Scalar function of scalar argument. Lectures vector-function of scalar argument. Assignment using a complete binary tree

and its differentiation.

One of the simplest ways to define a space curve is to define a vector equation:

Where is the radius vector of the point of the curve, and - parameter that determines the position of the point.

That. variable vector is a scalar function . Such functions in mathematical analysis are called vector functions of a scalar argument.

decomposing in terms of vectors, equation (1) can be given the form:

This decomposition makes it possible to pass to the parametric equation of the curve:

In other words, specifying a vector function is equivalent to specifying three scalar functions.

With respect to the vector function (1) that defines the given curve, the curve itself is called the hodograph of this function. The origin of coordinates is called in this case the hodograph pole.

Let now
And
- points of the curve defined by equation (1). And
, A
The radius vectors of these points will be

And
.

Vector
is called the increment of the vector function
corresponding to the increment
its argument, and denoted by
,

vector function
will be a continuous function , If

.

To find the derivative of
let's do it like this -

.

Set the direction now
. It's obvious that collinear with
and at
directed in the same direction as
and at
- in the opposite direction. But in the first case
and in the second
That. vector always directed along the secant of the hodograph
upward .

If we use the expansion And by orts, then

From here dividing (*) by
and going to the limit
For
we get

Based on (4), it can be shown that the following formulas are valid:

(5)

(6)

is a scalar function.

Proof (7).

We now examine some properties
. First of all, let's find its module:

.

Because we consider the hodograph arc to be rectifiable, then
is the length of the chord, and
- arc length. That's why

That. the modulus of the derivative of the vector function of the scalar argument is equal to the derivative of the hodograph arc with respect to the same argument.

Corollary 1. If - unit vector directed tangentially to the hodograph in the direction of increase , That

Corollary 2. If the length of the hodograph arc is taken as the argument of the vector function , That

(because
)

That. the derivative of the vector function along the length of the arc of the hodograph is equal to the unit vector of the tangent to the hodograph, directed in the direction of increasing the length of the arc.

Corollary 3. If the hodograph of a vector function is considered as a trajectory of a point, and - as the time of movement, counted from some , That
in magnitude and direction coincides with the velocity vector
.

Indeed, the scalar value of the speed is equal to the derivative of the path with respect to time:

In addition, the vector directed tangentially to the trajectory in the direction of motion, which corresponds to the direction of increase , i.e. corresponds to the direction .

That.
.

Consider now
, whose length is constant,
, i.e.

(*)
Where

Differentiating (*), we find:

Those.

In particular, the derived vector of any variable in the direction of the unit Always
.

Let now
the angle between the radii of the unit sphere drawn to the points
And
hodograph
. Then the chord length
from a triangle
will be equal to

The modulus of the derivative of a unit variable vector is equal to the angular velocity of rotation of this vector.

As for scalar functions, the differential of a vector function is written as

But even then

Curvature of a spatial curve.

Accompanying trihedron.

According to Corollary 2, for you can write the formula:

Change of direction , associated with a change in the tangent to the spatial curve, characterizes the curvature of the curve. For a measure of the curvature of a spatial curve, as for a flat one, they take the limit of the ratio of the angle of adjacency to the length of the arc, when

curvature,
adjacency angle,
arc length.

On the other side,
unit vector and its derivative vector perpendicular to it, and its modulus
differentiating By and introducing
unit vector with direction , we find:

Vector
the curvature vector of the space curve. Its direction, perpendicular to the direction of the tangent, is the direction of the normal of the space curve. But a space curve has at any point an innumerable set of normals, all of which lie in the plane passing through the given point of the curve and perpendicular to the tangent at the given point. This plane is called the normal plane of the spatial curve.

Definition. The normal of the curve along which the curvature vector of the curve is directed at a given point is the main normal of the spatial curve. That.
the unit vector of the principal normal.

Let us now construct the third unit vector equal to the vector product And

Vector , like also perpendicular those. lies in the normal plane. Its direction is called the direction of the binormal of the space curve at the given point. Vector
And make up a triple of mutually perpendicular unit vectors, the direction of which depends on the position of the point on the spatial curve and varies from point to point. These vectors form the so-called. the accompanying trihedron (the Frenet trihedron) of the spatial curve. Vector
And form a right triple, just like the unit vectors
in the right coordinate system.

