to all the problems I have mentioned, and is sufficiently simple and effective for their solution to be made completely achievable under practical conditions.
  • Leonid Kantorovich (1960) 'Mathematical Methods of Organizing and Planning Production.' Management Science, 6(4):366–422, 1960, p. 368); As cited in: Cockshott, W. Paul. 'Mises, Kantorovich and economic computation.' (2007).

• Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called induction. Newton was indebted to it for his theorem of the binomial and the principle of universal gravity.
  • Laplace, A Philosophical Essay on Probabilities, [Truscott and Emory] (New York 1902), p. 176.

M - R[edit]

• The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.
  • John von Neumann, originally published in Works of the Mind (edited by Robert B. Heywood, 1947) and reproduced in von Neumann, Collected Works (edited by A. H. Taub, 1961), Vol. I, 'The Mathematician', pp. 1–10.

• The analyst, who pursues a purely esthetic aim, helps create, just by that, a language more fit to satisfy the physicist.
  • Henri Poincaré, The Value of Science (1905) Ch. 5: Analysis and Physics, Tr.George Bruce Halsted (1907)

• L'analyse mathématique, n'est elle donc qu'un vain jeu d'esprit? Elle ne peut pas donner au physicien qu'un langage commode; n'est-ce pa là un médiocre service, dont on aurait pu se passer à la rigueur; et même n'est il pas à craindre que ce langage artificiel ne soit pas un voile interposé entre la réalité at l'oeil du physicien? Loin de là, sans ce langage, la pluspart des anaologies intimes des choses nous seraient demeurées à jamais inconnues; et nous aurions toujours ignoré l'harmonie interne du monde, qui est, nous le verrons, la seule véritable réalité objective.
  • So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn't that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist's eye? Far from that, without this language most of the intimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality.
    • Henri Poincaré in: Statistical Mechanics of Disordered Systems: A Mathematical Perspective, Cambridge University Press, 8 June 2006, p. 3

• The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.
  • Bertrand Russell, Principles of Mathematics (1903), Ch. I: Definition of Pure Mathematics, p. 5

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S - Z[edit]

• The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world wide states that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and to write.
  • H. G. Wells. Mankind in the Making, (London, 1904), pp. 191-192.

• As analysis was more cultivated, it gained a predominancy over geometry; being found to be a far more powerful instrument for obtaining results; and possessing a beauty and an evidence, which, though different from those of geometry, had great attractions for minds to which they became familiar. The person who did most to give to analysis the generality and symmetry which are now its pride, was also the person who made Mechanics analytical; I mean Euler.
  • William Whewell, History of the Inductive Sciences (1859) Vol. 1, pp. 363-363.

See also[edit]

External links[edit]

The paper pays attention to using in the preparation process of the future bachelors of mathematics an effective learning tool such as a computer. Its use is disclosed in the training materials of the course 'Mathematical Analysis', during the introduction of such concepts as the limit of a numerical sequence or function and their fixing. Information and communication technologies in teaching represent an effective means of increasing the productivity of the learning process, which allows finding new ways of solving urgent pedagogical problems and ensuring optimal management of the educational process. Information and communication technologies are indispensable tools by which teachers can qualitatively change the methods of teaching and organize the forms of the educational process in a new way, raise the level of the studied material, develop the individual traits of the trainees more qualitatively, and organize independent work.

Today we are witnessing how computerization penetrates into all aspects of our lives. There are practically no spheres of activity, wherever computers are not used: financial operations, engineering, communications and printing, modeling and construction of new facilities and more. It is difficult to imagine our life without these operating and multifunction tools. Therefore, it is important to teach various computer skills at all levels of the education system.

Using computer technology in education is a powerful new stimulus for the further development of the teaching science. There is a need for research to find new and effective techniques, methods of using computer technology that expand the possibilities of the educational process.

In recent years, pedagogical and methodological developments demonstrate the comprehensive implementation of information and communication tools in education and science. Today, many leading pedagogues turn increasingly to the use of these tools in the modern educational process, among them:

M.P. Lapchik, V.A. Dalinger, K.A. Zuev, N.V. Sholokhov, P.I. Sovertkov, I.V. Robert, S.I. Shapiro, Gershunskiy and others. Most of them take the consideration of various approaches to the use of computer technology as a learning

The computer can act as a subject of study, as a learning tool, and as a tool for scientific research. In this connection, it solves the following problems: execution of complex computing operations, analysis of the experimental results, the construction and interpretation of mathematical models, and so on.

