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All sciences. 3,2023. International Scientific Journal
Foziljon Oripovich Obidov

Sanjar Sadiqjonovich Odilov

Ibratjon Aliyev

Nurmuxammad Sultonaliyevich Toyirov

Sultonali Mukaramovich Abduraxmonov

Toira Abdusalyamova


The international scientific journal All Sciences, created at OOO Electron Laboratory and the Scientific School Electron, is a scientific publication that publishes the latest scientific results in various fields of science and technology, also representing a collection of publications on the above topics by a board of authors and reviewed by the editorial Board (academic Council) of the Scientific School Eletkron and on the Ridero platform monthly.





All sciences. 3,2023

International Scientific Journal



Authors: AliyevIbratjon, AbduraxmonovSultonaliMukaramovich, OdilovSanjarSadiqjonovich, ToyirovNurmuxammadSultonaliyevich, ObidovFoziljonOripovich, AbdusalyamovaToira



Editor-in-Chief Ibratjon Xatamovich Aliyev

Illustrator Ibratjon Xatamovich Aliyev

Illustrator Obbozjon Xokimovich Qo'ldashov

Illustrator Sultonali Mukaramovich Abduraxmonov

Cover design Ibratjon Xatamovich Aliyev

Cover design Ra'noxon Mukaramovna Aliyeva

Acting scientific supervisor Sultonali Mukaramovich Abduraxmonov

Economic Manager Farruh Murodjonovich Sharofutdinov

Economic consultant Botirali Rustamovich Jalolov

Proofreader Gulnoza Muxtarovna Sobirova

Proogreader Abdurasul Abdusoliyevich Ergashev

Proofreadfer Ekaterina Aleksandrovna Vavilova



Ibratjon Aliyev,2023

Sultonali Mukaramovich Abduraxmonov,2023

Sanjar Sadiqjonovich Odilov,2023

Nurmuxammad Sultonaliyevich Toyirov,2023

Foziljon Oripovich Obidov,2023

Toira Abdusalyamova,2023



ISBN978-5-0059-9245-1 (. 3)

ISBN978-5-0059-5900-3

Created with Ridero smart publishing system




PHYSICAL AND MATHEMATICAL SCIENCES





THE IMPORTANCE OFDIFFERENTIAL EQUATIONS INTHE STUDY OFGENERAL LAWS AND THE SIMPLEST CASES OFTRANSFORMATION





Aliev Ibratjon Khatamovich







2nd year student of the Faculty of Mathematics and Computer Science of Fergana State University







Ferghana State University, Ferghana, Uzbekistan


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Annotation. The study ofthe surrounding world directly reduces tothe need tomake certain forecasts, which are already reduced tothe importance ofestablishing for them the basic laws ofthe universe, which can be observed during the study ofcertain phenomena. At the same time, there is often the use ofphysical laws that are possible todescribe using not only ordinary equations, but also differential equations ofthe first and second orders, including alarge number ofpartial differential equations, quite often used inthis study and understanding.

Keywords: partial differential equations, ordinary differential equations, mathematical modeling, analogy, regularities.

Coming tothe study ofthe laws ofthe world inphysical science, certain laws were most often distinguished, the initial ones among which are the mechanical laws created byNewton and developed mathematically from his side, along with other scientists, among whom the figure ofLeibniz stands out vividly. For an example ofthis statement, we can give the differential forms ofthe basic equations ofmotion (1), which inturn are reduced tocertain values inthe formulas ofacceleration (2), force (3), work (4), power (5) and others.




























The present moments ofunderstanding can most often be considered precisely indifferential forms ofmeaning, for the reason that they can be numerically determined byintroducing some transformations, namely, byconverting (6) and taking acertain integral with the establishment ofcertain boundaries(7).













Such directions are developed not only in mechanical terms, but also in other branches of physics, electrostatics, electrodynamics, magnetostatics, magneto-dynamics and others can be a vivid example of this. To prove this, it is enough just to mention that the very concept of current strength is a derivative of the charge time, and voltage is a derivative of the work charge.

