Methodological development on the topic: mathematical research in mathematics lessons. The place of mathematical research methods in enterprise management

In 1774 L. Euler showed that if the function y(x) delivers a minimum to the linear integral J = J[y(x), then it must satisfy some differential equations, subsequently called Euler's equations. The discovery of this fact was an important achievement in mathematical modeling (building mathematical models). It became clear that the same mathematical model can be presented in two equivalent forms: in the form of a functional or in the form of a Euler differential equation (a system of differential equations). In this regard, the replacement of a differential equation with a functional is called inverse problem of the calculus of variations. Thus, the solution to the problem of an extremum of a functional can be considered in the same way as the solution to the Euler differential equation corresponding to this functional. Consequently, the mathematical formulation of the same physical problem can be presented either in the form of a functional with the corresponding boundary conditions (the extremum of this functional provides a solution to the physical problem), or in the form of the Euler differential equation corresponding to this functional with the same boundary conditions (the integration of this equation provides solution to the problem).

The widespread dissemination of variational methods in applied sciences was facilitated by the appearance in 1908 of a publication by W. Ritz, associated with the method of minimizing functionals, later called Ritz method. This method is considered the classical variational method. Its main idea is that the desired function y = y(x) y delivering the functional (A ) With boundary conditions y (a) = A, y (b) = IN minimum value, searched as a series

Where Cj (i = 0, yy) - unknown coefficients; (r/(d) (r = 0, P) - coordinate functions (algebraic or trigonometric polyp).

The coordinate functions are found in such a form that they exactly satisfy the boundary conditions of the problem.

Substituting (c) into (A), after determining the derivatives of the functional J from the unknowns C, (r = 0, r) with respect to the latter, a system of algebraic linear equations is obtained. After determining the coefficients C, the solution to the problem in closed form is found from (c).

When using a large number of series terms (c) (P- 5 ? °о) in principle it is possible to obtain a solution of the required accuracy. However, how show calculations of specific problems, matrix of coefficients C, (g = 0, P) is a filled square matrix with a large spread of coefficients in absolute value. Such matrices are close to singular and, as a rule, are ill-conditioned. This is because they do not satisfy any of the conditions under which matrices can be well-conditioned. Let's look at some of these conditions.

• 1. Positive definiteness of the matrix (terms located on the main diagonal must be positive and maximum).
• 2. Ribbon view of the matrix relative to the main diagonal with a minimum width of the tape (matrix coefficients located outside the tape are equal to zero).
• 3. Symmetricity of the matrix relative to the main diagonal.

In this regard, with increasing approximations in the Ritz method, the condition number of a matrix, determined by the ratio of its maximum to minimum eigenvalue, tends to an infinitely large value. And the accuracy of the resulting solution, due to the rapid accumulation of rounding errors when solving large systems of algebraic linear equations, may not improve, but worsen.

Along with the Ritz method, the related Galerkin method developed. In 1913, I. G. Bubnov established that algebraic linear equations with respect to unknowns C, (/ = 0, P) from (c) can be obtained without using a functional of the form (A). The mathematical formulation of the problem in this case includes a differential equation with appropriate boundary conditions. The solution, as in the Ritz method, is made in the form (c). Thanks to the special design of the coordinate functions φ,(x), solution (c) exactly satisfies the boundary conditions of the problem. To determine the unknown coefficients C, (g = 0, P) the discrepancy of the differential equation is compiled and the discrepancy is required to be orthogonal to all coordinate functions φ 7 Cr) (/ = i = 0, P). Determining recipients There are integrals with respect to the unknown coefficients C, (G= 0, r) we obtain a system of algebraic linear equations that completely coincides with the system of similar equations of the Ritz method. Thus, when solving the same problems using the same systems of coordinate functions, the Ritz and Bubnov-Galerkin methods lead to the same results.

Despite the identity of the results obtained, an important advantage of the Bubnov-Galerkin method compared to the Ritz method is that it does not require the construction of a variational analogue (functional) of the differential equation. Note that such an analogue cannot always be constructed. In connection with this Bubnov-Galerkin method, problems for which classical variational methods are not applicable can be solved.

Another method belonging to the variational group is the Kantorovich method. Its distinctive feature is that the unknown coefficients in linear combinations of type (c) are not constants, but functions that depend on one of the independent variables of the problem (for example, time). Here, as in the Bubnov-Galerkin method, the discrepancy of the differential equation is compiled and the discrepancy is required to be orthogonal to all coordinate functions (ру(дг) (j = i = 0, P). After defining integrals with respect to unknown functions fj(x) we will have a system of ordinary differential equations of the first order. Methods for solving such systems are well developed (standard computer programs are available).