Applications of the computer in the learning activities are mainly limited to its use as a ―big calculator‖, ―information textbook‖ tool for controlling students’ knowledge and less – as a performer of other functions. Moreover, its advantages and opportunities for learning are generally not fully disclosed. When using the computer [1], students master the learning material at their own speed, their knowledge are objectively evaluated, simultaneous feedback is carried out, they get a wide range of information to be provided in a variety of images. On the other hand, using the computer, teacher can take into account the individuality of students, pay more attention to them, get rid of routine work when searching for information, include various exercises, while spending a minimum of time for creating tasks, and as a result, spend more time for improving his intellectual level and professional skill.

No doubt, such use of computer solves many educational and psychological problems during the learning. However, the possibilities of this technical tool are much broader. In addition, information technologies have a significant impact not only on the learning process, but also on the development and education of students. They stimulate and create motivation, provide the dynamism of the learning process, stimulate learning and cognitive activity. Information technologies alone cannot always ensure efficiency of learning, but they can be a means of improving the current educational process [2].

The primary role in realization of these possibilities falls on the shoulders of teachers of higher educational institutions. In this connection, it is necessary to pay the due attention in pedagogical higher educational institutes not only to teachers of computer skills, but also to teachers of other disciplines (of humanitarian and natural–mathematical cycles) in their training the basics of modeling with the use of the application software. Teachers must be not only subject teachers, but also teachers having basics of didactics and new technologies of education.

M.P. Lapchik [3] noted that the computer science is not so much the content as the methodology. For this reason, all the teachers – as in general all the experts – must become teachers of computer skills; and computer literacy of young people must be formed in a constant communication with the computers in the classroom in all subjects, and during all the years of schooling. However, there is a need for advanced study of computer science as a specialty.

One can use such effective technical tool as the computer in all stages of study. Its use should not be limited to the laboratory studies; one can use it during lectures, practical lessons, tests, and coursework and diploma projects.

When the perception of the new academic content, computer can simulate the abstract objects, processes, phenomena, process the information quickly and accurately, etc.

In the process of understanding and generalization, computer can make a demonstrational analysis of the relationships, dependencies, etc.; it can monitor students’ independent work, formulate conclusions, hypotheses and prognoses.

When fixing the educational material, computer serves as a generator of problem situations; manages students’ independent work; analyzes the level of assimilation.

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In the application of knowledge, computer can serve as a simulator for formation of skills and abilities; provide differentiated support and control for the learners [4].

Turning on the computer in the learning process should not be chaotic and aimless. Each lesson must be thought-out; the involvement of information and communication tools must be justified by the objectives, demands of the subject, time expenditures. The development of skills in working with technical equipment must be carried out regularly throughout the course of study in high school.

The computer as a learning tool has another important advantage – its ability to represent various types of dependencies, numerical relations, and so on, in a visual form. At the lesson, the realization of obviousness principle takes place in a qualitatively different level. In addition, of particular interest is modeling of the phenomena that are inaccessible to direct observation, as well as phenomena whose description has not developed adequate mathematical apparatus, and which cannot be demonstrated within the classroom without the help of a computer.

Visualization has been not received the due attention in traditional higher education. It has been found that a person gets 90% of information about the surrounding reality through the visual analyzer, 9% – with the help of hearing, and 1% – with the help of touch. All this proves once again that the active use of the visual analyzer provides new, great possibilities to improve the quality and effectiveness of education [4]. This problem can also be solved with the help of computer tools involved in the learning process.

We give a statement: ―Who has not seen that out of 50 students at least 40 were disgusted and lost heart because of the abstract ideas, that are presented before they were becoming clear in the examples taken from everyday practice‖ [5]. The computer enters easily and naturally into the world of mathematical abstraction at the expense of their submission to the visual images, firmly imprinted in the minds of learners. Dynamic playback allows students to recall the difficult material studied earlier, it activates the formation of temporary connections (associations) in the cerebral cortex. When students see as mathematical formulations ―come alive‖ in the display, their motivation to the study increases, interest to mathematics grows, their ken expands.

