General characteristics of mathematical methods of analysis. Mathematical research

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 across 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 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). Mathematical formulation of the problem in 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 construction 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 it is required that the discrepancy 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. His hallmark is that the unknown coefficients in linear combinations of type (c) are not constants, but functions that depend on one of the independent problem variables(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 when solving boundary value problems is sharing exact (Fourier, integral transformations, etc.) and approximate (variational, weighted residuals, collocations, etc.) analytical methods. Such A complex approach allows the best way use positive sides these two most important apparatuses 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 on initial section parabolic coordinates must be performed big number 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 filled square matrix, is close to degenerate and turns out, as a rule, 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 steady state. This division of the thermal process into two stages in time allows for 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, finding 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 widely used is the integral method heat balance. 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 are successfully solved by the integral method, there is whole class tasks 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 physical features 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 they take as boundary properties linear 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 ones. 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 step in solving some other problems (thermal flexibility problems, inverse problems, control tasks, 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.

In the history of mathematics, we can roughly distinguish two main periods: elementary and modern mathematics. The milestone from which it is customary to count the era of new (sometimes called higher) mathematics was the 17th century - the century of the appearance of mathematical analysis. By the end of the 17th century. I. Newton, G. Leibniz and their predecessors created a new apparatus differential calculus and integral calculus, which forms the basis of mathematical analysis and even, perhaps, the mathematical basis of all modern natural science.

Mathematical analysis is a broad field of mathematics with a characteristic object of study (variable quantity), a unique research method (analysis by means of infinitesimals or by means of passages to limits), a certain system of basic concepts (function, limit, derivative, differential, integral, series) and constantly improving and a developing apparatus, the basis of which is differential and integral calculus.

Let's try to give an idea of ​​what kind of mathematical revolution took place in the 17th century, what characterizes the transition from elementary mathematics to what is now the subject of research in mathematical analysis and what explains its fundamental role in the entire modern system of theoretical and applied knowledge.

Imagine that in front of you is a beautifully executed color photography a stormy ocean wave rushing onto the shore: a powerful stooped back, a steep but slightly sunken chest, a head already tilted forward and ready to fall with a gray mane tormented by the wind. You stopped the moment, you managed to catch the wave, and you can now carefully study it in every detail without haste. A wave can be measured, and using the tools of elementary mathematics, you can draw many important conclusions about this wave, and therefore all its ocean sisters. But by stopping the wave, you deprived it of movement and life. Its origin, development, running, the force with which it hits the shore - all this turned out to be outside your field of vision, because you do not yet have either a language or a mathematical apparatus suitable for describing and studying not static, but developing, dynamic processes, variables and their relationships.

“Mathematical analysis is no less comprehensive than nature itself: it determines all tangible relationships, measures times, spaces, forces, temperatures.” J. Fourier

Movement, variables and their relationships surround us everywhere. Various types of motion and their patterns constitute the main object of study of specific sciences: physics, geology, biology, sociology, etc. Therefore, precise language and corresponding mathematical methods for describing and studying variable quantities turned out to be necessary in all areas of knowledge to approximately the same extent as numbers and arithmetic are necessary when describing quantitative relationships. So, mathematical analysis and forms the basis of the language and mathematical methods for describing variables and their relationships. Nowadays, without mathematical analysis it is impossible not only to calculate space trajectories, work nuclear reactors, the running of the ocean wave and the patterns of development of the cyclone, but also to economically manage production, resource distribution, organization technological processes, predict the course of chemical reactions or changes in the numbers of various interconnected species of animals and plants in nature, because all of these are dynamic processes.

Elementary math was mostly math constant values, she studied mainly the relationships between the elements of geometric figures, the arithmetic properties of numbers and algebraic equations. Its attitude to reality can to some extent be compared with an attentive, even thorough and complete study of each fixed frame of a film that captures the changing, developing living world in its movement, which, however, is not visible in a separate frame and which can only be observed by looking the tape as a whole. But just as cinema is unthinkable without photography, so too modern mathematics is impossible without that part of it that we conventionally call elementary, without the ideas and achievements of many outstanding scientists, sometimes separated by tens of centuries.

Mathematics is united, and the “higher” part of it is connected with the “elementary” part in much the same way as the next floor of a house under construction is connected with the previous one, and the width of the horizons that mathematics opens to us in the world, depends on which floor of this building we managed to climb to. Born in the 17th century. mathematical analysis has opened up possibilities for us scientific description, quantitative and qualitative study of variables and movement in the broad sense of the word.

What are the prerequisites for the emergence of mathematical analysis?

By the end of the 17th century. The following situation has arisen. Firstly, within the framework of mathematics itself, long years Some important classes of similar problems have accumulated (for example, problems of measuring areas and volumes of non-standard figures, problems of drawing tangents to curves) and methods for solving them in various special cases have appeared. Secondly, it turned out that these problems are closely related to the problems of describing arbitrary (not necessarily uniform) mechanical motion, and in particular with the calculation of its instantaneous characteristics (speed, acceleration at any time), as well as with finding the distance traveled for movement occurring at a given variable speed. The solution to these problems was necessary for the development of physics, astronomy, and technology.

Finally, thirdly, to mid-17th century V. the works of R. Descartes and P. Fermat laid the foundations analytical method coordinates (the so-called analytical geometry), which made it possible to formulate geometric and physical problems of heterogeneous origin in the general (analytical) language of numbers and numerical dependencies, or, as we now say, numerical functions.