This statement can be given for a large number of very different understandings, but the important fact is that such an approach, unlike classical mathematical regulation, becomes the only one when it is necessary to describe the gravitational characteristics of space on the scale of the entire space. An example of this kind of phenomena, where the use of derivatives and, accordingly, differential equations becomes known quantum physics.

However, on the scale ofphenomena where the classical mathematical apparatus can no longer perform its functions, it is not so much the usual classical derivatives that are reduced toordinary differential equations that are important, unless, ofcourse, the simplest cases are not taken into account, avivid example ofwhich is overcoming the potential well ofaparticle or describing its motion, or other similar trivial cases, only partial differential equations are more interesting.




Used literature


1.Pontryagin L. S. Ordinary differential equations. M.: Nauka, 1974.

2. Tikhonov A. N., Samarsky A. A. Equations of mathematical physics. M.: Nauka, 1972.

3. Tikhonov A. N., Vasilyeva A. B., Sveshnikov A. G. Differential equations. 4th ed. Fzimatlit, 2005.

4. Umnov A. E., Umnov E. A. Fundamentals of the theory of differential equations. Ed. 2nd 2007. 240 p.

5. Charles Henry Edwards, David E. Penny. Differential Equations and the problem of eigenvalues: Modeling and calculation using Mathematica, Maple and MATLAB = Differential Equations and Boundary Value Problems: Computing and Modeling. 3rd ed. M.: "Williams", 2007.

6. ElsholtsL.E.Differential Equations and Calculus ofVariations. M.: Science, 1969.




SOME OPERATIONS AND SPECIAL CASES OFMATHEMATICAL ANALYSIS INTHE EXPONENTIALSET





Aliev Ibratjon Khatamovich







2nd year student of the Faculty of Mathematics and Computer Science of Fergana State University







Ferghana State University, Ferghana, Uzbekistan


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Annotation. The importance ofdefining and converting exponential numbers and areal set is becoming more and more obvious every day, especially with the entry ofthis concept into mathematical physics, but as apurely mathematical object they are ofno small interest, although they also have practical applications. Inthis paper, methods ofperforming some algebraic operations with them are described, including using Eulers formula and integrals.

Keywords: inertial numbers, mathematical analysis, algebraic operations, Euler formula, integration, derivatives.

The very process oflogarithmization ofan exponential number ofageneral form can be seen in(1).








Thus, when logarithming, 2parts ofthe expression itself are formed the real one, as the natural logarithm ofthe coefficient ofthe ingential part and the logarithm ofthe ingential unit, which is defined in(2).








That is, inthis case, the question arises towhat degree it is necessary toraise the Euler number so that it gives an exponential unit. The answer is quite simple it is anegative logarithm ofzero (2) from this it follows that the logarithm ofthe exponential number is(3).








It is also interesting tosolve the Euler equation with atangential unit, and then with ageneral form ofan exponential number, which was described further, taking the expressions as unknowns. And for this, we can initially proceed from Taylor expansions (46).


















Which is easily proved, since when the unknown is zeroed, the sine in(5) is also zeroed, and the cosine in(6) is equal toone. And it already follows from this(7).








And the unknown in(7) can be all kinds ofnumbers, both complex, when substituting which the remarkable Euler equality follows, and exponential. And tobegin with, lets consider aspecial case with an exponential unit and perform the following transformations(8).








Based on this relation, we perform transformations in(9), leading toequation (10), while taking into account that this expression is identical, it is possible todifferentiate both parts ofthe equation in(11) byperforming the corresponding transformations.


















Since the final equality (11) can be represented as in(12), further carrying out additional differentiation, also introducing the condition that this is an identity, and in(13) the differentiation process for the right side ofequality is described indetail. And for the left part there is no need for adetailed painting.













When the differentiation is made, it is enough tomake elementary transformations, we get the trigonometric form ofthe special case (14).