Fast computers open new opportunities in teaching the foundations of mathematics (not only in pedagogical higher educational institutes). They help to disclose better the possibilities of mathematical methods and train specialists, who are able to use mathematical methods wider, fuller and faster. ―The most important problem in preparing the teacher of mathematics in pedagogical higher educational institute is the mastery by students of computer technology – writes [6] – and the ability to apply them in the study and teaching disciplines of the mathematical cycle‖.

Consider how one can use computer technology in relation to practical training in mathematical analysis in higher education.

Mathematical analysis in general takes one of the leading places in the mathematical training of teachers. The ideas and methods of analysis permeate explicitly or implicitly the whole course of algebra for grades 7-11. The course of the mathematical analysis in higher education is designed to equip students with mathematical methods of judgment, to raise their overall level of mathematical culture. It should be noted that the fundamentals of the analysis cause in general for students special difficulties in the perception of new concepts and, consequently, difficulties in the application of them to do exercises. Considering the high abstraction of educational material on mathematical analysis, it is necessary to increase the use of the visibility in its study, moreover not only in learning process of the theoretical materials, but also in the system of practice.

Principles Of Mathematical Analysis Table Of Contents

Let’s consider the theory of limits which is one of the main sections in mathematical analysis. Getting to know with it begins in elementary mathematics; where the passage to the limit is used to determine the length of circumference in a circle, volumes of bodies of rotation, and the sum of infinitely decreasing geometric progression, and so on. Operation of the passage to the limit is one of the basic operations of mathematical analysis.

The definition of the limit of a sequence or function can be carried out in stages. In addition, one can clear demonstrate these stages. It is expedient to use the theory of stage formation of mental actions.

To demonstrate the effects, in the implementation of which the object reaches its ―limit‖ position, let’s use the following dynamic images: swinging pendulum, movement of the ball when you hit it and the load suspended on a spring at the damped oscillations.

Geometrically, a sequence is represented on a number line as a series of points whose coordinates are equal to the corresponding elements of the sequence. For example, both the sequences 1/ n and

As we pointed earlier, one can use potential of computer not only at acquaintance with a new material but also in fixing the studied one. For example, it is useful in the ―theory of limits‖ to accompany the concept of the limit by graphic illustrations, in considering of which students should ―see’ whether there is a number that has the property f x A with any degree of the accuracy.

Many students at the end of this section remain unformed correspondence between the analytical limit

of a given function and its geometric image. In the educational literature and in practical classes, as a rule, problems are rarely considered in which there is a plotting of functions and definition of the limits by these images of those functions that are infinitely small, etc. In order to overcome this gap, it is expedient to use the following problems: from the depicted graphs of functions to determine which of them are infinitely small at x , at x , and at x , and which are not those (see the figure below), using the proposed images of graphs of functions, to determine those which meet the following limit relations.

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We tried to consider only one section studied in the course of mathematical analysis and to show on its example the possibility of presenting the theory with the use of information and communication tools as a means of visual and dynamic presentation.

This is a natural introduction of means in the learning process in higher educational institutes that allows students not only to learn the concepts being studied, and to fix them, to see over abstract visual images, but also equips the classroom experience in the use of your computer. The computerization of the educational process is essential for the improvement of education at all levels, provides great opportunities for deep study of the fundamentals of science, as well as intensification of the whole educational process.

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References:

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1. Gershunsky B.S. Computerization of education: problems and prospects. – M.: Nauka, 1987. – 264
2. Novoselov A.A. Formation of professional qualities of students of industrial college on integrative lessons of mathematics and informatics: diss. cand. ped. sciences. – Novosibirsk: Publish. House NSPU, 2000. – 22 p.
3. Computerization of the system of training teachers: Guidelines/Comp. MP Lapchik. – Omsk, 1988. – 30
4. Pavlov D. Tutorials on the use of computers in education. – Chelyabinsk Univ. Chelyab. region. Inst. improved. teachers. – 1990. – 140
5. Gnedenko B.V., Sirazhdinov S.H. Universities and scientific and technical progress // Coll. scientific and methodological articles on mathematics. – Moscow, 1987. – Vol. 14. – 3-11.
6. Dalinger V.A. Computer technology in teaching geometrical methodological recommendations. – Omsk Univer, OSPU, 2001. – 33
7. Bavrin I.I. Higher Mathematics. – M.: Publishing center “Academy”, 2002. – 616