Modern Soviet and foreign mathematicians in their works develop the ideas of N. N. Luzin. The confluence of these circumstances led to the fact that late XVII

V. two scientists - I. Newton and G. Leibniz - independently of each other managed to create a mathematical apparatus for solving these problems, summing up and generalizing individual results of their predecessors, including the ancient scientist Archimedes and contemporaries of Newton and Leibniz - B. Cavalieri, B. Pascal , D. Gregory, I. Barrow. This apparatus formed the basis of mathematical analysis - a new branch of mathematics that studies various developing processes, i.e. relationships between variables, which in mathematics are called functional dependencies or, in other words, functions. By the way, the term “function” itself was required and naturally arose precisely in the 17th century, and by now it has acquired not only general mathematical, but also general scientific significance.

Initial information about the basic concepts and mathematical apparatus of analysis is given in the articles “Differential calculus” and “Integral calculus”. In conclusion, I would like to dwell on only one principle of mathematical abstraction, common to all mathematics and characteristic of analysis, and in this regard to explain in what form mathematical analysis studies variable quantities and what is the secret of such universality of its methods for the study of all kinds of concrete and their relationships.

Let's look at a few illustrative examples and analogies.

Sometimes we no longer realize that, for example, a mathematical relation written not for apples, chairs or elephants, but in an abstract form abstracted from specific objects, is an outstanding scientific achievement. This is a mathematical law that, as experience shows, is applicable to various specific objects. This means that by studying in mathematics the general properties of abstract, abstract numbers, we thereby study the quantitative relationships of the real world.

For example, from a school mathematics course it is known that, therefore, in specific situation you could say: “If they don’t give me two six-ton ​​dump trucks to transport 12 tons of soil, then I can ask for three four-ton dump trucks and the job will be done, but if they give me only one four-ton dump truck, then it will have to make three trips.” Thus, the abstract numbers and numerical patterns that are now familiar to us are associated with their specific manifestations and applications.

The laws of change in specific variables and developing processes of nature are related in approximately the same way to the abstract, abstract form-function in which they appear and are studied in mathematical analysis.

For example, an abstract ratio may reflect the dependence of a cinema's box office on the number of tickets sold, if 20 is 20 kopecks - the price of one ticket. But if we are riding a bicycle on a highway, traveling 20 km per hour, then this same ratio can be interpreted as the relationship between the time (hours) of our cycling trip and the distance covered during this time (kilometers). You can always say that, for example, a change of several times leads to a proportional (i.e., the same number of times) change in the value of , and if , then the opposite conclusion is also true. This means, in particular, to double the box office of a movie theater, you will have to attract twice as many spectators, and in order to travel twice as far on a bicycle at the same speed, you will have to ride twice as long.

Mathematics studies both the simplest dependence and other, much more complex dependences in a general, abstract form, abstracted from a particular interpretation. The properties of a function or methods for studying these properties identified in such a study will be of the nature of general mathematical techniques, conclusions, laws and conclusions applicable to each specific phenomenon in which the function studied in abstract form occurs, regardless of what area of ​​knowledge this phenomenon belongs to .

So, mathematical analysis as a branch of mathematics took shape at the end of the 17th century. The subject of study in mathematical analysis (as it appears from modern positions) are functions, or, in other words, dependencies between variable quantities.

With the advent of mathematical analysis, mathematics became accessible to the study and reflection of developing processes in the real world; mathematics included variables and motion.

A project method that has enormous potential for creating universal educational activities, is becoming increasingly widespread in the school education system. But it is quite difficult to “fit” the project method into the classroom system. I include mini studies in regular lesson. This form of work opens up great opportunities for the formation cognitive activity and ensures that the individual characteristics of students are taken into account, paving the way for the development of skills on large projects.

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“If a student at school has not learned to create anything himself, then in life he will only imitate and copy, since there are few who, having learned to copy, would be able to independently apply this information.” L.N. Tolstoy.

Characteristic feature modern education is a sharp increase in the amount of information that students need to learn. The degree of development of the student is measured and assessed by his ability to independently acquire new knowledge and use it in educational and practical activities. Modern pedagogical process requires use innovative technologies in teaching.

The new generation Federal State Educational Standard requires use in educational process activity-type technologies, design and research methods are defined as one of the conditions for the implementation of the main educational program.

A special role is given to such activities in mathematics lessons, and this is not accidental. Mathematics is the key to understanding the world, the basis of scientific and technological progress and an important component of personal development. It is designed to cultivate in a person the ability to understand the meaning of the task assigned to him, the ability to reason logically, and to acquire algorithmic thinking skills.

It is quite difficult to fit the project method into the classroom system. I try to judiciously combine the traditional and learner-centered systems by incorporating elements of inquiry into the regular lesson. I will give a number of examples.

So, when studying the topic “Circle,” we conduct the following research with students.

Mathematical study "Circle".