Taken in pairs
define three planes passing through the same point on the curve and form the faces of the accompanying trihedron. Wherein And determine the contact plane (b.m. the arc of the curve in the vicinity of a given point is the arc of a flat curve in the contiguous plane, up to a b.m. of higher order);

And - straightening plane;

And is the normal plane.

Tangent, normal and binormal equations.

Equations of the planes of the accompanying trihedron.

Knowing
And , or any nonunit vectors collinear to them T, N And B we derive the equations named in this section.

For this, in the canonical equation of the line

and in the equation of the plane passing through the given point

take over
coordinates of a point selected on the curve, beyond
or respectively for
accept the coordinates of that of the vectors
or
, which determines the direction of the desired line or normal to the desired plane:

or - for a tangent or normal plane,

or - for the principal normal and rectifying plane,

or - for the binormal and the contiguous plane.

If the curve is given by the vector equation
or
then for the vector
tangent direction can be taken

For finding
And let's find the expansion first
by vectors
Earlier (corollary 1) we found that
Differentiating with respect to , we get:

But, because

Multiply now vector And

(*)

Based on (*) per vector , which has the direction of the binormal, we can take the vector

But then, for
you can take the vector product of these latter:

That. at any point of an arbitrary curve, we can determine all the elements of the accompanying trihedron.

Example. The equation of the tangent, normal and binormal to the right helix at any point.

Tangent

main normal

Binormal

Definition 1. The vector r is called a vector function of the scalar argument t if each value of the scalar from the range of acceptable values ​​corresponds to a certain value of the vector r. We will write it as follows: If the vector r is a function of the scalar argument t, then the x, y, z coordinates of the vector r are functions of the argument t: Vector function of the scalar argument. Hodograph. Limit and continuity of the vector function of a scalar argument Conversely, if the coordinates of the vector r are functions of t%, then the vector r itself will also be a function of t: Thus, specifying the vector function r(f) is equivalent to specifying three scalar functions y(t), z( t). Definition 2. The hodograph of the vector-function r(t) of a scalar argument is the locus of points that describes the end of the vector r(*) when the scalar t changes, when the beginning of the vector r(f) is placed at a fixed point O in space (Fig. I ). The hodograph of the radius vector r = r(*) moves 1 of the touching point will be the trajectory L of this point itself. The hodograph of the velocity v = v(J) of this point will be some other line L\ (Fig. 2). So, if a material point moves along a circle with a constant speed |v| = const, then its velocity hodograph is also a circle centered at the point 0\ and with a radius equal to |v|. Example 1. Construct the hodograph of the vector r = ti + t\ + t\. Solution. 1. This construction can be weighed by points, making a table: Fig.3 2i You can also do this. Denoting the coordinates of the vector V by x, y, z, we will have Hc And the key from these equations, the parameter 1U, we will obtain the equations of the surfaces y - z = x1, the line of intersection L of which will determine the hodograph of the vector r() (Fig. 3). D> Tasks for independent decision. Construct the hodographs of vectors: Let the vector function r = of the scalar argument t be defined in some neighborhood of the value to of the argument t, except, perhaps, for the value of extension 1. The constant vector A is called the limit of the vector r(t) at, if for any e > 0 there exists b > 0 such that for all t φ to satisfying condition 11 - the inequality is satisfied As in ordinary analysis, they write limr(0=A. And both in length and in direction (Fig. 4). definition 2. A vector a(t) is said to be infinitesimal as t -> to if a(t) has a limit as t -* to and this limit is equal to zero: The vector function of the scalar argument. Hodograph. The limit and continuity of the vector function of the scalar argument, or, which is the same, for any e there exists 6 > 0 such that for all t ↦ to satisfying the condition, the inequality |a(t)| example 1. Show that the vector is an infinitely scarlet vector for t -* 0. Solution. We have where it is clear that if for any e 0 we take 6 = ~, then at -0| we will mark |. According to the definition, this means that a(t) is an infinitely scarlet vector as t 0. 1> problems for an independent solution of r. Show that the limit of the modulus of the vector is equal to the modulus of its limit if the latter limit exists. . Prove that in order for the vector function r(*) to have the limit A for to, it is necessary and sufficient that r( can be represented in the form t) is a vector endlessly for t -* t0 14. The vector function a + b(*) is continuous for t = t0 Does it follow that the vectors a(t) and b(J) are also continuous for t - to 15. Prove that if a( are continuous vector functions, then their scalar product (a(*),b(f)) and vector product |a(f),b(t)] are also continuous.