One of the directions in solving boundary value problems is the joint use of exact (Fourier, integral transformations, etc.) and approximate (variational, weighted residuals, collocations, etc.) analytical methods. Such an integrated approach makes it possible to make the best use of the positive aspects of these two most important devices of applied mathematics, since it becomes possible, without carrying out subtle and cumbersome mathematical calculations, to obtain expressions in a simple form that are equivalent to the main part of the exact solution, consisting of an infinite functional series. For practical calculations, as a rule, this partial sum of several terms is used. When using such methods, to obtain more accurate results in the initial section of the parabolic coordinate, it is necessary to perform a large number of approximations. However, with large P coordinate functions with adjacent indices lead to algebraic equations related by an almost linear relationship. The coefficient matrix in this case, being a filled square matrix, is close to singular and, as a rule, turns out to be ill-conditioned. And when P- 3? °° the approximate solution may not converge even to a weakly accurate solution. Solving systems of algebraic linear equations with ill-conditioned matrices presents significant technical difficulties due to the rapid accumulation of rounding errors. Therefore, such systems of equations must be solved with high accuracy of intermediate calculations.

A special place among the approximate analytical methods that make it possible to obtain analytical solutions in the initial section of the time (parabolic) coordinate is occupied by methods that use the concept front of temperature disturbance. According to these methods, the entire process of heating or cooling bodies is formally divided into two stages. The first of them is characterized by the gradual propagation of the front of temperature disturbance from the surface of the body to its center, and the second by a change in temperature throughout the entire volume of the body until the onset of a stationary state. This division of the thermal process into two stages in time makes it possible to carry out a step-by-step solution of problems of non-stationary thermal conductivity and for each stage separately, as a rule, already in the first approximation, find calculation formulas that are satisfactory in accuracy, quite simple and convenient in engineering applications. These methods also have a significant drawback, which is the need for an a priori choice of the coordinate dependence of the desired temperature function. Usually quadratic or cubic parabolas are accepted. This ambiguity of the solution gives rise to the problem of accuracy, since, assuming in advance one or another profile of the temperature field, each time we will obtain different final results.

Among the methods that use the idea of ​​a finite speed of movement of the front of a temperature disturbance, the most widespread is the integral heat balance method. With its help, a partial differential equation can be reduced to an ordinary differential equation with given initial conditions, the solution of which can often be obtained in a closed analytical form. The integral method, for example, can be used to approximately solve problems when the thermophysical properties are not constant, but are determined by a complex functional dependence, and problems in which, along with thermal conductivity, convection must also be taken into account. The integral method also has the disadvantage noted above - an a priori choice of the temperature profile, which gives rise to the problem of uniqueness of the solution and leads to its low accuracy.

Numerous examples of the application of the integral method to solving heat conduction problems are given in the work of T. Goodman. In this work, along with an illustration of great possibilities, its limitations are also shown. Thus, despite the fact that many problems can be successfully solved by the integral method, there is a whole class of problems for which this method is practically not applicable. These are, for example, problems with impulse changes in input functions. The reason is that the temperature profile in the form of a quadratic or cubic parabola does not correspond to the true temperature profile for such problems. Therefore, if the true temperature distribution in the body under study takes the form of a nonmonotonic function, then it is impossible to obtain a satisfactory solution consistent with the physical meaning of the problem under any circumstances.

An obvious way to improve the accuracy of the integral method is to use polynomial temperature functions of a higher order. In this case, the main boundary conditions and smoothness conditions at the front of the temperature disturbance are not sufficient to determine the coefficients of such polynomials. In this regard, there is a need to search for the missing boundary conditions, which, together with the given ones, would allow us to determine the coefficients of the optimal temperature profile of a higher order, taking into account all the physical features of the problem under study. Such additional boundary conditions can be obtained from the main boundary conditions and the original differential equation by differentiating them at boundary points in spatial coordinates and in time.

When studying various heat transfer problems, it is assumed that the thermophysical properties do not depend on temperature, and linear conditions are taken as boundary conditions. However, if the body temperature varies over a wide range, then, due to the dependence of thermophysical properties on temperature, the heat conduction equation becomes nonlinear. Its solution becomes much more complicated, and known accurate analytical methods turn out to be ineffective. The integral heat balance method allows one to overcome some difficulties associated with the nonlinearity of the problem. For example, it reduces a partial differential equation with nonlinear boundary conditions to an ordinary differential equation with given initial conditions, the solution of which can often be obtained in closed analytical form.