Now, when the general form for the doubly differentiated case is obtained, it is necessary toreturn tothe primordial ones, because this is the identity, resulting inthe following equalities (1516).













And indeed, this value is close tothe most potential value, so this expression can be considered the second kind ofwriting ofthe exponential unit. Now, it is possible toproceed tothe solution ofthe Euler equation for the general form ofthe intentional numbers, having carried out the first substitution and the usual replacement operations at stage (17) and (18) at the beginning.













When the necessary transformations come toan end, and other actions no longer take place, it is also sufficient todifferentiate both parts ofequality as avalid identity (19).








Differentiating the first part ofthe equality, we can come tothe result in(20), and for the second part, the calculations will continue throughout (21).













Then, applying (2225), one can come tothe form (26).




























As aresult, it is enough toequalize both results in(20) and (26), since these are two parts ofthe identity, and then get (27) with the necessary simplification, and already in(28) with additional simplification and differentiation as an identity.













At the same time, the differentiation ofthe first part ofequality is obvious in(29), as well as the second in(30), after which equality and the resulting transformations can be introduced into (31).


















As aresult, equalities are formed that need tobe integrated twice, because their derivatives were taken earlier, getting (32).








Integrating the first part, aseparate result is obtained in(33) and integrating the second part in(34).













Thus, it is possible toarrive at equality (35), from where it is possible toarrive at another equality inthe same equation.








The result is really quite surprising, but this is equality (35), which came out after substituting the general form ofan ingential number into Eulers formula and the solution for this case is the ingential number (36). Thus, this is the first full-fledged equation, the solution ofwhich was an intentional number.








Although the complex numbers themselves are located on the axis ofnumbers, this interval can also be expressed on the tangential plane. This coordinate system has an axis starting from infinity as the ordinate, and the abscissa has all real numbers. Thus, all exponential numbers can be represented on such arectangular coordinate system, inthe case ofadding complex numbers already inspace.




Used literature


1.I. V. Bargatin, B. A. Grishanin, V. N. Zadkov. Entangled quantum states of atomic systems. Editorial office named after Lomonosov. 2001.

2. G. Kane. Modern elementary particle physics. Publishing house Mir. 1990.

3. S. Hawking. The theory of everything. From singularity to infinity: the origin and fate of the universe. Publishing house AST. 2006.

4. S. Hawking, L. Mlodinov. The supreme plan. A physicist's view of the creation of the world. Publishing house AST. 2010.

5. T. D'amour. The world according to Einstein. From relativity theory to string theory. Moscow Publishing House. 2016.

6. S. Hawking, L. Mlodinov. The shortest history oftime. Amphora Publishing House. 2011.




ABOUT RESEARCH ON THE COLLATZ HYPOTHESIS INTHE FACE OFAMATHEMATICAL PHENOMENON





Aliev Ibratjon Khatamovich







2nd year student of the Faculty of Mathematics and Computer Science of Fergana State University







Ferghana State University, Ferghana, Uzbekistan


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Annotation. When young mathematicians are told about this problem, they are immediately warned that it is not worth taking up its solution, because it seems impossible. Asimple-looking hypothesis could not be proved bythe best minds ofmankind. For comparison, the famous mathematician Paul Erdos said: Mathematics is not yet ripe for such questions. However, it is worth studying this hypothesis inmore detail, which is investigated inthis paper.

Keywords: Collatz hypothesis, hailstone numbers, series, algorithm, sequences, proofs.

In short, its essence is as follows. A certain number is selected and if it is not even, it is multiplied by 3 and 1 is added, if it is even, then divided by 2.

We can give an algorithm of this series for the number 7:

7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1

Next, a cycle is obtained:

1 4 2 1 etc.