1. Think about how to build a circle, what tools are needed for this. Circle symbol.
2. In order to define a circle, let’s look at what properties this geometric figure has. Connect the center of the circle with a point belonging to the circle. Let's measure the length of this segment. Let's repeat the experiment three times. Let's draw a conclusion.
3. The segment connecting the center of the circle with any point on it is called the radius of the circle. This is the definition of radius. Radius designation. Using this definition, construct a circle with a radius of 2cm5mm.
4. Construct a circle of arbitrary radius. Construct a radius and measure it. Record your measurements. Construct three more different radii. How many radii can be drawn in a circle?
5. Let us try, knowing the property of the points of a circle, to give its definition.
6. Construct a circle of arbitrary radius. Connect two points on the circle so that this segment passes through the center of the circle. This segment is called the diameter. Let's define diameter. Diameter designation. Construct three more diameters. How many diameters does a circle have?
7. Construct a circle of arbitrary radius. Measure the diameter and radius. Compare them. Repeat the experiment three more times with different circles. Draw a conclusion.
8. Connect any two points on the circle. The resulting segment is called a chord. Let us define a chord. Build three more chords. How many chords does a circle have?
9. Is the radius a chord? Prove it.
10. Is the diameter a chord? Prove it.

Works research nature may be propaedeutic in nature. Having examined the circle, we can consider the series interesting properties, which students can formulate at the level of a hypothesis, and then prove this hypothesis. For example, the following study:

"Mathematical Research"

1. Construct a circle of radius 3 cm and draw its diameter. Connect the ends of the diameter to an arbitrary point on the circle and measure the angle formed by the chords. Carry out the same constructions for two more circles. What do you notice?
2. Repeat the experiment for a circle of arbitrary radius and formulate a hypothesis. Can it be considered proven using the constructions and measurements carried out?

When studying the topic “The relative position of lines on a plane”, mathematical research is carried out in groups.

Tasks for groups:

1. group.

1. In one coordinate system, construct graphs of the function

Y = 2x, y = 2x+7, y = 2x+3, y = 2x-4, y = 2x-6.

2.Answer the questions by filling out the table:

The content of the article

MATHEMATICAL ANALYSIS, a branch of mathematics that provides methods for the quantitative study of various processes of change; deals with the study of the rate of change (differential calculus) and the determination of the lengths of curves, areas and volumes of figures bounded by curved contours and surfaces (integral calculus). It is typical for problems of mathematical analysis that their solution is associated with the concept of a limit.

The beginning of mathematical analysis was laid in 1665 by I. Newton and (around 1675) independently by G. Leibniz, although important preparatory work was carried out by I. Kepler (1571–1630), F. Cavalieri (1598–1647), P. Fermat (1601– 1665), J. Wallis (1616–1703) and I. Barrow (1630–1677).

To make the presentation more vivid, we will resort to the language of graphics. Therefore, it may be useful for the reader to look into the article ANALYTICAL GEOMETRY before starting to read this article.

DIFFERENTIAL CALCULUS

Tangents.

In Fig. 1 shows a fragment of the curve y = 2x – x 2, enclosed between x= –1 and x= 3. Sufficiently small segments of this curve look straight. In other words, if R is an arbitrary point of this curve, then there is a certain straight line passing through this point and which is an approximation of the curve in a small neighborhood of the point R, and the smaller the neighborhood, the better the approximation. Such a line is called tangent to the curve at the point R. The main task of differential calculus is to construct a general method that allows one to find the direction of a tangent at any point on a curve at which a tangent exists. It is not difficult to imagine a curve with a sharp break (Fig. 2). If R is the top of such a break, then we can construct an approximating straight line P.T. 1 – to the right of the point R and another approximating straight line RT 2 – to the left of the point R. But there is no single straight line passing through a point R, which approached the curve equally well in the vicinity of the point P both on the right and on the left, therefore the tangent at the point P does not exist.

In Fig. 1 tangent FROM drawn through the origin ABOUT= (0,0). The slope of this line is 2, i.e. when the abscissa changes by 1, the ordinate increases by 2. If x And y– coordinates of an arbitrary point on FROM, then, moving away from ABOUT to a distance X units to the right, we are moving away from ABOUT on 2 y units up. Hence, y/x= 2, or y = 2x. This is the tangent equation FROM to the curve y = 2x – x 2 at point ABOUT.

It is now necessary to explain why, out of the set of lines passing through the point ABOUT, the straight line is chosen FROM. How does a straight line with a slope of 2 differ from other straight lines? There is one simple answer, and it is difficult to resist the temptation to give it using the analogy of a tangent to a circle: the tangent FROM has only one common point with the curve, while any other non-vertical line passing through the point ABOUT, intersects the curve twice. This can be verified as follows.

Since the expression y = 2x – x 2 can be obtained by subtraction X 2 of y = 2x(equations of straight line FROM), then the values y there is less knowledge for the graph y for a straight line at all points except the point x= 0. Therefore, the graph is everywhere except the point ABOUT, located below FROM, and this line and the graph have only one common point. Moreover, if y = mx- equation of some other line passing through a point ABOUT, then there will definitely be two points of intersection. Really, mx = 2x – x 2 not only when x= 0, but also at x = 2 – m. And only when m= 2 both intersection points coincide. In Fig. 3 shows the case when m is less than 2, so to the right of ABOUT a second intersection point appears.

What FROM– the only non-vertical straight line passing through a point ABOUT and having only one common point with the graph, not its most important property. Indeed, if we turn to other graphs, it will soon become clear that the property of the tangent in the general case is not executed. For example, from Fig. 4 it is clear that near the point (1,1) the graph of the curve y = x 3 is well approximated by a straight line RT which, however, has more than one common point with it. However, we would like to consider RT tangent to this graph at point R. Therefore, it is necessary to find some other way to highlight the tangent than the one that served us so well in the first example.