It is known that exact analytical solutions have currently been obtained only for problems in a simplified mathematical formulation, when many important characteristics of processes are not taken into account (nonlinearity, variability of properties and boundary conditions, etc.). All this leads to a significant deviation of mathematical models from real physical processes occurring in specific power plants. In addition, exact solutions are expressed by complex infinite functional series, which in the vicinity of boundary points and for small values ​​of the time coordinate are slowly converging. Such solutions are of little use for engineering applications, and especially in cases where solving a temperature problem is an intermediate stage in solving some other problems (thermal flexibility problems, inverse problems, control problems, etc.). In this regard, the methods of applied mathematics listed above are of great interest, making it possible to obtain solutions, although approximate, in an analytical form, with an accuracy in many cases sufficient for engineering applications. These methods make it possible to significantly expand the range of problems for which analytical solutions can be obtained in comparison with classical methods.

Let's compare the methodology for using mathematics in practical research with the methodology of other natural sciences. Sciences such as physics, chemistry, and biology directly study the real object itself (possibly on a reduced scale and in laboratory conditions). Scientific results, after the necessary verification, can also be directly applied in practice. Mathematics studies not the objects themselves, but their models. The description of the object and the formulation of the problem are translated from ordinary language into the “language of mathematics” (formalized), resulting in a mathematical model. This model is further studied as a mathematical problem. The obtained scientific results are not immediately applied in practice, since they are formulated in mathematical language. Therefore, the reverse process is carried out - a meaningful interpretation (in the language of the original problem) of the obtained mathematical results. Only after this the question of their application in practice is decided.

An integral part of the methodology of applied mathematics is a comprehensive analysis of a real problem, preceding its mathematical modeling. In general, a systemic analysis of the problem involves performing the following steps:

· humanitarian (pre-mathematical) analysis of the problem;

· mathematical study of the problem;

· application of the obtained results in practice.

Carrying out such a systematic analysis of each specific problem should be carried out by a research group, including economists (as problem makers or customers), mathematicians, lawyers, sociologists, psychologists, ecologists, etc. Moreover, mathematicians, as the main researchers, should participate not only in “solving » task, but also in its formulation, as well as in the implementation of the results in practice.

To carry out mathematical research of an economic problem, the following main steps are required:

1. study of the subject area and determination of the purpose of the research;

2. formulation of the problem;

3. collection of data (statistical, expert and others);

4. construction of a mathematical model;

5. selection (or development) of a computational method and construction of an algorithm for solving the problem;

6. programming the algorithm and debugging the program;

7. checking the quality of the model using a test example;

8. implementation of results in practice.

Stages 1 -3 relate to the pre-mathematical part of the study. The subject area must be thoroughly studied by economists themselves so that they, as customers, can clearly formulate the problem and define goals for researchers. Researchers must be provided with all necessary documentary and statistical data in a comprehensive manner. Mathematicians organize, store, analyze and process data provided to them in a convenient (electronic) form by customers.

Stages 4 -7 relate to the mathematical part of the research. The result of this stage is the formulation of the original problem in the form of a rigorous mathematical problem. A mathematical model can rarely be “selected” from among the available, known models (Fig. 1.1). The process of selecting model parameters so that it matches the object being studied is called model identification. Based on the nature of the resulting model (task) and the purpose of the study, either a known method is chosen, or a known method is adapted (modified), or a new one is developed. After this, an algorithm (the procedure for solving the problem) and a computer program are compiled. The results obtained using this program are analyzed: test problems are solved, the necessary changes and corrections are introduced into the algorithm and program.

If for “pure” mathematics it is traditional to select a mathematical model once and formulate assumptions once at the very beginning of the study, then in applied work it is often useful to return to the model and make corrections to it after the first round of trial calculations has already been carried out. Moreover, comparison of models often turns out to be fruitful when the same phenomenon is described not by one, but by several models. If the conclusions turn out to be (approximately) the same for different models and different research methods, this is evidence of the correctness of the calculations, the adequacy of the model to the object itself, and the objectivity of the recommendations given.

The final stage 8 carried out jointly by customers and model developers.

The results of mathematical (as well as any scientific) research are only recommendations for use in practice. The final decision on this issue - whether to apply the model or not - depends on the customer, i.e., on the person responsible for the outcome and the consequences to which the application of the recommended results will lead.

To build a mathematical model of a specific economic task (problem), it is recommended to perform the following sequence of work:

1. determination of known and unknown quantities, as well as existing conditions and prerequisites (what is given and what needs to be found?);

2. identifying the most important factors of the problem;

3. identification of controllable and uncontrollable parameters;

4. mathematical description through equations, inequalities, functions and other relationships between model elements (parameters, variables), based on the content of the problem under consideration.

