Differential calculi of a function of one and several variables

Differential calculus is a section of mathematical analysis that studies the derivative, differentials and their use in the study of a function.

History of appearance

The differential calculus was separated into an independent discipline in the second half of the 17th century, thanks to the works of Newton and Leibniz, who formulated the basic propositions in the calculus of differentials and noticed the connections between integration and differentiation. From that moment, the discipline developed along with the calculus of integrals, thus forming the basis of mathematical analysis. The appearance of these calculi opened a new modern period in the mathematical world and caused the emergence of new disciplines in science. Also expanded the possibility of applying mathematical science in natural science and technology.

Basic concepts

Differential calculus is based on the fundamental concepts of mathematics. They are: a real number, continuity, function and limit. After a while they took a modern look, thanks to integral and differential calculi.

Process of creation

Formation of differential calculus in the form of applied, and then scientific method, occurred before the emergence of the philosophical theory, which was created by Nikolai Kuzansky. His works are considered evolutionary development from the judgments of ancient science. Despite the fact that the philosopher himself was not a mathematician, his contribution to the development of mathematical science is undeniable. Kuzansky was one of the first to abandon the consideration of arithmetic as the most accurate field of science, placing mathematics of that time in doubt.

In ancient mathematicians, the universal criterion was a unit, while the philosopher offered as a new measure infinity in place of an exact number. In this connection, the representation of accuracy in mathematical science is inverted. Scientific knowledge, according to him, is divided into rational and intellectual. The second is more accurate, according to the scientist, since the first gives only an approximate result.

Idea

The basic idea and concept in differential calculus are related to the function in small neighborhoods of certain points. For this, it is necessary to create a mathematical apparatus for the study of a function whose behavior in a small neighborhood of the established points is close to the behavior of a polynomial or a linear function. This is based on the definition of the derivative and the differential.

The emergence of the concept of derivative was caused by a large number of problems from the natural sciences and mathematics, which led to the finding of values of limits of one type.

One of the main tasks that are given as an example, starting with the high school classes, is the determination of the speed of a point in a straight line and the construction of a tangent line to this curve. The differential is related to this, since it is possible to approximate the function in a small neighborhood of the point of the linear function in question.

In comparison with the concept of the derivative of a function of a real variable, the definition of differentials simply goes over to a function of a general nature, in particular, to the image of one Euclidean space onto another.

Derivative

Let the point move along the direction of the Oy axis, in time we take x, which is measured from a certain beginning of the moment. Describe such a movement by the function y = f (x), which corresponds to each time moment x of the coordinate of the moved point. This function in mechanics should be called the law of motion. The main characteristic of motion, in particular uneven, is the instantaneous velocity. When the point moves along the axis Oy according to the law of mechanics, then at a random time x it acquires the coordinate f (x). At the time moment x + Δx, where Δx denotes the time increment, its cadinate will be f (x + Δx). This is how the formula Δy = f (x + Δx) - f (x) is formed, which is called the increment of the function. It represents a path traversed in a time from x to x + Δx.

In connection with the appearance of this velocity, a derivative is introduced at the instant of time. In an arbitrary function the derivative at a fixed point is called the limit (under the condition of its existence). It can be designated by certain symbols:

F '(x), y', ý, df / dx, dy / dx, Df (x).

The process of calculating a derivative is called differentiation.

Differential calculus of a function of several variables

This method of calculus is used in the study of a function with several variables. In the presence of two variables x and y, the partial derivative with respect to x at the point A is called the derivative of this function with respect to x with fixed y.

Can be denoted by the following characters:

F '(x) (x, y), u' (x), ∂u / ∂x, or ∂f (x, y) '/ ∂x.

Required skills

To successfully learn and be able to solve diffusers, skills in integration and differentiation are required. To make it easier to understand the differential equations, one should well understand the subject of the derivative and the indefinite integral. It also does not hurt to learn to look for a derivative of an implicitly defined function. This is due to the fact that in the process of studying it is often necessary to use integrals and differentiation.

Types of differential equations

Practically in all control works related to first-order differential equations, there are 3 types of equations: homogeneous, with separable variables, linear non-homogeneous.

There are also more rare varieties of equations: with complete differentials, Bernoulli's equations and others.

Solution Basics

To begin with, one should remember algebraic equations from the school course. They contain variables and numbers. To solve the ordinary equation, it is necessary to find a set of numbers satisfying the given condition. As a rule, such equations had one root, and to verify the correctness it was only necessary to substitute this value for the place of the unknown.