Let us assume that through the point ABOUT and an arbitrary point Q = (h,k) on the curve graph y = 2x – x 2 (Fig. 5) a straight line (called a secant) is drawn. Substituting the values ​​into the equation of the curve x = h And y = k, we get that k = 2h – h 2, therefore, the angular coefficient of the secant is equal to

At very small h meaning m close to 2. Moreover, choosing h close enough to 0 we can do m arbitrarily close to 2. We can say that m"tends to the limit" equal to 2 when h tends to zero, or whatever the limit m equals 2 at h tending to zero. Symbolically it is written like this:

Then the tangent to the graph at the point ABOUT is defined as a straight line passing through a point ABOUT, with a slope equal to this limit. This definition of a tangent is applicable in the general case.

Let's show the advantages of this approach with one more example: let's find the slope of the tangent to the graph of the curve y = 2x – x 2 at any point P = (x,y), not limited to the simplest case when P = (0,0).

Let Q = (x + h, y + k) – the second point on the graph, located at a distance h to the right of R(Fig. 6). We need to find the slope k/h secant PQ. Dot Q is at a distance

above the axis X.

Opening the brackets, we find:

Subtracting from this equation y = 2x – x 2, find the vertical distance from the point R to the point Q:

Therefore, the slope m secant PQ equals

Now that h tends to zero, m tends to 2 – 2 x; We will take the last value as the angular coefficient of the tangent P.T.. (The same result will occur if h accepts negative values, which corresponds to the choice of point Q on the left of P.) Note that when x= 0 the result obtained coincides with the previous one.

Expression 2 – 2 x called the derivative of 2 x – x 2. In the old days, the derivative was also called "differential ratio" and " differential coefficient" If by expression 2 x – x 2 designate f(x), i.e.

then the derivative can be denoted

In order to find out the slope of the tangent to the graph of the function y = f(x) at some point, it is necessary to substitute in fў ( x) value corresponding to this point X. Thus, the slope fў (0) = 2 at X = 0, fў (0) = 0 at X= 1 and fў (2) = –2 at X = 2.

The derivative is also denoted atў , dy/dx, D x y And Du.

The fact that the curve y = 2x – x 2 near a given point is practically indistinguishable from its tangent at this point, allows us to speak of the angular coefficient of the tangent as the “angular coefficient of the curve” at the point of tangency. Thus, we can say that the slope of the curve we are considering has a slope of 2 at the point (0,0). We can also say that when x= 0 rate of change y relatively x is equal to 2. At point (2,0) the slope of the tangent (and the curve) is –2. (The minus sign means that as we increase x variable y decreases.) At the point (1,1) the tangent is horizontal. We say it's a curve y = 2x – x 2 has a stationary value at this point.

Highs and lows.

We have just shown that the curve f(x) = 2x – x 2 is stationary at point (1,1). Because fў ( x) = 2 – 2x = 2(1 – x), it is clear that when x, less than 1, fў ( x) is positive, and therefore y increases; at x, large 1, fў ( x) is negative, and therefore y decreases. Thus, in the vicinity of the point (1,1), indicated in Fig. 6 letter M, meaning at grows to a point M, stationary at point M and decreases after the point M. This point is called the “maximum” because the value at at this point exceeds any of its values ​​in a sufficiently small neighborhood. Similarly, the “minimum” is defined as the point in the vicinity of which all values y exceed the value at at this very point. It may also happen that although the derivative of f(x) at a certain point and vanishes; its sign in the vicinity of this point does not change. Such a point, which is neither a maximum nor a minimum, is called an inflection point.

As an example, let's find stationary point crooked

The derivative of this function is equal to

and goes to zero at x = 0, X= 1 and X= –1; those. at points (0,0), (1, –2/15) and (–1, 2/15). If X a little less than –1, then fў ( x) is negative; If X a little more than –1, then fў ( x) is positive. Therefore, the point (–1, 2/15) is the maximum. Similarly, it can be shown that the point (1, –2/15) is a minimum. But the derivative fў ( x) is negative both before the point (0,0) and after it. Therefore, (0,0) is the inflection point.

The study of the shape of the curve, as well as the fact that the curve intersects the axis X at f(x) = 0 (i.e. when X= 0 or ) allow us to present its graph approximately as shown in Fig. 7.

In general, if we exclude unusual cases(curves containing straight segments or an infinite number of bends), there are four options relative position curve and tangent in the vicinity of the tangent point R. (Cm. rice. 8, on which the tangent has a positive slope.)

1) On both sides of the point R the curve lies above the tangent (Fig. 8, A). In this case they say that the curve at the point R convex down or concave.

2) On both sides of the point R the curve is located below the tangent (Fig. 8, b). In this case, the curve is said to be convex upward or simply convex.

3) and 4) The curve is located above the tangent on one side of the point R and below - on the other. In this case R– inflection point.

Comparing values fў ( x) on both sides of R with its value at the point R, one can determine which of these four cases one has to deal with in a particular problem.

Applications.

All of the above has important applications in various areas. For example, if a body is thrown vertically upward from initial speed 200 feet per second then altitude s, on which they will be located through t seconds compared to starting point will be

Proceeding in the same way as in the examples we considered, we find

this quantity goes to zero at c. Derivative fў ( x) is positive up to the value c and negative after this time. Hence, s increases to , then becomes stationary, and then decreases. That's how it is general description movements of a body thrown upward. From it we know when the body reaches highest point. Next, substituting t= 25/4 V f(t), we get 625 feet, the maximum lift height. In this problem fў ( t) has a physical meaning. This derivative shows the speed at which the body is moving at an instant t.