The known parameters of the problem relative to its mathematical model are considered external(given a priori, i.e. before building the model). In economic literature they are called exogenous variables. The values ​​of initially unknown variables are calculated as a result of studying the model, so in relation to the model they are considered internal. In economic literature they are called endogenous variables.

IN § 2 the most important are understood as factors that play a significant role in the task itself and which, one way or another, influence the final result. IN § 3 controllable are those parameters of a problem that can be given arbitrary numerical values ​​based on the conditions of the problem; uncontrollable are those parameters whose value is fixed and cannot be changed.

From the point of view of purpose, we can distinguish descriptive models And decision making models. Descriptive Models reflect the content and basic properties of economic objects as such. With their help, numerical values ​​of economic factors and indicators are calculated.

Decision-making models help to find the best options for planned indicators or management decisions. Among them, the least complex are optimization models, through which tasks such as planning are described (modeled), and the most complex are game models that describe problems of a conflicting nature, taking into account the intersection of various interests. These models differ from descriptive models in that they have the ability to select the values ​​of control parameters (which is absent in descriptive models).

Examples of drawing up mathematical models

Example 1.1. Let a certain economic region produce several types of products exclusively on its own and only for the population of this region. It is assumed that the technological process has been worked out, and the population's demand for these goods has been studied. It is necessary to determine the annual volume of product output, taking into account the fact that this volume must provide both final and industrial consumption.

Let's create a mathematical model of this problem. According to the condition, the following are given: types of products, demand for them and the technological process; it is required to find the volume of output of each type of product. Let us denote the known quantities: - population demand for the th product; - the quantity of the i-th product required to produce a unit of the i-th product using this technology . Let us denote the unknown quantities: - volume of output of the th product . Totality is called the demand vector, the numbers are called technological coefficients, and the totality - release vector. According to the conditions of the problem, the vector is distributed into two parts: for final consumption (vector) and for reproduction (vector). Let's calculate that part of the vector that goes to reproduction. By virtue of the notation, for the production of the quantity of the -th product, the quantity of the -th product is used. Then the amount shows the amount of the -good that is needed for the entire output . Therefore, the equality must be satisfied:

Generalizing this reasoning to all types of products, we arrive at the desired model:

By solving the resulting system of linear equations, we find the required release vector.

In order to write this model in a more compact (vector) form, we introduce the following notation:

A square matrix A (of size) is called a technological matrix. Obviously, the model can be written as: or

We received the classic “Input-Output” model, authored by the famous American economist V. Leontiev.

Example 1.2. The oil refinery has two grades of oil: grade - 10 units, grade - 15 units. When refining oil, two materials are obtained: gasoline () and fuel oil (). There are three options for the processing technology process:

I: 1 unit A+ 2 units IN gives 3 units. B+ 2 units M;

II:2 units A+ 1 unit IN gives 1 unit. B+ 5 units M;

III:2 units A+ 2 units IN gives 1 unit. B+ 2 units M.

The price of gasoline is $10 per unit, fuel oil is $1 per unit. It is necessary to determine the most advantageous combination of technological processes for processing the available amount of oil.

Before modeling, let us clarify the following points. From the conditions of the problem it follows that the “profitability” of the technological process for the plant should be understood in the sense of obtaining maximum income from the sale of its finished products (gasoline and fuel oil). In this regard, it is clear that the plant’s “choice (making) decision” consists of determining which technology to apply and how many times. Obviously, there are quite a lot of such possible options.

Let us denote the unknown quantities: - the amount of use of the th technological process. Other model parameters (oil reserves, gasoline and fuel oil prices) known.

Then one specific decision of the plant comes down to choosing one vector for which the plant’s revenue is equal to dollars. Here, 32 dollars is the income received from one application of the first technological process ($10 3 units. B+ 1 dollar 2 units M= $32). Coefficients 15 and 12 for the second and third technological processes, respectively, have a similar meaning. Accounting for oil reserves leads to the following conditions:

for variety A: ,

for variety IN: ,

where in the first inequality coefficients 1, 2, 2 are the consumption rates of oil grade A for one-time use of technological processes I, II, III respectively. The coefficients of the second inequality have a similar meaning for oil grade IN.

The mathematical model as a whole has the form:

Find a vector such that

maximize

subject to the following conditions:

,

,

.

The shortened form of this entry is:

under restrictions

, (1.4.2)

,

We got the so-called linear programming problem. Model (1.4.2.) is an example of an optimization model of a deterministic type (with well-defined elements).