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DIFFERENTIAL GEOMETRY

I. VECTOR FUNCTION OF SCALAR ARGUMENT

Vector-function (definition 1.1), ways to define it.

Radius vector and hodograph, parametric definition of hodograph.

Derivative of a vector function (Definition 1.6).

The geometric meaning of the derivative of a vector function.

Rules for differentiating vector functions.

1.1. DEFINITION OF A VECTOR FUNCTION

Definition 1.1If each value of the scalar argumentaligned vector
three-dimensional space R3 , then we say that the vector function (or vector function) of the scalar argument is given on the set Xt .

If in space R3 given Cartesian coordinate systemABOUT xyz , then the task is vector - functions
,
is equivalent to specifying three scalar functionsX( t ), y ( t ), z ( t ) – coordinates of the vector :

= { x ( t ), y ( t ), z ( t )} (1.1)

or , (1.2)

Where
are the coordinate vectors.

1.2. A SPATIAL LINE AS A HODOGRAPH OF A RADIUS-VECTOR

Definition 1.2 If the beginning of all vectors ,placed at the origin, they are called radius vectors.

Definition 1.3 The line, which is the locus of the ends of the radius vectors , , is called the hodograph of the vector function , and their common beginning is called the hodograph pole.

If the parameter t is time, and is the radius vector of the moving point, then the hodograph of the function is the trajectory of the moving point.

The hodograph equation can be written in vector form (1.2) or in parametric form:

(1.3)

In particular, if the vector functionwith a change in the argument, only its module changes, and the direction does not change (), then the hodograph of such a vector function will be a rectilinear ray emanating from the origin; if only the direction of the vector changes, and its modulus remains unchanged (
), then the hodograph of the vector function will be a curve located on a sphere with a center at the pole and a radius equal to the constant modulus of the vector.

Picture 1.

1.3. LIMIT, CONTINUITY, AND DERIVATIVE OF THE VECTOR–FUNCTION

Definition 1. 4 Vector is called the limit of the vector functionat
, If

. (1.4)

Definition 1.5 The vector function is called continuous at a pointt 0, if it has a limit at this point equal to the value of the vector function at this point:

. (1.5)

Definition 1.6Derivative vector function at the point t is called the limit of the ratio of the increment of the vector function to the increment of the argument
at
:

(1.6)

1.4. GEOMETRIC AND MECHANICAL MEANING OF THE FIRST DERIVATIVE VECTOR-FUNCTION

The geometric meaning of the first derivative of the vector function of the scalar argument is that this derivative is a new vector directed tangentially to the hodograph:
. Let's show it.

Figure 2

We will assume that the hodograph of the considered vector function is a continuous line that has a tangent at any of its points.

Let's give an argument t increment, then geometrically the ratio
is some vector
lying on the secant MM'. With this vector rotates and turns into a vector
, lying on a tangent and directed in the direction of increaset . So the vector

(1.7)

will be the unit vector of the tangent, oriented in the direction of increasing parametert .

Therefore, the vector
can be taken as the direction vector of the tangent to the curve at the point ), (or
), and write the tangent equation as follows:

(1.8)

If t – time, and is the radius vector of the point
moving in three-dimensional space, then aboutratio is called the average speed of a point on the segment [t; t+t].

mechanical sensethe first derivative of the vector function is that this derivative is the speed of the point M at the momentt :

Rules for differentiating vector functions

Let us prove rule 1 using the rules for subtracting vectors and dividing a vector by a number:

The proof of the rest of the rules is based on rule 1 and the rules for operations with vectors.

Example 1.1: Given a vector function .Construct its hodograph and formulate its tangent equation at an arbitrary point.