Let us now consider an application of another type (Fig. 9). From a sheet of cardboard with an area of ​​75 cm2, you need to make a box with a square bottom. What should be the dimensions of this box in order for it to have maximum volume? If X– side of the base of the box and h is its height, then the volume of the box is V = x 2 h, and the surface area is 75 = x 2 + 4xh. Transforming the equation, we get:

Derivative of V turns out to be equal

and goes to zero at X= 5. Then

And V= 125/2. Graph of a function V = (75x – x 3)/4 is shown in Fig. 10 (negative values X omitted as not having physical meaning in this problem).

Derivatives.

An important task of differential calculus is the creation of methods that allow you to quickly and conveniently find derivatives. For example, it is easy to calculate that

(The derivative of a constant is, of course, zero.) It is not difficult to derive a general rule:

Where n– any whole number or fraction. For example,

(This example shows how useful fractional indicators degrees.)

Here are some of the most important formulas:

There are also the following rules: 1) if each of the two functions g(x) And f(x) has derivatives, then the derivative of their sum is equal to the sum of the derivatives of these functions, and the derivative of the difference is equal to the difference of the derivatives, i.e.

2) the derivative of the product of two functions is calculated by the formula:

3) the derivative of the ratio of two functions has the form

4) the derivative of a function multiplied by a constant is equal to the constant multiplied by the derivative of this function, i.e.

It often happens that the values ​​of a function have to be calculated step by step. For example, to calculate sin x 2, we first need to find u = x 2, and then calculate the sine of the number u. We find the derivative of such complex functions using the so-called “chain rule”:

In our example f(u) = sin u, fў ( u) = cos u, hence,

These and other similar rules allow you to immediately write down derivatives of many functions.

Linear approximations.

The fact that, knowing the derivative, we can in many cases replace the graph of a function near a certain point of its tangent at this point, has great value, since straight lines are easier to work with.

This idea finds direct application in calculating approximate values ​​of functions. For example, it is quite difficult to calculate the value when x= 1.033. But you can use the fact that the number 1.033 is close to 1 and that . Up close x= 1 we can replace the graph with a tangent curve without making any serious mistakes. The slope of such a tangent equal to the value derivative ( x 1/3)ў = (1/3) x–2/3 at x = 1, i.e. 1/3. Since point (1,1) lies on the curve and the angular coefficient of the tangent to the curve at this point is equal to 1/3, the tangent equation has the form

On this straight line X = 1,033

Received value y should be very close to the true value y; and, indeed, it is only 0.00012 more than the true one. In mathematical analysis, methods have been developed that make it possible to increase the accuracy of this kind of linear approximations. These methods ensure the reliability of our approximate calculations.

The procedure just described suggests a useful notation. Let P– point corresponding to the function graph f variable X, and let the function f(x) is differentiable. Let's replace the graph of the curve near the point R tangent to it drawn at this point. If X change by value h, then the ordinate of the tangent will change by the amount h H f ў ( x). If h is very small, then the latter value serves as a good approximation to the true change in the ordinate y graphic arts. If instead h we will write the symbol dx(this is not a product!), but a change in ordinate y let's denote dy, then we get dy = f ў ( x)dx, or dy/dx = f ў ( x) (cm. rice. eleven). Therefore, instead of Dy or f ў ( x) the symbol is often used to denote a derivative dy/dx. The convenience of this notation depends mainly on the explicit appearance of the chain rule (differentiation complex function); in the new notation this formula looks like this:

where it is implied that at depends on u, A u in turn depends on X.

Magnitude dy called differential at; in reality it depends on two variables, namely: from X and increments dx. When the increment dx very small size dy is close to the corresponding change in value y. But assume that the increment dx little, no need.

Derivative of a function y = f(x) we designated f ў ( x) or dy/dx. It is often possible to take the derivative of the derivative. The result is called the second derivative of f (x) and is denoted f ўў ( x) or d 2 y/dx 2. For example, if f(x) = x 3 – 3x 2, then f ў ( x) = 3x 2 – 6x And f ўў ( x) = 6x– 6. Similar notation is used for higher order derivatives. However, to avoid large quantity strokes (equal to the order of the derivative), the fourth derivative (for example) can be written as f (4) (x), and the derivative n-th order as f (n) (x).

It can be shown that the curve at a point is convex downward if the second derivative is positive, and convex upward if the second derivative is negative.

If a function has a second derivative, then the change in value y, corresponding to the increment dx variable X, can be approximately calculated using the formula

This approximation is usually better than that given by the differential fў ( x)dx. It corresponds to replacing part of the curve not with a straight line, but with a parabola.

If the function f(x) there are derivatives of higher orders, then

The remainder term has the form

Where x- some number between x And x + dx. The above result is called Taylor's formula with remainder term. If f(x) has derivatives of all orders, then usually Rn® 0 at n ® Ґ .

INTEGRAL CALCULUS

Squares.

When studying the areas of curvilinear flat figures New aspects of mathematical analysis are opening up. The ancient Greeks tried to solve problems of this kind, for whom determining, for example, the area of ​​a circle was one of the most difficult tasks. Much success Archimedes achieved a solution to this problem, who also managed to find the area of ​​a parabolic segment (Fig. 12). Using very complex reasoning, Archimedes proved that the area of ​​a parabolic segment is 2/3 of the area of ​​the circumscribed rectangle and, therefore, in this case is equal to (2/3)(16) = 32/3. As we will see later, this result can be easily obtained by methods of mathematical analysis.