Example 1.3. An investor needs to determine the best mix of stocks, bonds and other securities to purchase for a certain amount in order to obtain a certain profit with minimal risk to himself. The profit for every dollar invested in a security of this type is characterized by two indicators: expected profit and actual profit. For an investor, it is desirable that the expected profit per dollar of investment is not lower than a given value for the entire set of securities. Note that to correctly model this problem, a mathematician is required to have certain basic knowledge in the field of portfolio theory of securities. Let us denote the known parameters of the problem: - the number of types of securities; - actual profit (random number) from the -th type of security - expected profit from the -th type of security. Let us denote the unknown quantities: - funds allocated for the acquisition of securities of the type . Due to notation, the entire invested amount is defined as . To simplify the model, we introduce new quantities

Thus, this is the share of all funds allocated for the acquisition of securities of the type. It's obvious that . From the conditions of the problem it is clear that the investor’s goal is to achieve a certain level of profit with minimal risk. In essence, risk is a measure of the deviation of actual profit from the expected one. Therefore it can be identified with covariance

profits for securities of type and type. Here M- designation of mathematical expectation. The mathematical model of the original problem has the form:

(1.4.3)

We obtained the well-known Markowitz model for optimizing the structure of a securities portfolio. Model (1.4.3.) is an example of an optimization model of the stochastic type (with elements of randomness).

Example 1.4. On the basis of a trade organization there are types of one of the minimum assortment products. Only one type of a given product must be brought into the store. You need to choose the type of product that is appropriate to bring into the store. If a product of this type is in demand, then the store will make a profit from its sale, but if it is not in demand, it will make a loss.

The new calculus as a system was fully created by Newton, who, however, did not publish his discoveries for a long time.

The official date of birth of differential calculus can be considered May, when Leibniz published his first article "A New Method of Highs and Lows...". This article, in a concise and inaccessible form, set out the principles of a new method called differential calculus.

Leibniz and his students

These definitions are explained geometrically, while in Fig. infinitesimal increments are depicted as finite. The consideration is based on two requirements (axioms). First:

It is required that two quantities that differ from each other only by an infinitesimal amount can be taken [when simplifying expressions?] indifferently one instead of the other.

The continuation of each such line is called a tangent to the curve. Investigating the tangent passing through the point, L'Hopital attaches great importance to the quantity

reaching extreme values ​​at the inflection points of the curve, while the relation to is not given any special significance.

It is noteworthy to find extremum points. If, with a continuous increase in diameter, the ordinate first increases and then decreases, then the differential is first positive compared to , and then negative.

But any continuously increasing or decreasing value cannot turn from positive to negative without passing through infinity or zero... It follows that the differential of the largest and smallest value must be equal to zero or infinity.

This formulation is probably not flawless, if we remember the first requirement: let, say, , then by virtue of the first requirement

at zero, the right hand side is zero and the left hand side is not. Apparently it should have been said that it can be transformed in accordance with the first requirement so that at the maximum point . . In the examples, everything is self-explanatory, and only in the theory of inflection points does L'Hopital write that it is equal to zero at the maximum point, being divided by .

Further, with the help of differentials alone, extremum conditions are formulated and a large number of complex problems related mainly to differential geometry on the plane are considered. At the end of the book, in chap. 10, sets out what is now called L'Hopital's rule, although in an unusual form. Let the ordinate of the curve be expressed as a fraction, the numerator and denominator of which vanish at . Then the point of the curve c has an ordinate equal to the ratio of the differential of the numerator to the differential of the denominator taken at .

According to L'Hôpital's plan, what he wrote constituted the first part of Analysis, while the second was supposed to contain integral calculus, that is, a method of finding the connection between variables based on the known connection of their differentials. Its first presentation was given by Johann Bernoulli in his Mathematical lectures on the integral method. Here a method is given for taking most elementary integrals and methods for solving many first-order differential equations are indicated.

Pointing to the practical usefulness and simplicity of the new method, Leibniz wrote:

What a person versed in this calculus can obtain directly in three lines, other learned men were forced to look for by following complex detours.

Euler

The changes that took place over the next half century are reflected in Euler's extensive treatise. The presentation of the analysis opens with a two-volume “Introduction”, which contains research on various representations of elementary functions. The term “function” first appears only in Leibniz, but it was Euler who put it in the first place. The original interpretation of the concept of a function was that a function is an expression for counting (German. Rechnungsausdrϋck) or analytical expression.

A variable quantity function is an analytical expression composed in some way from this variable quantity and numbers or constant quantities.