This leads to the hypothesis that if you take any positive integer, if you follow the algorithm, it necessarily falls into the cycle 4, 2, 1. The hypothesis is named after Lothar Collatz, who is believed to have come to this hypothesis in the 30s of the last century, but this problem has many names, it is also known as the Ulam hypothesis, Kakutani's theorem, Toitz's hypothesis, Hass's algorithm, the Sikazuz sequence, or simply as "3n+1".

How did this hypothesis gain such fame? It is worth noting that in the professional environment, the fame of such a hypothesis is very bad, so the very fact that someone is working on this hypothesis may lead to the fact that this researcher will be called crazy or ignorant.

The numbers themselves that are obtained during this transformation are called hailstones, because, like hail in the clouds, the numbers then fall, then rise, but sooner or later, all fall to one, at least so it is believed. For convenience, we can make an analogy that the values entered into this algorithm are altitude above sea level. So, if you take the number 26, then it first sharply decreases, then rises to 40, after which it drops to 1 in 10 steps. Here you can give a series for 26:

26 13 40 20 10 5 16 8 4 2 1

However, if we take the neighboring number 27, it will jump at avariety ofheights, reaching the mark of9,232, which, continuing the analogy, is higher than Mount Everest, but even this number is destined tocollapse tothe Ground, although it will take 111steps toreach 1and get stuck inthe same loop. The same interesting numbers can be numbers 31, 41, 47, 54, 55, 62, 63, 71, 73, 82and others. For comparison, we can analyze the table (Table 1) and the graph (Fig. 1) for these interesting numbers.






Table 1. Aseries oflong numbers for interesting values ofnumbers-granules (the first line is the original value)






Table 1. Aseries oflong numbers for interesting values ofnumbers-granules (the first line is the original value)






Table 1. Aseries oflong numbers for interesting values ofnumbers-granules (the first line is the original value)






Table 1. Aseries oflong numbers for interesting values ofnumbers-granules (the first line is the original value)






Table 1. Aseries oflong numbers for interesting values ofnumbers-granules (the first line is the original value)






Table 1. Aseries oflong numbers for interesting values ofnumbers-granules (the first line is the original value)






Table 1. Aseries oflong numbers for interesting values ofnumbers-granules (the first line is the original value)






Table 1. Aseries oflong numbers for interesting values ofnumbers-granules (the first line is the original value)






Table 1. Aseries oflong numbers for interesting values ofnumbers-granules (the first line is the original value)






Table 1. Aseries oflong numbers for interesting values ofnumbers-granules (the first line is the original value)






Table 1. Aseries oflong numbers for interesting values ofnumbers-granules (the first line is the original value)






Table 1. Aseries oflong numbers for interesting values ofnumbers-granules (the first line is the original value)






Table 1. Aseries oflong numbers for interesting values ofnumbers-granules (the first line is the original value)






Fig. 1. Graph ofvalues for interesting numbers-granules ofthe algorithm



When the path of one number is so much different even from the neighboring one, how do you even approach the proof of such a hypothesis? Of course, all mathematicians were at a loss and absolutely no one could solve this problem. So Jeffrey Lagarias is a world expert on this problem, and he said that no one should take up this problem if he wants to become a mathematician. A large-scale work was carried out and a huge number of hailstones were studied, trying to find a pattern. Here it can be argued that all values come to one, however, what can be said about the path that all numbers take? The interesting thing is that this path is absolutely random.

For example, we can give agraph ofall the values ofthis algorithm from 1to100(Fig.2).






Fig. 2. Graph ofvalues for numbers-granules from 1to100



As you can see, most often growth begins initially and after a sharp decline, while the value of the number is simply not considered, however, if you make the graph logarithmic, there is a downward trend in its fluctuations. It can also be observed on the stock market on the day of the collapse, which is not accidental, because these are examples of geometric Brownian motion, that is, if you take logarithms and calculate the linear component, the fluctuations seem random, as if a coin was thrown at each step. And if we consider this function analysis as part of mathematical analysis, then there begins to be an obvious connection with probability theory. From where it turns out that when heads are obtained, the line goes up, and when tails go down, from where a special graph is obtained.

