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MATHEMATICS PART 2 University textbook Almaty

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The textbook is intended for first-year students of technical universities and educational institutes studying mathematics, who have already studied basic mathematics and need to study the methods of higher mathematics. I dedicate this series of texts to the memory of my parents - Dina Uonovna Kim, a mathematician, and Evgeny Andreevich Kim, a civil engineer. The text series is intended for first-year students of technical universities and educational institutes studying mathematics, who have already studied basic mathematics and need to study the methods of higher mathematics.

The series of texts can also be used by students for independent study of the relevant material; it is the basis for preparing for exams in higher mathematics. This series of textbooks contains the necessary material for all sections of the higher mathematics course and additional information necessary for the study of specific courses.

  • Introduction
  • Limit of functions of several variables
  • Continuity of functions of several variables
  • Partial derivatives of function of several variables
  • Partial derivatives of higher orders. Mixed partial derivatives
  • Differentiability and total differential
  • Higher order differentials
  • Derivative of composite functions
  • Derivative of implicit functions
  • Equations of normal and tangent plane
  • Maximum and minimum of functions of two variables
  • Conditional extremum of two variables function

Thus, by exchanging the order of a passage to the limit, we obtained the same results. The increases of the independent variables x and y are called the differences of the independent variables x and y (dx and dy). Assume that at some point М0(х0, у0, z0) there exist partial derivatives of z on the surface, that is, the surface is smooth enough near the point М0.

If the function z = f(x, y) reaches an extremum at М0(x0, y0), then all partial derivatives of the first order at these values ​​of the arguments are equal to zero or do not exist. The considered method is used to study the conditional extremum of a function of any number of variables.

Double integral

  • Basic concepts
  • Properties of double integral
  • Iterated integral
  • Evaluation of double integrals
  • Substitution method in double integrals
  • Applications of double integral

If the function z = f(x, y) is continuous in the closed domain D, then it can be integrated in this domain. The regular area is the area that is regular in the direction of the x-axis and in the direction of the y-axis. To evaluate the double integral, we first take the inner integral with respect to y, assuming x is a constant, and then we take the outer integral, i.e. the result of the first integration Ф(х) is integrated over the variable х in the range from a to b.

To evaluate the double integral of this form, we first take the inner integral with respect to x, assuming y to be a constant, then we take the outer integral, i.e. the result of the first integration Ф(y) is integrated over the variable y in the range from c to d. A special case of substitution of variables in the double integral is the transition to polar coordinates.

Triple integral

  • Basic concepts
  • Three-tuple integral
  • Evaluation of triple integrals
  • Substitution method in triple integrals
  • Applications of triple integral

V is projected onto the Oxy plane into the regular (two-dimensional) region D; .. 3) any part of the region V, cut by a plane parallel to any of the coordinate planes (Oxy, Oxz, Oyz), also has properties 1) and 2). Then the triple integral of the function f(x, y, z) in the region V is defined as follows:. If the regular region V is divided into n regions V1, V2, …, Vn according to planes, parallel to any coordinate plane, then.

The first octant is the octant in which all three coordinates are positive. The moments Sxy, Sxz, Syz of the body with respect to the coordinate planes Oxy, Oxz, Oyz are evaluated using formulas.

Basic concepts

Any DE solution obtained from the general solution, if specific values ​​are assigned to arbitrary constants, is called a particular DE solution у = (х, С1, С2, …, Сп); and correspondingly F(х, С1, С2, …, Сп) = 0 is called a particular integral. The number of initial conditions required for a given differential equation depends on the order of the differential equation.

The first order differential equations

  • Separable equations
  • Homogeneous differential equations
  • Linear equations
  • The Bernoulli differential equations
  • Exact differential equations

By a direct substitution into the initial equation, we are convinced that y = 0 is a special solution, since it is not included in the general solution. However, this particular solution does not satisfy the initial condition; therefore, it cannot be a solution to the Cauchy problem. The equation М(x, y)dx + N(x, y)dy = 0 is the homogeneous equation of the first order if and only if М(x, y) and N(x, y) are homogeneous functions of the same power .

The first-order linear differential equation is the first-order equation linear with respect to the unknown function and its derivative. If the right side is equal to zero, i.e. g(x) ≡ 0, then equation (11.2.6) is called a homogeneous linear equation; otherwise it is called a non-homogeneous linear equation. Thus, the general solution of the first-order linear DE is (11.2.6). b) The Lagrange method (the variations of constant).

First note that if п = 0 or п = 1 the equation is linear and in these cases we already know how to solve it. Solving this equation and substituting the expression y1–n in place of z, we find у(х) as the solution of equation (11.2.11).