Emphasizing that “the main difference between functions lies in the way they are composed of variable and constant,” Euler lists the actions “through which quantities can be combined and mixed with each other; these actions are: addition and subtraction, multiplication and division, exponentiation and extraction of roots; This should also include the solution of [algebraic] equations. In addition to these operations, called algebraic, there are many others, transcendental, such as: exponential, logarithmic and countless others, delivered by integral calculus.” This interpretation made it possible to easily handle multi-valued functions and did not require an explanation of which field the function was being considered over: the counting expression was defined for complex values ​​of variables even when this was not necessary for the problem under consideration.

Operations in the expression were allowed only in finite numbers, and the transcendental penetrated with the help of an infinitely large number. In expressions, this number is used along with natural numbers. For example, such an expression for the exponent is considered acceptable

in which only later authors saw the ultimate transition. Various transformations were carried out with analytical expressions, which allowed Euler to find representations for elementary functions in the form of series, infinite products, etc. Euler transforms expressions for counting as they do in algebra, without paying attention to the possibility of calculating the value of a function at a point for each from written formulas.

Unlike L'Hopital, Euler examines in detail transcendental functions and in particular their two most studied classes - exponential and trigonometric. He discovers that all elementary functions can be expressed using arithmetic operations and two operations - taking the logarithm and the exponent.

The proof itself perfectly demonstrates the technique of using the infinitely large. Having defined sine and cosine using the trigonometric circle, Euler derived the following from the addition formulas:

Assuming and , he gets

discarding infinitesimal quantities of higher order. Using this and a similar expression, Euler obtained his famous formula

Having indicated various expressions for functions that are now called elementary, Euler moves on to consider curves on a plane drawn by free movement of the hand. In his opinion, it is not possible to find a single analytical expression for every such curve (see also the String Dispute). In the 19th century, at the instigation of Casorati, this statement was considered erroneous: according to Weierstrass’s theorem, any continuous curve in the modern sense can be approximately described by polynomials. In fact, Euler was hardly convinced by this, because he still needed to rewrite the passage to the limit using the symbol.

Euler begins his presentation of differential calculus with the theory of finite differences, followed in the third chapter by a philosophical explanation that “an infinitesimal quantity is exactly zero,” which most of all did not suit Euler’s contemporaries. Then, differentials are formed from finite differences at an infinitesimal increment, and from Newton's interpolation formula - Taylor's formula. This method essentially goes back to the work of Taylor (1715). In this case, Euler has a stable relation , which, however, is considered as a relation of two infinitesimals. The last chapters are devoted to approximate calculation using series.

In the three-volume integral calculus, Euler interprets and introduces the concept of integral as follows:

The function whose differential is called its integral and is denoted by the sign placed in front.

In general, this part of Euler’s treatise is devoted to a more general, from a modern point of view, problem of the integration of differential equations. At the same time, Euler finds a number of integrals and differential equations that lead to new functions, for example, -functions, elliptic functions, etc. A rigorous proof of their non-elementary nature was given in the 1830s by Jacobi for elliptic functions and by Liouville (see elementary functions).

Lagrange

The next major work that played a significant role in the development of the concept of analysis was Theory of analytic functions Lagrange and Lacroix's extensive retelling of Lagrange's work in a somewhat eclectic manner.

Wanting to get rid of the infinitesimal altogether, Lagrange reversed the connection between derivatives and the Taylor series. By analytic function Lagrange understood an arbitrary function studied by analytical methods. He designated the function itself as , giving a graphical way to write the dependence - earlier Euler made do with only variables. To apply analysis methods, according to Lagrange, it is necessary that the function be expanded into a series

whose coefficients will be new functions. It remains to call it a derivative (differential coefficient) and denote it as . Thus, the concept of derivative is introduced on the second page of the treatise and without the help of infinitesimals. It remains to be noted that

therefore the coefficient is twice the derivative of the derivative, that is

This approach to the interpretation of the concept of derivative is used in modern algebra and served as the basis for the creation of Weierstrass's theory of analytic functions.

Lagrange operated with such series as formal ones and obtained a number of remarkable theorems. In particular, for the first time and quite rigorously he proved the solvability of the initial problem for ordinary differential equations in formal power series.

The question of assessing the accuracy of approximations provided by partial sums of the Taylor series was first posed by Lagrange: in the end Theories of analytic functions he derived what is now called Taylor's formula with a remainder term in Lagrange form. However, in contrast to modern authors, Lagrange did not see the need to use this result to justify the convergence of the Taylor series.