Solution. For any point ( x , y , z ) hodograph vector - functions we have:x = cost ; y = asint ; z = bt and therefore, for any
equalityx 2 + y 2 = a 2 , and the generatrix is ​​parallel to the axis Oz. If the parameter t interpreted as time, then with uniform motion around the circumference of the projection of the end of the radius vector onto the planeOxy its projection onto the axisOz will move uniformly and in a straight line at a speedb . In other words, the applicate of the hodograph point of the vector function grows in proportion to the angle of rotation of its projection onto the planeOxy . Therefore, the desired hodograph will have the form shown in Fig. 3 and it is called a helix. To find the tangents to the hodograph (helix), we find the derivative of the vector function.

Solution. Because the, then and

Let the set of values ​​of the vector function of the scalar argument be reduced to a common origin at the point 0. Let us combine the origin of the Cartesian coordinate system with this point. Then for any vector can be expanded in terms of orts

Thus, specifying a vector function of a scalar argument means specifying three scalar functions When the value of the argument changes, the end of the vector will describe a curve in space, which is called the hodograph of the vector

Let there be a close value for Then the derivative of the vector function to the scalar argument is called

№17 Speed ​​and acceleration of a point in curvilinear motion

Speed

Velocity is entered as a characteristic of the movement of a material point. Velocity is a vector quantity, which is characterized by both the speed of movement (the modulus of the velocity vector) and its direction (the direction of the velocity vector) at a given time. Let a material point move along some curvilinear trajectory, and at the time t it corresponds to the radius vector r0 (Fig. 1). For a small time interval Δt, the point will make a path Δs and at the same time receive an elementary (infinitely small) displacement Δr.

Average speed vector is the ratio of the increment Δr of the radius-vector of the point to the time interval Δt:

The direction of the average velocity vector coincides with the direction of Δr. With an infinite decrease in Δt, the average speed tends to a value called the instantaneous speed v:

Hence, the instantaneous speed v is a vector quantity, which is equal to the first derivative of the radius-vector of the moving point with respect to time. Because in the limit, the secant coincides with the tangent, then the velocity vector v is directed tangentially to the trajectory in the direction of motion (Fig. 2).

Fig.2

As Δt decreases, Δs will increasingly approach |Δr|, so the instantaneous velocity modulus

This means that the module of instantaneous speed is equal to the first derivative of the path with respect to time:

With uneven motion, the instantaneous velocity modulus is different at different times. In this case, the scalar value is used - average speed of uneven movement:

If we integrate over time within the range from t to t + Δt the expression ds=vdt (see formula (2)), then we will find the length of the path traveled by the point during the time Δt:

In the case of uniform motion, the numerical value of the instantaneous speed is constant; Then expression (3) takes the form

The length of the path traveled by a point in the time interval from t1 to t2 is given by the integral

ACCELERATION

With uneven movement, it is often necessary to know how quickly the speed changes over time. The physical quantity that characterizes the rate of change of speed in absolute value and direction is called acceleration. Consider a plane motion - a motion in which the trajectories of each point of the system under consideration lie in the same plane. Let the vector v be the speed of point A at time t. During the time Δt, the point moved to position B and received a speed different from v both in modulus and direction and equal to v1 + Δv. We transfer the vector v1 to point A and find Δv (Fig. 1).

The average acceleration of non-uniform movement in the interval from t to t + Δt is a vector quantity equal to the ratio of the change in speed Δv to the time interval Δt:

The instantaneous acceleration a (acceleration) of a material point at time t will be a vector quantity:

equal to the first derivative of the speed with respect to time.

Let us decompose the vector Δv into two components. To do this, from point A (Fig. 1) in the direction of velocity v, we set aside the vector AD, modulo equal to v1. Obviously, the vector CD, equal to Δvτ, determines the change in speed over time Δt modulo: Δvτ=v1-v. The second component Δvn of the vector Δv characterizes the change in velocity over time Δt in direction.

Tangential acceleration component:

i.e., equal to the first time derivative of the modulus of speed, thereby determining the rate of change of speed modulo.

We are looking for the second component of acceleration. We assume that point B is very close to point A, so Δs can be considered an arc of a circle of some radius r, slightly different from the chord AB. Triangle AOB is similar to triangle EAD, which implies Δvn/AB=v1/r, but since AB=vΔt, then

In the limit at Δt→0 we get v1→v.