The differential equation is similar to this. In general, this first-order equation includes:

• Independent variable.
• The derivative of the first function.
• Function or dependent variable.

In some cases, one of the unknowns, x or y, may be absent, but this is not so important, since it is necessary to have the first derivative, without derivatives of higher orders, so that the solution and the differential calculus are correct.

To solve the differential equation is to find the set of all functions that fit the given expression. Such a set of functions is often called the general solution of DW.

Integral calculus

The integral calculus is one of the sections of mathematical analysis that studies the concept of an integral, the properties and methods of its computation.

Often, the calculation of the integral occurs when calculating the area of a curvilinear figure. By this area is meant the limit to which the area of a polygon inscribed in a given figure tends to gradually increase its side, while the sides can be performed less than any previously specified arbitrary small value.

The main idea in calculating the area of an arbitrary geometric figure is to calculate the area of a rectangle, that is, to prove that its area is equal to the product of length and width. When it comes to geometry, then all constructions are made using a ruler and a compass, and then the ratio of length to width is a rational value. When calculating the area of a rectangular triangle, you can determine that if you put the same triangle next to it, a rectangle is formed. In the parallelogram, the area is calculated by a similar, but slightly more complicated method, through a rectangle and a triangle. In polygons, the area is counted through the triangles that enter it.

When determining the mercy of an arbitrary curve, this method does not work. If you break it into single squares, then there will be unfilled seats. In this case, try to use two covers, with rectangles on top and bottom, as a result those include a function graph and do not include. Important is the way to break these rectangles. Also, if we take more and more breakdowns, then the area from above and below must converge on a certain value.

It is necessary to return to the method of dividing into rectangles. There are two popular methods.

Riemann formalized the definition of the integral, created by Leibniz and Newton, as the area of the subgraph. In this case, we considered the figures consisting of a number of vertical rectangles and obtained by dividing the segment. When there is a limit to the reduction of the breaking, to which the area of such a figure reduces, this limit is called the Riemann integral of a function on a given interval.

The second method is the construction of the Lebesgue integral, which consists in dividing its range of values into intervals for dividing the domain into parts of the integrand and then compiling the integral sum from the values obtained in these parts, and then summing it with the corresponding measures of the preimages of these integrals.

Modern benefits

One of the main manuals on the study of differential and integral calculus was written by Fichtenholz, "The Course of Differential and Integral Calculus." His textbook is a fundamental aid in the study of mathematical analysis, which has withstood many publications and translations into other languages. It was created for university students and has long been used in a variety of educational institutions as one of the main study guides. Gives theoretical data and practical skills. It was first published in 1948.

Algorithm of function research

To investigate the methods of differential calculus function, it is necessary to follow the algorithm already defined:

1. Find the domain of the function.
2. Find the roots of a given equation.
3. Calculate extrema. To do this, calculate the derivative and the points where it equals zero.
4. We substitute the obtained value into the equation.

Varieties of differential equations

DU of the first order (in other words, the differential calculus of one variable) and their types:

• Equation with separating variables: f (y) dy = g (x) dx.
• The simplest equations, or the differential calculus of a function of one variable, have the formula: y '= f (x).
• A linear inhomogeneous DN of the first order: y '+ P (x) y = Q (x).
• The Bernoulli differential equation: y '+ P (x) y = Q (x) y ^a .
• Equation with total differentials: P (x, y) dx + Q (x, y) dy = 0.

Differential equations of the second order and their types:

• A linear homogeneous differential equation of the second order with constant values of the coefficient: y ⁿ + py '+ qy = 0 p, q belongs to R.
• A linear nonhomogeneous differential equation of the second order with a constant value of the coefficients: y ⁿ + py '+ qy = f (x).
• Linear homogeneous differential equation: y ⁿ + p (x) y '+ q (x) y = 0, and a nonhomogeneous equation of the second order: y ⁿ + p (x) y' + q (x) y = f (x).

Differential equations of higher orders and their types:

• Differential equations that allow decreasing order: F (x, y ^(k) , y ^{(k + 1)} , .., y ⁽ⁿ⁾ = 0.
• The higher-order linear equation is homogeneous: y ⁽ⁿ⁾ + f _(n-1) y ^(n-1) + ... + f ₁ y '+ f ₀ y = 0 , and non-uniform: y ⁽ⁿ⁾ + f _{(n -1)} y ^(n-1) + ... + f ₁ y '+ f ₀ y = f (x) .