If we consider this chart when compared with the same exchange, then it is more likely in a short-term analysis, although in the long term, stocks are still growing, and "3x+1" is falling. You can also pay attention to the highest digit of the gradient numbers this means a histogram, which is obtained by counting the number of digits from which the numbers begin in a number of granules for a particular number of the algorithm. If you add these values each time, for 1, 2, 3, etc., more and more data is obtained, while the ratio of the height of the columns becomes more and more ordered.

So for the first billion sequences, the most frequent value is one, 29.94% of all cases, 2 17,47%, 3 12,09%, 4 10,63%, 5 7,94%, 6 6,16%, 7 5,76%, 8 5,31%, 9 4,7% and the larger the figure, the less often it turns out to be ahead.

This arrangement is typical not only for hailstone numbers, there are many examples, these are the population of countries, the value of companies, all physical constants or Fibonacci numbers, and much more. This law is called Benford's law. Surprisingly, if you trace the violation of the Benford law in tax returns, you can even determine the fact of fraud. This law also helps to identify anomalies in the counting of votes in elections or many other things.

The best effect of this law occurs when the numbers entered in it have a spread of several orders of magnitude, as in this case, but Benford's law, unfortunately, cannot tell whether all the numbers end up in the 4-2-1 cycle. To do this, you need to use a different method. Initially, it is strange that this algorithm reduces all numbers to 1, given that even and non-even numbers equally and non-even numbers increase by more than 3 times, and even numbers decrease by 2 times.

This suggests the conclusion that all sequences should, in theory, go up, not down. But it is worth paying attention to the fact that whenever an operation is performed with a non-even number, that is, when it is multiplied by 3 and 1 is added, it necessarily turns into an even number, therefore, the next step it will always be divided by 2. It turns out that non-even numbers are not tripled, but multiplied by (3x+1) /2 or more precisely by 1.5, because 0.5 for large numbers can be ignored. So the maximum growth from this is exactly 1.5.

A graph has already been given for all numbers from 1 to 100, but it is worth considering a small case for all non-even numbers. As you know, in the second step they turn into even values, and then exactly half of them are immediately reduced, after dividing again to non-even. But every 4 numbers will have to be divided by 2 twice, which means that these non-even numbers are ? of the previous one. Every 8th number will have to be divided by 2 three times to get an even number. Every 16 four times, etc.

So taking the geometric mean, you can see that in order to get from one non-even number to another through all even numbers, you need to multiply it by ?, which is less than one, hence it turns out that statistically, this sequence decreases more often than it grows.

Let's give an example for a large number, for example 341. Its row looks like this:

341 1024 512 256 128 64 32 16 8 4 2 1.

He had only one non-even and all even numbers, which is why this series is remarkable. However, they can be depicted both in the form of graphs and in the form of trees, showing how one of the numbers is connected with the next in its sequence, creating a graph.

And if the hypothesis is correct, then any number should be in this huge graph, consisting of an infinite number of "streams" forming 4-2-1 cycles in one stream. There is an interesting visualization of such a graph, which uses an algorithm that on noneven numbers, it rotates clockwise at the selected angle, and counterclockwise on even numbers.

As a result, an interesting curved structure is obtained, more often in one direction. Resembling coral, algae or a tree in the wind. But this is only for a small number of numbers, for huge arrays, changing the angles of rotation, you can create huge and dazzlingly beautiful figures, as if generated by nature.

The hypothesis seems to be incorrect only in 2 cases:

1. If a number is found that will give infinity in the algorithm, that is, for some unknown reason, this "force of attraction" to 4-2-1 should not act on it;

2. Somewhere there is a sequence that would form its own closed loop, and all the numbers in it should be outside the main graph.

However, none of these options has been found yet, although all numbers up to 2 to the 68th power have already been checked by a simple search, which equals 295,147,905,179,352,825,856 numbers. It is known for sure that all the numbers from these values come to the 4-2-1 cycle. Moreover, based on these data, it is calculated that even if there is such a special data cycle, it should consist of at least 186 billion numbers. And it turns out that all the works indicate that the hypothesis is true, but still does not prove it.