Because v1→v, the angle EAD tends to zero, and since triangle EAD is isosceles, then the angle ADE between v and Δvn tends to a right angle. Therefore, as Δt→0, the vectors Δvn and v become mutually perpendicular. Because the velocity vector is directed tangentially to the trajectory, then the vector Δvn, perpendicular to the velocity vector, is directed to the center of curvature of the point trajectory. The second component of acceleration, equal to

is called the normal component of acceleration and is directed along a straight line perpendicular to the tangent to the trajectory (called the normal) to the center of its curvature (therefore, it is also called centripetal acceleration).

The total acceleration of the body is the geometric sum of the tangential and normal components (Fig. 2):

This means that the tangential component of acceleration is a characteristic of the rate of change of speed in absolute value (directed tangentially to the trajectory), and the normal component of acceleration is a characteristic of the rate of change of speed in direction (directed towards the center of curvature of the trajectory). Depending on the tangential and normal components of acceleration, motion can be classified as follows:

1)aτ=0, an=0 - rectilinear uniform motion;

2) aτ=an=const, аn=0 - rectilinear uniform motion. With this type of movement

If the initial moment of time t1 = 0, and the initial speed v1 = v0, then, denoting t2=t and v2 = v, we get a=(v-v0)/t, whence

Integrating this formula from zero to an arbitrary time t, we find that the length of the path traveled by the point in the case of uniformly variable motion

3)aτ=f(t), an=0 - rectilinear motion with variable acceleration;

4)aτ=0, an=const. When aτ=0, the modulo speed does not change, but changes in direction. From the formula an=v2/r it follows that the radius of curvature must be constant. Therefore, circular motion is uniform; uniform curvilinear motion;

5)aτ=0, an≠0 uniform curvilinear motion;

6)aτ=const, an≠0 - curvilinear uniform motion;

7)aτ=f(t), an≠0 - curvilinear motion with variable acceleration.

#18 Tangent plane and surface normal equations

Definition. Let a function of two variables z =f(х,у), M0(x0;y0) be an interior point of D, M(x0+Δx;y+Δy) be a point from D "neighboring" to M0.

Consider the full increment of the function:

If Δz is represented as:

where A, B are constants (independent of Δx, Δy), - distance between M and M0, α(Δx,Δy) - infinitely small at Δx 0, Δy 0; then the function z = f(x, y) is called differentiable at the point M0, and the expression

is called the total differential of the function z = f(x; y) at the point M0.

Theorem 1.1. If z =f(x;y) is differentiable at the point M0, then

Proof

Since in (1.16) Δx, Δy are arbitrary infinitesimal, we can take Δy =0, Δx≠0, Δx 0, then

after which it follows from (1.16)

Similarly, it is proved that

and Theorem 1.1. proven.

Remark: the differentiability of z = f(x, y) at the point M0 implies the existence of partial derivatives. The converse is not true (the existence of partial derivatives at the point M0 does not imply differentiability at the point M0).

As a result, taking into account Theorem 1.1, formula (1.18) takes the form:

Consequence. The function differentiable at the point M0 is continuous at this point (because it follows from (1.17) that for Δx 0, Δy 0: Δz 0, z(M) z(M0)).

Note: Similarly for the case of three or more variables. Expression (1.17) will take the form:

Using the geometric meaning (Fig. 1.3) of partial derivatives and we can obtain the following equation (1.24) of the tangent plane πcass to the surface: z = f (x, y) at the point C0 (x0, y0, z0), z0 = z (M):

Comparing (1.24) and (1.21) we obtain the geometric meaning of the total differential of a function of two variables:

Increment of the applicate z during the movement of the point C along the tangent plane from the point C0 to the point

where is from (1.24).

The equation of the normal Ln to the surface: z \u003d f (x, y) at the point C0 is obtained as the equation of a straight line passing through C0 perpendicular to the tangent plane:

No. 19 Derivative in direction. Gradient

Let the function and dot . Let us draw a vector from the point, the direction cosines of which . On the vector , at a distance from its origin, consider the point , i.e. .

We will assume that the function and its first-order partial derivatives are continuous in the domain.

The limit of the relation at is called the derivative of the function at the point in the direction of the vector and is denoted by , i.e. .

To find the derivative of a function at a given point in the direction of the vector use the formula:

Where are the direction cosines of the vector , which are calculated by the formulas:
.

Let the function .

The vector whose projections on the coordinate axes are the values ​​of the partial derivatives of this function at the corresponding point is called the gradient of the function and is denoted or (read "nabla u"): .

