Differential calculi of the function of one or several variables

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

Appearance story

Differential calculus stood out as an independent discipline in the second half of the 17th century, thanks to the work of Newton and Leibniz, who formulated the main provisions in the calculus of differentials and noticed the connection between integration and differentiation. From that moment, the discipline developed along with the calculus of integrals, thus constituting 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. It also expanded the possibility of applying mathematical science in science and technology.

Basic concepts

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

Process of creation

The formation of differential calculus in the form of an applied and then a scientific method occurred before the emergence of the philosophical theory that Nikolai Kuzansky created. 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, putting mathematics of that time in doubt.

For ancient mathematicians, the universal criterion was unity, while the philosopher proposed to offer infinity instead of the exact number as a new measure. In this regard, the representation of accuracy in mathematical science is inverted. Scientific knowledge, in his opinion, 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 is connected with a function in small neighborhoods of certain points. For this, it is necessary to create a mathematical apparatus for investigating a function whose behavior in a small neighborhood of set points is close to the behavior of a polynomial or linear function. This is based on the definition of the derivative and differential.

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

One of the main tasks that are given as an example, starting with the senior grades of the school, is to determine the speed of a point in a straight line and construct a tangent line to this curve. The differential is associated with this, since it is possible to approximate a function in a small neighborhood of the considered point of a linear function.

Compared 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 on another.

Derivative

Let the point move in the direction of the Oy axis, for the time we take x, which is measured from a certain beginning of the moment. Such a movement can be described by the function y = f (x), which is associated with each time moment x of the coordinate of the moved point. To take this function in mechanics as the law of motion. The main characteristic of motion, especially uneven motion, is instantaneous speed. When a point moves along the axis Oy according to the law of mechanics, then at a random time moment x it acquires the coordinate f (x). At the time moment x + Δx, where Δx denotes the increment of time, its coordinate will be f (x + Δx). So the formula Δy = f (x + Δx) - f (x) is formed, which is called the increment of the function. It represents the distance traveled by a point during the time from x to x + Δx.

In connection with the occurrence of this velocity, a derivative is introduced at a time instant. In an arbitrary function, the derivative at a fixed point is called the limit (provided it exists). It can be designated by certain symbols:

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

The process of calculating the derivative is called differentiation.

Differential calculus of the function of several variables

This method of calculus will be used to study a function with several variables. If there are two variables x and y, the partial derivative with respect to x at point A is called the derivative of this function with respect to x with fixed y.

It can be indicated by the following symbols:

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

Essential Skills

To successfully learn and be able to solve diffuses, integration and differentiation skills are required. To make it easier to understand differential equations, you should have a good understanding of the derivative and the indefinite integral. It also does not hurt to learn how to look for a derivative of an implicitly specified function. This is due to the fact that in the process of studying one will often have to use integrals and differentiation.

Types of Differential Equations

In almost all control papers related to first-order differential equations, there are 3 types of equations: homogeneous, with separable variables, linear inhomogeneous.

There are more rare varieties of equations: with full differentials, Bernoulli equations and others.

Solution Basics

To begin with, we should recall the algebraic equations from the school course. They contain variables and numbers. To solve the usual equation, one should find a set of numbers satisfying the given condition. As a rule, such equations had the same root, and to verify the correctness, one had only to substitute this value in the place of the unknown.

The differential equation is similar to this. In the general case, such a 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, however, this is not so important, since it is necessary to have the first derivative, without derivatives of higher orders, so that the solution and differential calculus are correct.

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

Integral calculus

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

Often, the calculation of the integral occurs when calculating the area of ​​a curved figure. By this area is meant the limit to which the area of ​​the polygon inscribed in the given shape tends with the gradual increase of its side, while these sides can be made less than any previously specified small value.

The main idea in calculating the area of ​​an arbitrary geometric figure is to calculate the area of ​​the rectangle, that is, proof that its area is equal to the product of length and width. When it comes to geometry, all constructions are made using a ruler and compass, and then the ratio of length to width is a rational value. When calculating the area of ​​a right-angled triangle, you can determine that if you put the same triangle side by side, a rectangle is formed. In a 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 entering it.

When determining the mercy of an arbitrary curve, this method will not work. If you break it into single squares, then there will be empty spaces. In this case, they try to use two coverings, with rectangles above and below, as a result, they include a graph of the function and do not include. Important here remains the way of breaking into these rectangles. Also, if we take the breakdowns more and more decreasing, then the area above and below should converge at a certain value.

You should return to the method of division into rectangles. There are two popular methods.

Riemann formalized the definition of an integral created by Leibniz and Newton as the area of ​​a subgraph. In this case, shapes consisting of a number of vertical rectangles and obtained by dividing a segment were considered. When there is a limit to reducing the partitioning to which the area of ​​such a figure is reduced, 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 the fact that for the place of dividing the defined region into parts of the integrand and then compiling the integral sum of the obtained values ​​in these parts, its range of values ​​is divided into intervals, and then it is summed up with the corresponding measures of the inverse images of these integrals.