Steps of solving a problem with a differential equation

With the help of the DU, not only mathematical or physical questions are solved, but also various problems from biology, economics, sociology and others. Despite a wide variety of topics, one should follow a single logical sequence when solving such problems:

1. Drawing up the DM. One of the most difficult stages, which requires maximum accuracy, since any error will lead to completely incorrect results. It is necessary to take into account all factors influencing the process, and to determine the initial conditions. It should also be based on facts and logical inferences.
2. Solution of the compiled equation. This process is simpler than the first point, since it requires only strict mathematical calculations.
3. Analysis and evaluation of the results. The resulting solution should be evaluated to establish the practical and theoretical value of the result.

An example of the use of differential equations in medicine

The use of DM in the field of medicine is encountered in the construction of an epidemiological mathematical model. Moreover, it should not be forgotten that these equations also occur in biology and chemistry, which are close to medicine, because the study of different biological populations and chemical processes in the human body plays an important role in it.

In the example given with the epidemic, it is possible to consider the spread of infection in an isolated society. The inhabitants are divided into three types:

• Infected, number x (t), consisting of individuals, carriers of infection, each of which is contagious (incubation period is short).
• The second species includes susceptible individuals of y (t), capable of contracting when in contact with the infected.
• The third species includes the non-susceptible individuals z (t), who are immune or died due to illness.

The number of individuals is constant, birth records, natural deaths and migration are not taken into account. There will be two hypotheses in the basis.

The percentage of morbidity at a certain time is x (t) y (t) (the assumption is based on the theory that the number of cases is proportional to the number of intersections between patients and susceptible representatives, which in the first approximation will be proportional to x (t) y (t), in Due to this, the number of cases increases, and the number of susceptible people decreases with the speed, which is calculated by the formula ax (t) y (t) (a> 0).

The number of insensitive individuals who acquired immunity or died increases at a rate proportional to the number of cases, bx (t) (b> 0).

As a result, it is possible to compile a system of equations taking into account all three indicators and draw conclusions on its basis.

Example of use in the economy

Differential calculus is often used in economic analysis. The main task in economic analysis is the study of quantities from the economy, which are written in the form of a function. This is used to solve problems such as income changes immediately after tax increases, the introduction of duties, changes in the company's revenue when the value of production changes, in what proportion can the replaced employees be replaced by new equipment. To solve such questions, it is required to construct a link function from the incoming variables, which are then studied using differential calculus.

In the economic sphere, it is often necessary to find the most optimal indicators: maximum labor productivity, the highest income, the lowest costs, and so on. Each such indicator is a function of one or more arguments. For example, production can be considered as a function of the outlays of labor and capital. In this connection, finding an appropriate value can be reduced to finding the maximum or minimum of a function from one or more variables.

Such problems create a class of extremal problems in the economic field, for which a differential calculus is necessary. When the economic indicator needs to be minimized or maximized as a function of another indicator, then at the maximum point the ratio of the increment of the function to the arguments will tend to zero if the increment of the argument tends to zero. Otherwise, when such an attitude tends to some positive or negative value, the specified point is not suitable, because with increasing or decreasing the argument it is possible to change the dependent value in the necessary direction. In the terminology of the differential calculus this means that the required condition for the maximum of the function is the zero value of its derivative.

In the economy there are often problems of finding an extremum of a function with several variables, because the economic indicators are made up of many factors. Similar questions are well studied in the theory of functions of several variables that apply methods of differential computation. Such tasks include not only maximized and minimized functions, but also limitations. Similar questions pertain to mathematical programming, and they are solved with the help of specially developed methods, also based on this section of science.

Among the methods of differential calculus used in economics, an important section is the marginal analysis. In the economic sphere, this term refers to a set of methods for studying variable indicators and results when changing the volumes of creation, consumption, based on an analysis of their limits. The limiting index is the derivative or partial derivatives with several variables.

The differential calculus of several variables is an important topic from the field of mathematical analysis. For a detailed study, various teaching aids for higher education institutions can be used. One of the most famous created Fichtenholz - "The course of differential and integral calculus." As is obvious from the title, the skills in working with integrals are of considerable importance for solving differential equations. When the differential calculus of a function of one variable takes place, the solution becomes simpler. Although, it should be noted, it obeys the same basic rules. In order to practice the function in differential calculus, it is sufficient to follow the already existing algorithm, which is given in the upper grades of the school and is only slightly complicated when new variables are entered.