In this case, we say that a vector field of gradients is defined in the domain.

To find the gradient of a function at a given point use the formula: .

№22 basic properties of the indefinite integral

Indefinite integral

where F is the antiderivative of the function f (on the interval); C is an arbitrary constant.

Basic properties

1.

2.

3. If That

24)

25)

28)

This method is used in cases where the integrand is a product or quotient of heterogeneous functions. In this case, V'(x) is taken to be the part that is easily integrated.

29)

32) Decomposition of a rational fraction into simple fractions.

Every proper rational fraction
can be represented as a sum of a finite number of simple rational fractions of the first - fourth types. For decomposition
the denominator must be decomposed into simple fractions Q m (x) into linear and square factors, for which you need to solve the equation:

- (5)

Theorem.Proper rational fraction
, Where
, can be expanded in a unique way into the sum of simple fractions:

- (6)

(A 1 , A 2 , …, A k , B 1 , B 2 , …, B 1 , M 1 , N 1 , M 2 , M 2 , …, M s , N s are some real numbers).

33) Decomposition of a proper fraction into simpler fractions with complex roots of the denominator

Formulation of the problem. Find the indefinite integral

1 . Let us introduce the notation:

Compare the powers of the numerator and denominator.

If the integrand is an improper rational fraction, i.e. numerator degreen greater than or equal to the power of the denominatorm , then we first select the integer part of the rational function by dividing the numerator by the denominator:

Here the polynomial is the remainder of the division by and the degreepk(x) less degreeQm

2 . Expanding a proper rational fraction

into elementary fractions.

If its denominator has simple complex roots i.e.

then the decomposition has the form

3 . To calculate uncertain coefficients,A1,A2,A3...B1,B1,B3... we reduce the fraction on the right side of the identity to a common denominator, after which we equate the coefficients at the same powersX in numerators on the left and right. Let's get the system 2 S equations with 2 S unknown, which has a unique solution.

4 We integrate elementary fractions of the form

47) If there is a finite limit I of the integral sum as λ → 0, and it does not depend on the way the points ξ i are chosen, the way the segment is split, then this limit is called the definite integral of the function f (x) over the segment and is denoted as follows:

In this case, the function f (x) is called integrable on . The numbers a and b are called the lower and upper limits of integration, respectively, f (x) - the integrand, x - the integration variable. It should be noted that it does not matter what letter denotes the integration variable of a definite integral

since changing the notation of this kind does not affect the behavior of the integral sum in any way. Despite the similarity in notation and terminology, the definite and indefinite integrals are different.

48) Theorem on the existence of a definite integral

Let us divide the segment into parts by points x1,x2,x3... so that

Denote by deltaX the length of the i-th piece and by the maximum of these lengths.

Let us choose some point arbitrarily on each segment so that (it is called the “middle point”), and compose

quantity, which is called the integral sum

Let's find the limit

Definition. If it exists and it does not depend on

a) a method for dividing a segment into parts and from

b) the method of choosing the midpoint,

is a definite integral of the function f(x) over the segment .

The function f(x) is called in this case integrable on the segment . The values ​​a and b are called the lower and upper limits of integration, respectively.

50) Basic properties of a definite integral

1) If the interval of integration is divided into a finite number of partial intervals, then the definite integral taken over the interval is equal to the sum of definite integrals taken over all its partial intervals.

2) the mean value theorem.

Let the function y = f(x) be integrable on the segment ,m=min f(x) and M=max f(x) , then there is such a number

Consequence.

If the function y = f(x) is continuous on the segment , then there is a number such that.

3) When the limits of integration are rearranged, the definite integral changes its sign to the opposite.

4) A definite integral with the same limits of integration is equal to zero.

5) Function module integration

If the function f(x) is integrable, then its modulus is also integrable on the interval.

6) Inequality integration

If f(x) and q(x) are integrable on an interval and x belongs to

That

7) Linearity

The constant factor can be taken out of the sign of a definite integral

if f(x) exists and is integrable on the interval , A=const

If the function y=f(x) is continuous on the interval and F(x) is any of its antiderivatives on (F’(x)=f(x)), then the formula

Let the substitution x=α(t) be made to calculate the integral of a continuous function.