Modern benefits

One of the main manuals on the study of differential and integral calculus was written by Fichtenholtz - "The Course of Differential and Integral Calculus". His textbook is a fundamental tool for the study of mathematical analysis, which has survived many publications and translations into other languages. Designed for university students and has long been used in many educational institutions as one of the main study guides. Provides theoretical data and practical skills. First published in 1948.

Functional Research Algorithm

In order to investigate a function by methods of differential calculus, it is necessary to follow an already given algorithm:

1. Find the scope of the function.
2. Find the roots of a given equation.
3. Count extremes. To do this, calculate the derivative and the points where it is equal to zero.
4. Substitute the obtained value in the equation.

Varieties of differential equations

First-order differential equations (in other words, differential calculus of one variable) and their types:

• Equation with separable variables: f (y) dy = g (x) dx.
• The simplest equations, or differential calculus of a function of one variable, having the formula: y '= f (x).
• First-order linear non-uniform DE: y '+ P (x) y = Q (x).
• Bernoulli differential equation: y '+ P (x) y = Q (x) y ^a .
• Equation with full differentials: P (x, y) dx + Q (x, y) dy = 0.

Second-order differential equations and their types:

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

Differential equations of higher orders and their types:

• A differential equation that can be reduced in 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 heterogeneous: y ⁽ⁿ⁾ + f _{(n -1)} y ^(n-1) + ... + f ₁ y '+ f ₀ y = f (x) .

Stages of solving a problem with a differential equation

With the help of remote control, not only mathematical or physical problems are solved, but also various problems from biology, economics, sociology and other things. Despite the wide variety of topics, you should adhere to a single logical sequence in solving such problems:

1. Drawing up the remote control. One of the most difficult stages, which requires maximum accuracy, since any error will lead to completely incorrect results. All factors affecting the process should be considered and initial conditions determined. It should also be based on facts and logical conclusions.
2. The solution of the composed equation. This process is simpler than the first point, since it requires only rigorous implementation of mathematical calculations.
3. Analysis and evaluation of the results. The derived solution should be evaluated to establish the practical and theoretical value of the result.

An example of using differential equations in medicine

The use of remote control in the field of medicine is found in the construction of an epidemiological mathematical model. It should not be forgotten that these equations are also found 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 above epidemic example, the spread of infection in an isolated society can be considered. The inhabitants are divided into three types:

• Infected, the number x (t), consisting of individuals, carriers of infection, each of which is contagious (the incubation period is short).
• The second species includes susceptible individuals y (t) that can become infected by contact with infected individuals.
• The third species includes immune z (t) individuals that are immune or died due to illness.

The number of individuals is constant, accounting for birth, natural deaths and migration is not taken into account. There will be two hypotheses at the core.

The percentage of incidence at a certain time moment 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 a first approximation will be proportional to x (t) y (t)), in In this regard, the number of patients increases, and the number of susceptibles decreases with speed, which is calculated by the formula ax (t) y (t) (a> 0).

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

As a result, we can draw up a system of equations taking into account all three indicators and draw conclusions on its basis.

Case study in economics

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 changes in income immediately after an increase in taxes, input of duties, changes in the company's revenue when the cost of production changes, in which proportion can the retired employees be replaced with new equipment. To solve such problems, it is required to construct a connection function from the input 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, highest income, 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 labor and capital expenditures. In this regard, finding a suitable value can be reduced to finding the maximum or minimum of a function from one or more variables.

Such problems create a class of extreme problems in the economic field, the solution of which requires differential calculus. When an economic indicator is required 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 a relation tends to some positive or negative value, the indicated point is not suitable, because when you increase or decrease the argument, you can change the dependent value in the necessary direction. In the terminology of differential calculus, this will mean that the required condition for the maximum of the function is the zero value of its derivative.

In economics, there are often problems of finding the extremum of a function with several variables, because economic indicators are made up of many factors. Similar questions have been well studied in the theory of functions of several variables, using differential calculation methods. Similar tasks include not only maximized and minimized functions, but also constraints. Similar questions relate to mathematical programming, and they are solved with the help of specially developed methods, also based on this branch of science.

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

Differential calculus of several variables is an important topic from the field of mathematical analysis. For a detailed study, you can use various textbooks for higher education. One of the most famous was created by Fichtenholtz - "The Course of Differential and Integral Calculus". As the name implies, skills in working with integrals are of considerable importance for solving differential equations. When there is a differential calculus of the function of one variable, the solution becomes simpler. Although, it should be noted, it obeys the same basic rules. In practice, to study the function by differential calculus, it is enough to follow the existing algorithm, which is given in high school and is only a little complicated when entering new variables.