The predecessors of Newton and Leibniz, mainly Kepler and Cavalieri, solved problems of calculating the areas of curvilinear figures using a method that can hardly be called logically sound, but which turned out to be extremely fruitful. When Wallis in 1655 combined the methods of Kepler and Cavalieri with the methods of Descartes ( analytical geometry) and took advantage of the newly born algebra, the stage was completely set for the appearance of Newton.

Wallis divided the figure, the area of ​​which needed to be calculated, into very narrow strips, each of which he approximately considered a rectangle. Then he added up the areas of the approximating rectangles and in the simplest cases obtained the value to which the sum of the areas of the rectangles tended when the number of strips tended to infinity. In Fig. Figure 13 shows rectangles corresponding to some division into strips of the area under the curve y = x 2 .

Main theorem.

The great discovery of Newton and Leibniz made it possible to eliminate the laborious process of going to the limit of the sum of areas. This was done thanks to a new look at the concept of area. The point is that we must imagine the area under the curve as generated by an ordinate moving from left to right and ask at what rate the area swept by the ordinates changes. We will get the key to answering this question if we consider two special cases in which the area is known in advance.

Let's start with the area under the graph of a linear function y = 1 + x, since in this case the area can be calculated using elementary geometry.

Let A(x) – part of the plane enclosed between the straight line y = 1 + x and a segment OQ(Fig. 14). When driving QP right area A(x) increases. At what speed? It is not difficult to answer this question, since we know that the area of ​​a trapezoid is equal to the product of its height and half the sum of its bases. Hence,

Rate of area change A(x) is determined by its derivative

We see that Aў ( x) coincides with the ordinate at points R. Is this a coincidence? Let's try to check on the parabola shown in Fig. 15. Area A (x) under the parabola at = X 2 in the range from 0 to X equal to A(x) = (1 / 3)(x)(x 2) = x 3/3. The rate of change of this area is determined by the expression

which exactly coincides with the ordinate at moving point R.

If we assume that this rule holds in the general case such that

is the rate of change of the area under the graph of the function y = f(x), then this can be used for calculations and other areas. In fact, the ratio Aў ( x) = f(x) expresses a fundamental theorem that could be stated as follows: the derivative, or rate of change of area as a function of X, equal to the function value f (x) at point X.

For example, to find the area under the graph of a function y = x 3 from 0 to X(Fig. 16), let's put

A possible answer reads:

since the derivative of X 4 /4 is really equal X 3. Besides, A(x) is equal to zero at X= 0, as it should be if A(x) is indeed an area.

Mathematical analysis proves that there is no other answer other than the above expression for A(x), does not exist. Let us show that this statement is plausible using the following heuristic (non-rigorous) reasoning. Suppose there is some second solution IN(x). If A(x) And IN(x) “start” simultaneously from zero value at X= 0 and change at the same rate all the time, then their values ​​​​cannot be X cannot become different. They must coincide everywhere; therefore, there is a unique solution.

How can you justify the relationship? Aў ( x) = f(x) in general? This question can only be answered by studying the rate of change of area as a function of X in general. Let m – smallest value functions f (x) in the range from X before ( x + h), A M – highest value this function in the same interval. Then the increase in area when going from X To ( x + h) must be enclosed between the areas of two rectangles (Fig. 17). The bases of both rectangles are equal h. The smaller rectangle has a height m and area mh, larger, respectively, M And Mh. On the graph of area versus X(Fig. 18) it is clear that when the abscissa changes to h, the ordinate value (i.e. area) increases by the amount between mh And Mh. The secant slope on this graph is between m And M. what happens when h tends to zero? If the graph of a function y = f(x) is continuous (i.e. does not contain discontinuities), then M, And m tend to f(x). Therefore, the slope Aў ( x) graph of area as a function of X equals f(x). This is precisely the conclusion that needed to be reached.

Leibniz proposed for the area under a curve y = f(x) from 0 to A designation

In a rigorous approach, this so-called definite integral should be defined as the limit of certain sums in the manner of Wallis. Considering the result obtained above, it is clear that this integral is calculated provided that we can find such a function A(x), which vanishes when X= 0 and has a derivative Aў ( x), equal to f (x). Finding such a function is usually called integration, although it would be more appropriate to call this operation anti-differentiation, meaning that it is in some sense the inverse of differentiation. In the case of a polynomial, integration is simple. For example, if

which is easy to verify by differentiating A(x).

To calculate the area A 1 under the curve y = 1 + x + x 2 /2, enclosed between ordinates 0 and 1, we simply write

and, substituting X= 1, we get A 1 = 1 + 1/2 + 1/6 = 5/3. Square A(x) from 0 to 2 is equal to A 2 = 2 + 4/2 + 8/6 = 16/3. As can be seen from Fig. 19, the area enclosed between ordinates 1 and 2 is equal to A 2 – A 1 = 11/3. It is usually written as a definite integral

Volumes.

Similar reasoning makes it surprisingly easy to calculate the volumes of bodies of revolution. Let's demonstrate this using the example of calculating the volume of a ball, another classical problem, which the ancient Greeks, using methods known to them, managed to solve with great difficulty.