1) The function x=α(t) and its derivative x’=α’(t) are continuous for t belonging to

2) The set of values ​​of the function x=α(t) with t belonging is the segment

3) A α(c)=a and α(v)=b

Let the function f(x) be continuous on the interval and have an infinite discontinuity at x=b. If there is a limit, then it is called an improper integral of the second kind and denoted by .

Thus, by definition,

If the limit on the right-hand side exists, then the improper integral converges. If the indicated limit does not exist or is infinite, then the integral is said to be diverges.

Example 2 Consider, for example, a function of three variables f(X,at,z), having the following truth table:

With the lexicographic order of the location of the vectors of variable values  X n they can be omitted and the function will be completely defined by its own vector of truth values f= (10110110).

Matrix method

It means that many variables  X n splits into two parts  at m And  z n–m such that all possible truth values ​​of the vector  at m are plotted along the rows of the matrix, and all possible truth values ​​of the vector  z n-m— by columns. Function truth values f on every set   n = ( 1 , ...,  m ,  m+ 1 ,...,  n) are placed in the cells formed by the intersection of the line ( 1 , ...,  m) and column ( m+ 1 ,...,  n).

In the Example 2 considered above, in the case of splitting variables ( x, y, z) into subsets ( X) And ( y, z) the matrix takes the form:

	y,z

An essential feature of the matrix method of setting is that the complete sets of variables  X n, corresponding to adjacent (both vertically and horizontally) cells, differ in one coordinate.

Assignment using a complete binary tree

For description n-local function f( X n) uses the height binary tree property n, which consists in the fact that each hanging vertex in it one-to-one corresponds to a certain set of values ​​of the vector  X n. Accordingly, this hanging vertex can be assigned the same truth value that the function has on this set f. As an example (Fig. 1.3), we present the task with the help of a binary tree of the three-place function considered above f=(10110110).

The first row of digits assigned to the hanging vertices of the tree denotes the lexicographic number of the set, the second is the set itself, and the third is the value of the function on it.

Job withn - dimensional unit cubeIN n

Because the tops IN n can also be mapped one-to-one to the set of all sets  X n, That n-local function f(X n) can be specified by assigning its truth values ​​to the corresponding vertices of the cube IN n . Figure 1.4 shows the task of the function f= (10110110) in Cuba IN 3 . Truth values ​​are assigned to the vertices of the cube.

Definition . Algebra of logic name the set of Boolean constants and variables together with the logical connectives introduced on them.

formula task

Logic algebra functions can be given as analytic expressions.

Definition. Let X― alphabet of variables and constants used in the algebra of logic, F― the set of notations for all elementary functions and their generalizations for the number of variables exceeding 2.

Formula over X, F(logical algebra formula) let's name all records of the form:

A) X, Where X X;

b)  F 1 , F 1 &F 2 ,F 1 F 2 , F 1 F 2 , F 1  F 2 , F 1 F 2 ,F 1 F 2 ,F 1 F 2 , Where F 1 , F 2 are formulas over X, F;

V) h(F 1 , … ,F n ), Where n > 2, F 1 ,… ,F n are formulas over X,F, h ― the notation for the generalized threshold function from F .

As follows from the definition, for binary elementary functions, the infix form is used, in which the function symbol is placed between the arguments, for negation and generalized functions, the prefix form is used, in which the function symbol is placed before the argument list.

Example 3

1. Expressions X(atz); ( x, y, z  u) are formulas of the algebra of logic, since they satisfy the definition given above.

2. Expression  X (at z) is not a formula of the algebra of logic, because the operation is incorrectly applied  .

Definition. The function realized by the formula F, is the function obtained by substituting the values ​​of the variables in F. Let's denote it f(F).

Example 4 Consider the formula F=hu (Xz). In order to build the truth table of the implemented function, it is necessary to sequentially, taking into account the strength of logical connectives, perform logical multiplication hu, then the implication ( Xz), then add the obtained truth values ​​modulo 2. The result of the actions is shown in the table:

				X z

The formula representation of functions makes it possible to estimate a priori many properties of functions. The transition from a formula task to a truth table can always be performed by successive substitutions of truth values ​​into the elementary functions included in the formula. The reverse transition is ambiguous, since the same function can be represented by different formulas. It requires separate consideration.