The question of whether the functions used in analysis can really be expanded into a power series subsequently became the subject of debate. Of course, Lagrange knew that at some points elementary functions may not be expanded into a power series, but at these points they are not differentiable in any sense. Cauchy in his Algebraic analysis cited the function as a counterexample

extended by zero at zero. This function is smooth everywhere on the real axis and at zero it has a zero Maclaurin series, which, therefore, does not converge to the value . Against this example, Poisson objected that Lagrange defined the function as a single analytical expression, while in Cauchy’s example the function is defined differently at zero and at . Only at the end of the 19th century did Pringsheim prove that there is an infinitely differentiable function, given by a single expression, for which the Maclaurin series diverges. An example of such a function is the expression

Further development

In the last third of the 19th century, Weierstrass arithmetized the analysis, considering the geometric justification to be insufficient, and proposed a classical definition of the limit through the ε-δ language. He also created the first rigorous theory of the set of real numbers. At the same time, attempts to improve the Riemann integrability theorem led to the creation of a classification of discontinuity of real functions. “Pathological” examples were also discovered (continuous functions that are nowhere differentiable, space-filling curves). In this regard, Jordan developed measure theory, and Cantor developed set theory, and at the beginning of the 20th century, mathematical analysis was formalized with their help. Another important development of the 20th century was the development of non-standard analysis as an alternative approach to justifying analysis.

Sections of mathematical analysis

• Metric space, Topological space

see also

Bibliography

Encyclopedic articles

• // Encyclopedic Lexicon: St. Petersburg: type. A. Plushara, 1835-1841. Volume 1-17.

• // Encyclopedic Dictionary of Brockhaus and Efron: In 86 volumes (82 volumes and 4 additional ones). - St. Petersburg. , 1890-1907.

Educational literature

Standard textbooks

For many years, the following textbooks have been popular in Russia:

• Courant, R. Course of differential and integral calculus (in two volumes). The main methodological discovery of the course: first, the main ideas are simply stated, and then they are given rigorous evidence. Written by Courant while he was a professor at the University of Göttingen in the 1920s under the influence of Klein’s ideas, then transferred to American soil in the 1930s. The Russian translation of 1934 and its reprints gives the text based on the German edition, the translation of the 1960s (the so-called 4th edition) is a compilation from the German and American versions of the textbook and is therefore very verbose.
• Fikhtengolts G. M. A course in differential and integral calculus (in three volumes) and a problem book.
• Demidovich B. P. Collection of problems and exercises in mathematical analysis.
• Lyashko I. I. et al. Reference book for higher mathematics, vol. 1-5.

Some universities have their own analysis guides:

• MSU, Mechanics and Mat:

• Arkhipov G. I., Sadovnichy V. A., Chubarikov V. N. Lectures on math. analysis.
• Zorich V. A. Mathematical analysis. Part I. M.: Nauka, 1981. 544 p.
• Zorich V. A. Mathematical analysis. Part II. M.: Nauka, 1984. 640 p.
• Kamynin L. I. Course of mathematical analysis (in two volumes). M.: Moscow University Publishing House, 2001.

• V. A. Ilyin, V. A. Sadovnichy, Bl. H. Sendov. Mathematical analysis / Ed. A. N. Tikhonova. - 3rd ed. , processed and additional - M.: Prospekt, 2006. - ISBN 5-482-00445-7

• Moscow State University, physics department:

• Ilyin V. A., Poznyak E. G. Fundamentals of mathematical analysis (in two parts). - M.: Fizmatlit, 2005. - 648 p. - ISBN 5-9221-0536-1
• Butuzov V.F. et al. Mat. analysis in questions and tasks

• Mathematics at a technical university Collection of textbooks in 21 volumes.

• St. Petersburg State University, Faculty of Physics:

• Smirnov V.I. Course of higher mathematics, in 5 volumes. M.: Nauka, 1981 (6th edition), BHV-Petersburg, 2008 (24th edition).

• NSU, ​​Mechanics and Mathematics:

• Reshetnyak Yu. G. Course of mathematical analysis. Part I. Book 1. Introduction to mathematical analysis. Differential calculus of functions of one variable. Novosibirsk: Publishing House of the Institute of Mathematics, 1999. 454 with ISBN 5-86134-066-8.
• Reshetnyak Yu. G. Course of mathematical analysis. Part I. Book 2. Integral calculus of functions of one variable. Differential calculus of functions of several variables. Novosibirsk: Publishing House of the Institute of Mathematics, 1999. 512 with ISBN 5-86134-067-6.
• Reshetnyak Yu. G. Course of mathematical analysis. Part II. Book 1. Fundamentals of smooth analysis in multidimensional spaces. Series theory. Novosibirsk: Publishing House of the Institute of Mathematics, 2000. 440 with ISBN 5-86134-086-2.
• Reshetnyak Yu. G. Course of mathematical analysis. Part II. Book 2. Integral calculus of functions of several variables. Integral calculus on manifolds. External differential forms. Novosibirsk: Publishing House of the Institute of Mathematics, 2001. 444 with ISBN 5-86134-089-7.
• Shvedov I. A. Compact course of mathematical analysis,: Part 1. Functions of one variable, Part 2. Differential calculus of functions of several variables.