Let's rotate part of the plane contained inside a quarter circle of radius r, at an angle of 360° around the axis X. As a result, we get a hemisphere (Fig. 20), the volume of which we denote V(x). We need to determine the rate at which it increases V(x) with increasing x. Moving from X To X + h, it is easy to verify that the increment in volume is less than the volume p(r 2 – x 2)h circular cylinder with radius and height h, and more than volume p[r 2 – (x + h) 2 ]h cylinder radius and height h. Therefore, on the graph of the function V(x) the angular coefficient of the secant is enclosed between p(r 2 – x 2) and p[r 2 – (x + h) 2 ]. When h tends to zero, the slope tends to

At x = r we get

for the volume of the hemisphere, and therefore 4 p r 3/3 for the volume of the entire ball.

A similar method allows one to find the lengths of curves and the areas of curved surfaces. For example, if a(x) – arc length PR in Fig. 21, then our task is to calculate aў( x). Let us use at the heuristic level a technique that allows us not to resort to the usual passage to the limit necessary when strict proof result. Let us assume that the rate of change of the function A(x) at point R the same as it would be if the curve were replaced by its tangent P.T. at the point P. But from Fig. 21 is directly visible when stepping h to the right or left of the point X along RT meaning A(x) changes to

Therefore, the rate of change of the function a(x) is

To find the function itself a(x), you just need to integrate the expression on the right side of the equality. It turns out that integration is quite difficult for most functions. Therefore, the development of methods of integral calculus is most mathematical analysis.

Antiderivatives.

Every function whose derivative is equal to the given function f(x), is called antiderivative (or primitive) for f(x). For example, X 3 /3 – antiderivative for the function X 2 since ( x 3 /3)ў = x 2. Of course X 3 /3 is not the only antiderivative of the function X 2 because x 3 /3 + C is also a derivative for X 2 for any constant WITH. However, in what follows we agree to omit such additive constants. In general

Where n is a positive integer, since ( x n + 1/(n+ 1))ў = x n. Relation (1) is satisfied in an even more general sense if n replace with any rational number k, except –1.

An arbitrary antiderivative function for given function f(x) is usually called the indefinite integral of f(x) and denote it in the form

For example, since (sin x)ў = cos x, the formula is valid

In many cases where there is a formula for the indefinite integral of a given function, it can be found in numerous widely published tables of indefinite integrals. The integrals of elementary functions(these include powers, logarithms, exponential function, trigonometric functions, inverse trigonometric functions, as well as their finite combinations obtained using the operations of addition, subtraction, multiplication and division). Using table integrals you can calculate integrals of more complex functions. There are many ways to calculate indefinite integrals; The most common of these is the variable substitution or substitution method. It consists in the fact that if we want to replace in the indefinite integral (2) x to some differentiable function x = g(u), then for the integral to remain unchanged, it is necessary x replaced by gў ( u)du. In other words, the equality

(substitution 2 x = u, from where 2 dx = du).

Let us present another integration method - the method of integration by parts. It is based on the already known formula

By integrating the left and right sides, and taking into account that

This formula is called the integration by parts formula.

Example 2. You need to find . Since cos x= (sin x)ў , we can write that

From (5), assuming u = x And v= sin x, we get

And since (–cos x)ў = sin x we find that

It should be emphasized that we limited ourselves to only very a brief introduction into a very vast subject, in which numerous ingenious techniques have been accumulated.

Functions of two variables.

Due to the curve y = f(x) we considered two problems.

1) Find the angular coefficient of the tangent to the curve at a given point. This problem is solved by calculating the value of the derivative fў ( x) at the specified point.

2) Find the area under the curve above the axis segment X, limited vertical lines X = A And X = b. This problem is solved by calculating a definite integral.

Each of these problems has an analogue in the case of a surface z = f(x,y).

1) Find the tangent plane to the surface at a given point.

2) Find the volume under the surface above the part of the plane xy, bounded by a curve WITH, and from the side – perpendicular to the plane xy passing through the points of the boundary curve WITH (cm. rice. 22).

The following examples show how these problems are solved.

Example 4. Find the tangent plane to the surface

at point (0,0,2).

A plane is defined if two intersecting lines lying in it are given. One of these straight lines ( l 1) we get in the plane xz (at= 0), second ( l 2) – in the plane yz (x = 0) (cm. rice. 23).

First of all, if at= 0, then z = f(x,0) = 2 – 2x – 3x 2. Derivative with respect to X, denoted fў x(x,0) = –2 – 6x, at X= 0 has a value of –2. Straight l 1 given by the equations z = 2 – 2x, at= 0 – tangent to WITH 1, lines of intersection of the surface with the plane at= 0. Similarly, if X= 0, then f(0,y) = 2 – y – y 2 , and the derivative with respect to at looks like

Because fў y(0,0) = –1, curve WITH 2 – line of intersection of the surface with the plane yz– has a tangent l 2 given by the equations z = 2 – y, X= 0. The desired tangent plane contains both lines l 1 and l 2 and is written by the equation

This is the equation of the plane. In addition, we receive direct l 1 and l 2, assuming, accordingly, at= 0 and X = 0.

The fact that equation (7) really defines a tangent plane can be verified at a heuristic level by noting that this equation contains first-order terms included in equation (6), and that second-order terms can be represented in the form -. Since this expression is negative for all values X And at, except X = at= 0, surface (6) lies below plane (7) everywhere, except for the point R= (0,0,0). We can say that surface (6) is convex upward at the point R.