Another way was also chosen. A scattering graph was constructed by taking the numbers themselves on one axis and the values on the other. If it can be proved that in any sequence of the algorithm there is a number smaller than the original one, it is possible to confirm the Collatz hypothesis. But any initial number is reduced to a smaller number, which in its own sequence will lead to a number even smaller, etc., up to 1.

That is, the only possible outcome for this particular case is the 4-2-1 cycle, but it has not yet been possible to prove this.

Although in 1976, Riho Terras showed that almost all sequences include values below the original one. In 1979, it was shown that the values would be less than the original ones by these values raised to the power of 0.869. Later, in 1994, the degree became more precise 0.7925. Here, almost all the numbers mean that when the initial values tend to infinity, the proportion of the limiting function tends to 1. In 2019, mathematician Terry Tao was able to prove that this algorithm obeys even stricter restrictions.

He managed toshow that all numbers will be less than the values ofthe function at any point, provided that the limit ofthe function, when the variable tends toinfinity, will be equal toinfinity. Inthis case, the function can grow arbitrarily slowly, the same logarithm, or logarithm logarithm, or logarithm logarithm, etc. This allows us toassert that arbitrarily small numbers exist inthe series ofany initial number. And as it was said in2020, only adirect proof ofthe hypothesis can be better than this.




Used literature


1.Hayes, Brian. The ups and downs of hailstone numbers. American. 1984. No. 3. pp. 102-107.

2. Stewart, Ian. The greatest mathematical problems. M.: Alpina non-fiction, 2015. 460 p.

3. Jeff Lagarias. The 3x+1and its generalizations. American Mathematical Monthly. 1985. Vol. 92. P. 323.




GENERAL IDEA OFTHE CONCEPT AND USE OFDIFFERENTIAL EQUATIONS INTHE DESCRIPTION OFSOME DYNAMIC PHENOMENA





Aliev Ibratjon Khatamovich







2nd year student of the Faculty of Mathematics and Computer Science of Fergana State University







Ferghana State University, Ferghana, Uzbekistan


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Annotation. As Stephen Strogatz once said: Since the time ofNewton, mankind has come torealize that physicists are expressed inthe language ofdifferential equations. Ofcourse, this language is used far beyond physics and the talent touse it, exactly as toperceive it, gives new colors when studying the surrounding world. This paper describes ageneral idea ofthis method and the process ofstudyingit.

Keywords: differential equations, calculus, algorithms, mathematical physics.

By themselves, differential equations arise every time it is easier to describe a change than absolute values. For example, it is easier to describe the nature of an increase or decrease in the growth or decline of the population or population of a particular species than to describe certain values at a certain point in time. In physics, more precisely in Newtonian mechanics, motion is described by force, and force is determined by constant mass and changing acceleration, which is a statement of change.

Differential equations are divided into 2 large categories ordinary differential equations or ODES involving functions with one variable, most often in the face of time and partial differential equations with several variables. If partial differential equations describe more complex characteristics, for example, temperature changes at different points in space, then ordinary differential equations describe more static characteristics that change over time.

As agood example, we can consider the process offalling ofan object. As you know, the gravitational acceleration is 9.81m/s2, which means that if you analyze the position ofthe body at every second and translate this state into vectors, they will accumulate an additional downward acceleration of9.81m/s2 every second. This gives an example ofthe simplest differential equation, the solution ofwhich is the function y (t), the derivative ofwhich gives the vertical component, and the velocity gives the vertical component ofacceleration(1).








This equation can be solved byallocating (2) for speed and (3) for path.













Another interesting moment is when it is possible todescribe the movement ofcelestial objects on this scale due tothe force ofgravity. So, two bodies are given whose attraction is directed towards each other with aforce inversely proportional tothe square ofthe distance between them(4).