• MIPT, Moscow

• Kudryavtsev L. D. Course of mathematical analysis (in three volumes).

• BSU, physics department:

• Bogdanov Yu. S. Lectures on mathematical analysis (in two parts). - Minsk: BSU, 1974. - 357 p.

Advanced textbooks

Textbooks:

• Rudin U. Fundamentals of mathematical analysis. M., 1976 - a small book, written very clearly and concisely.

Problems of increased complexity:

• G. Polia, G. Szege, Problems and theorems from analysis. Part 1, Part 2, 1978. (Most of the material relates to TFKP)
• Pascal, E.(Napoli). Esercizii, 1895; 2 ed., 1909 // Internet Archive

Textbooks for humanities

• A. M. Akhtyamov Mathematics for sociologists and economists. - M.: Fizmatlit, 2004.
• N. Sh. Kremer and others. Higher mathematics for economists. Textbook. 3rd ed. - M.: Unity, 2010

Problem books

• G. N. Berman. Collection of problems for the course of mathematical analysis: Textbook for universities. - 20th ed. M.: Science. Main editorial office of physical and mathematical literature, 1985. - 384 p.
• P. E. Danko, A. G. Popov, T. Ya. Kozhevnikov. Higher mathematics in exercises and problems. (In 2 parts) - M.: Vyssh.shk, 1986.
• G. I. Zaporozhets Guide to solving problems in mathematical analysis. - M.: Higher School, 1966.
• I. A. Kaplan. Practical lessons in higher mathematics, in 5 parts.. - Kharkov, Publishing house. Kharkov State Univ., 1967, 1971, 1972.
• A. K. Boyarchuk, G. P. Golovach. Differential equations in examples and problems. Moscow. Editorial URSS, 2001.
• A. V. Panteleev, A. S. Yakimova, A. V. Bosov. Ordinary differential equations in examples and problems. "MAI", 2000
• A. M. Samoilenko, S. A. Krivosheya, N. A. Perestyuk. Differential equations: examples and problems. VS, 1989.
• K. N. Lungu, V. P. Norin, D. T. Pismenny, Yu. A. Shevchenko. Collection of problems in higher mathematics. 1 course. - 7th ed. - M.: Iris-press, 2008.
• I. A. Maron. Differential and integral calculus in examples and problems (Functions of one variable). - M., Fizmatlit, 1970.
• V. D. Chernenko. Higher mathematics in examples and problems: Textbook for universities. In 3 volumes - St. Petersburg: Politekhnika, 2003.

Directories

Classic works

Essays on the history of analysis

• Kestner, Abraham Gottgelf. Geschichte der Mathematik . 4 volumes, Göttingen, 1796-1800
• Kantor, Moritz. Vorlesungen über geschichte der mathematik Leipzig: B. G. Teubner, - . Bd. 1, Bd. 2, Bd. 3, Bd. 4
• History of mathematics edited by A. P. Yushkevich (in three volumes):

• Volume 1 From ancient times to the beginning of modern times. (1970)
• Volume 2 Mathematics of the 17th century. (1970)
• Volume 3 Mathematics of the 18th century. (1972)

• Markushevich A.I. Essays on the history of the theory of analytic functions. 1951
• Vileitner G. History of mathematics from Descartes to the middle of the 19th century. 1960

Notes

1. Wed., e.g. Cornell Un course
2. Newton I. Mathematical works. M, 1937.
3. Leibniz //Acta Eroditorum, 1684. L.M.S., vol. V, p. 220-226. Rus. Transl.: Uspekhi Mat. Sciences, vol. 3, v. 1 (23), p. 166-173.
4. L'Hopital. Infinitesimal Analysis. M.-L.: GTTI, 1935. (Hereinafter: L'Hopital) // Mat. analysis on EqWorld
5. L'Hopital, ch. 1, def. 2.
6. L'Hopital, ch. 4, def. 1.
7. L'Hopital, ch. 1, requirement 1.
8. L'Hopital, ch. 1, requirement 2.
9. L'Hopital, ch. 2, def.
10. L'Hopital, § 46.
11. L'Hopital is worried about something else: for him the length of a segment and it is necessary to explain what its negativity means. The remark made in § 8-10 can even be understood to mean that when decreasing with increasing one should write , but this is not used further.