Example 5. Find the tangent plane to the surface z = f(x,y) = x 2 – y 2 at origin 0.

On surface at= 0 we have: z = f(x,0) = x 2 and fў x(x,0) = 2x. On WITH 1, intersection lines, z = x 2. At the point O the slope is equal to fў x(0,0) = 0. On the plane X= 0 we have: z = f(0,y) = –y 2 and fў y(0,y) = –2y. On WITH 2, intersection lines, z = –y 2. At the point O curve slope WITH 2 is equal fў y(0,0) = 0. Since the tangents to WITH 1 and WITH 2 are axes X And at, the tangent plane containing them is the plane z = 0.

However, in the neighborhood of the origin, our surface is not on the same side of the tangent plane. Indeed, a curve WITH 1 everywhere, except point 0, lies above the tangent plane, and the curve WITH 2 – respectively below it. Surface intersects tangent plane z= 0 in straight lines at = X And at = –X. Such a surface is said to have a saddle point at the origin (Fig. 24).

Partial derivatives.

In previous examples we used derivatives of f (x,y) By X and by at. Let us now consider such derivatives in more in general terms. If we have a function of two variables, for example, F(x,y) = x 2 – xy, then we can determine at each point two of its “partial derivatives”, one by differentiating the function with respect to X and fixing at, the other – differentiating by at and fixing X. The first of these derivatives is denoted as fў x(x,y) or ¶ f/¶ x; second - how f f ў y. If both mixed derivatives (by X And at, By at And X) are continuous, then ¶ 2 f/¶ x¶ y= ¶ 2 f/¶ y¶ x; in our example ¶ 2 f/¶ x¶ y= ¶ 2 f/¶ y¶ x = –1.

Partial derivative fў x(x,y) indicates the rate of change of the function f at point ( x,y) in the direction of increasing X, A fў y(x,y) – rate of change of function f in the direction of increasing at. Rate of change of function f at point ( X,at) in the direction of a straight line making an angle q with positive axis direction X, is called the derivative of the function f towards; its value is a combination of two partial derivatives of the function f in the tangent plane is almost equal (at small dx And dy) true change z on the surface, but calculating the differential is usually easier.

The formula we have already considered from the change of variable method, known as the derivative of a complex function or the chain rule, in the one-dimensional case when at depends on X, A X depends on t, has the form:

For functions of two variables, a similar formula has the form:

The concepts and notations of partial differentiation are easy to generalize to higher dimensions. In particular, if the surface is specified implicitly by the equation f(x,y,z) = 0, the equation of the tangent plane to the surface can be given a more symmetrical form: the equation of the tangent plane at the point ( x(x 2 /4)], then integrated over X from 0 to 1. The final result is 3/4.

Formula (10) can also be interpreted as a so-called double integral, i.e. as the limit of the sum of the volumes of elementary “cells”. Each such cell has a base D x D y and height, equal to height surface above some point of the rectangular base ( cm. rice. 26). It can be shown that both points of view on formula (10) are equivalent. Double integrals are used to find centers of gravity and numerous moments encountered in mechanics.

A more rigorous justification of the mathematical apparatus.

So far we have presented the concepts and methods of mathematical analysis on an intuitive level and did not hesitate to resort to geometric shapes. It remains for us to briefly consider more strict methods, which appeared in the 19th and 20th centuries.

At the beginning of the 19th century, when the era of storm and pressure in the “creation of mathematical analysis” ended, questions of its justification came to the fore. In the works of Abel, Cauchy and a number of other outstanding mathematicians, the concepts of “limit”, “continuous function”, “convergent series” were precisely defined. This was necessary in order to introduce logical order into the basis of mathematical analysis in order to make it a reliable research tool. The need for a thorough justification became even more obvious after the discovery in 1872 by Weierstrass of functions that were everywhere continuous but nowhere differentiable (the graph of such functions has a kink at each point). This result had a stunning effect on mathematicians, since it clearly contradicted their geometric intuition. An even more striking example of the unreliability of geometric intuition was the continuous curve constructed by D. Peano, which completely fills a certain square, i.e. passing through all its points. These and other discoveries gave rise to the program of “arithmetization” of mathematics, i.e. making it more reliable by justifying all mathematical concepts using the concept of number. The almost puritanical abstinence from clarity in works on the foundations of mathematics had its historical justification.

According to modern canons of logical rigor, it is unacceptable to talk about the area under the curve y = f(x) and above the axis segment X, even f – continuous function without first defining exact meaning the term “area” without establishing that the area thus defined actually exists. This problem was successfully solved in 1854 by B. Riemann, who gave precise definition the concept of a definite integral. Since then, the idea of ​​summation behind the concept of a definite integral has been the subject of many in-depth studies and generalizations. As a result, today it is possible to give meaning definite integral, even if the integrand is discontinuous everywhere. New concepts of integration, to the creation of which A. Lebesgue (1875–1941) and other mathematicians made a great contribution, increased the power and beauty of modern mathematical analysis.

It would hardly be appropriate to go into detail about all these and other concepts. We will limit ourselves only to giving strict definitions of the limit and the definite integral.

In conclusion, let us say that mathematical analysis, being an extremely valuable tool in the hands of a scientist and engineer, still attracts the attention of mathematicians today as a source of fruitful ideas. In the same time modern development seems to indicate that mathematical analysis is increasingly being absorbed by those dominant in the 20th century. branches of mathematics such as abstract algebra and topology.