It is known that the derivative ofthe coordinate is velocity, the derivative ofvelocity is acceleration and it is necessary toobtain afunction for motion, but according toequation (4), only the equation for acceleration (5) is known.








It may be strange here that the derivative is equal to the same function, but this is a common phenomenon when the derivative of the first or higher orders is determined by the values of themselves. But in practice, it is more often necessary to work with second-order differential equations, as can be seen in the previous examples.

However, there are also differential equations with third (6) or fourth (7) derivatives or higher (8) derivatives, which are considered higher-order differential equations.


















In a way, it turns out that you need to find infinitely many numbers, one for each moment of time, but in general this coincides with the description of the function. And most often, even if in many cases it is possible to apply the classical description, then to a greater extent the use of the technology of ordinary mathematical transformations no longer meets the requirements. The usual description of the characteristics of a mathematical pendulum can be a proof of this.

Considering the real and idealized case, it can be noted that idealization works only at small angles of deflection of the pendulum, but when the angle becomes large enough, for example, equal to a semicircle, then the graph describing its oscillations as a whole ceases to be similar to the graphs of sine or cosine. The reason for this is the need to describe its motion exclusively using not partial, but general equations of harmonic oscillations with second-order differential equations.

This analogy can be applied tomany other physical, most often real phenomena.




Used literature


1.Pontryagin L. S. Ordinary differential equations. M.: Nauka, 1974.

2. Tikhonov A. N., Samarsky A. A. Equations of mathematical physics. M.: Nauka, 1972.

3. Tikhonov A. N., Vasilyeva A. B., Sveshnikov A. G. Differential equations. 4th ed. Fzimatlit, 2005.

4. Umnov A. E., Umnov E. A. Fundamentals of the theory of differential equations. Ed. 2nd 2007. 240 p.

5. Charles Henry Edwards, David E. Penny. Differential Equations and the problem of eigenvalues: Modeling and calculation using Mathematica, Maple and MATLAB = Differential Equations and Boundary Value Problems: Computing and Modeling. 3rd ed. M.: "Williams", 2007.

6. ElsholtsL.E.Differential Equations and Calculus ofVariations. M.: Science, 1969.




ON THE ISSUES OFTHE STUDY OFTRIPLE INTEGRALS





Aliev Ibratjon Khatamovich







2nd year student of the Faculty of Mathematics and Computer Science of Fergana State University







Ferghana State University, Ferghana, Uzbekistan


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Annotation. The study ofamathematical theory capable ofdescribing the observable world as aseparate tool is very necessary not only for researchers inthis field, but also for all representatives ofhuman civilization, not tomention those who want toproduce descriptions ofmathematical theories for the sake ofthese same mathematical theories. Special inthis part is such amethod ofresearch as the use ofatriple or triple integral, along with separate operations with elements under the sign ofthe same triple integral.

Keywords: triple integrals, mathematical analysis, research, definition, calculation, scientific novelty.

The need to determine the volume, therefore, based on density and mass, along with the number of components and other derivatives of these quantities for objects that cannot be determined practically, a vivid example of this is the determination of the mass of a mountain, huge details for which it is not possible or too costly to use the Archimedes method, but it is necessary to calculate the volume or even when determining parameters of the whole planet.

The object of the study is a three-fold integral. The subject of the study is the process of calculating a triple integral, as well as operating with it when replacing variables of this triple integral. As a result of the study, some hypothesis is put forward that asserts the possibility of using triple integrals in calculating the parameters of linear accelerators relative to the internal particles of smooth electromagnetic waveguides of linear accelerators.

The objectives of the study are:

Definition of the concept of a triple integral;

Study of the process of replacing variables in a triple integral;

Showing a model for calculating parameters by means of a three-fold integral for a linear accelerator waveguide.

The objectives of the study are:

Study of the concept of a three-fold integral in the general understanding and some coordinate systems, indicating general concepts in a brief form;




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