Higher orders differential equations

  • Basic concepts
  • The equations that allow reduction of the order
  • Linear differential equations (LDE)
    • Homogeneous LDE of the second order
    • Homogeneous LDE of n-th order
    • Homogeneous LDE of the second order
    • Homogeneous LDE of n-th order with
    • Nonhomogeneous LDE of the second order
    • Nonhomogeneous LDE of the second order with
    • Nonhomogeneous LDE of higher order
    • Nonhomogeneous LDE of higher order with constant

The equations that allow reduction of order We consider the special forms of DE that allow reduction of order. Thus we get the first order DE, i.e. we have reduced the order of the given DE. Differential equation of nth order is called linear if it is a polynomial of the first degree with respect to the unknown function y and its derivatives y, y,.

If the multiplicity of the root is equal to one (mk = 1), then the root is called a simple root. All the roots of the characteristic equation are real, but not all are simple (there are roots with multiplicity m > 1). Assume that уpartik is a particular solution of the equation ygsh is the general solution of the associated homogeneous equation (11.3.18).

Then the general solution of the non-homogeneous LDE (11.3.17) can be represented as the sum of ygsh. Thus, to solve a non-homogeneous differential equation, we need to solve the corresponding homogeneous differential equation and find some solution of the non-homogeneous equation. Second-order non-homogeneous LDE with constant coefficients and with a special right-hand side with constant coefficients and with a special right-hand side. We consider the equation of form.

By comparing the left and right sides of the obtained equality, we can form the following system:. If we know the general solution ygsh of the homogeneous equation, then a particular solution уpart of the n-th order inhomogeneous LDE can be found by the method of variation of parameters. Suppose that the right-hand side is f(x) = Pn(x)∙eαx, where Pn(x) is a polynomial of the nth degree, then: . a) if the number  is not a root of the characteristic equation, then ypartis = Qn(x)∙eαx,.

System of differential equations

  • The first order system of DE
  • System of LDE with constant coefficients

Substituting these expressions into the last equation in (11.4.3), we obtain the nth-order DE with respect to a function у1:. Let us consider another method of integration of the normal system of equations (11.4.1) in the case when it is a first-order LDE system with constant coefficients, i.e., a system of the form

Numerical series

  • Basic concepts
  • Series with positive terms
  • Alternating series

Note well that the converse is not true: if lim п 0,. then the series does not necessarily converge. If all the terms of the series are positive, then the series is called a series of positive terms or a series with positive terms. In many cases, the convergence and divergence of the series can be determined with the help of the so-called sufficient criteria.

The root test is often useful when n appears as an exponent in the general term of the series. The point of all this is that we do not need to require that the terms of the series be decreasing for all n. Leibniz's theorem is illustrated geometrically: if we mark the partial sums on the number line (Figure 12.1.1): then the points corresponding to the partial sums will approach the point s, which corresponds to the sum of the series.

In the general case, the alternating series is the series whose terms can be both positive and negative, i.e. The alternating series (12.1.7) is called absolutely convergent if the series of absolute values ​​of its terms (12.1.8) converges. If the alternating series (12.1.7) converges, and the series of absolute terms (12.1.8) diverges, then the given alternating series (12.1.7) is called conditionally (or not absolutely) convergent.

If the series converges absolutely, it is absolutely convergent for any rearrangement of its terms. Here the sum of the series does not depend on the order of its terms. If the series converges conditionally, then, for any number A, we can rearrange the terms of this series so that its sum will be equal to A.

Functional series

  • Basic concepts
  • Power series
  • Taylor and Maclaurin series
  • Applications of power series
  • Fourier series

The region of convergence of a power series is always an interval which, in particular, can degenerate into a point. In order to find the radius of convergence of the power series (12.2.3), we can proceed as follows. The number R is the radius of convergence of the set, it is calculated as in the previous example.

Assume that f(x) is a function with derivatives up to the (n + 1)th order, in the vicinity of the point x = a. One application of power series (with the occasional use of Taylor series) is in the field of ordinary differential equations when finding series solutions to differential equations. One of the more commonly used methods in that subject makes use of Fourier series.

A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. The calculation and study of Fourier series is known as harmonic analysis and is extremely useful as a way of breaking down an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution of the original problem or an approximation to it with any desired or practical accuracy. Fourier series has many applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, etc.

If f(x) is a periodic function of period 2π, bounded and piecewise monotonic in the interval [–π; π], then we can expand f(x) to the Fourier series. The value of this series at the point of discontinuity of the function f(x) is equal to the arithmetic mean of the left-hand and right-hand limits, i.e.

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