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MATHEMATICAL ANALYSIS, a branch of mathematics that provides methods for the quantitative study of various processes of change; deals with the study of the rate of change (differential calculus) and the determination of the lengths of curves, areas and volumes of figures bounded by curved contours and surfaces (integral calculus). It is typical for problems of mathematical analysis that their solution is associated with the concept of a limit.
The beginning of mathematical analysis was laid in 1665 by I. Newton and (around 1675) independently by G. Leibniz, although important preparatory work was carried out by I. Kepler (1571–1630), F. Cavalieri (1598–1647), P. Fermat (1601– 1665), J. Wallis (1616–1703) and I. Barrow (1630–1677).
To make the presentation more vivid, we will resort to the language of graphics. Therefore, it may be useful for the reader to look into the article ANALYTICAL GEOMETRY before starting to read this article.
DIFFERENTIAL CALCULUS
Tangents.
In Fig. 1 shows a fragment of the curve y = 2x – x 2, enclosed between x= –1 and x= 3. Sufficiently small segments of this curve look straight. In other words, if R is an arbitrary point of this curve, then there is a certain straight line passing through this point and which is an approximation of the curve in a small neighborhood of the point R, and the smaller the neighborhood, the better the approximation. Such a line is called tangent to the curve at the point R. The main task of differential calculus is to construct a general method that allows one to find the direction of a tangent at any point on a curve at which a tangent exists. It is not difficult to imagine a curve with a sharp break (Fig. 2). If R is the top of such a break, then we can construct an approximating straight line P.T. 1 – to the right of the point R and another approximating straight line RT 2 – to the left of the point R. But there is no single straight line passing through a point R, which approached the curve equally well in the vicinity of the point P both on the right and on the left, therefore the tangent at the point P does not exist.
In Fig. 1 tangent FROM drawn through the origin ABOUT= (0,0). The slope of this line is 2, i.e. when the abscissa changes by 1, the ordinate increases by 2. If x And y– coordinates of an arbitrary point on FROM, then, moving away from ABOUT to a distance X units to the right, we are moving away from ABOUT on 2 y units up. Hence, y/x= 2, or y = 2x. This is the tangent equation FROM to the curve y = 2x – x 2 at point ABOUT.
It is now necessary to explain why, out of the set of lines passing through the point ABOUT, the straight line is chosen FROM. How does a straight line with a slope of 2 differ from other straight lines? There is one simple answer, and it is difficult to resist the temptation to give it using the analogy of a tangent to a circle: the tangent FROM has only one common point with the curve, while any other non-vertical line passing through the point ABOUT, intersects the curve twice. This can be verified as follows.
Since the expression y = 2x – x 2 can be obtained by subtraction X 2 of y = 2x(equations of straight line FROM), then the values y there is less knowledge for the graph y for a straight line at all points except the point x= 0. Therefore, the graph is everywhere except the point ABOUT, located below FROM, and this line and the graph have only one common point. Moreover, if y = mx- equation of some other line passing through a point ABOUT, then there will definitely be two points of intersection. Really, mx = 2x – x 2 not only when x= 0, but also at x = 2 – m. And only when m= 2 both intersection points coincide. In Fig. 3 shows the case when m is less than 2, so to the right of ABOUT a second intersection point appears.
What FROM– the only non-vertical straight line passing through a point ABOUT and having only one common point with the graph, not its most important property. Indeed, if we turn to other graphs, it will soon become clear that the tangent property we noted is not satisfied in the general case. For example, from Fig. 4 it is clear that near the point (1,1) the graph of the curve y = x 3 is well approximated by a straight line RT which, however, has more than one common point with it. However, we would like to consider RT tangent to this graph at point R. Therefore, it is necessary to find some other way to highlight the tangent than the one that served us so well in the first example.
Let us assume that through the point ABOUT and an arbitrary point Q = (h,k) on the curve graph y = 2x – x 2 (Fig. 5) a straight line (called a secant) is drawn. Substituting the values into the equation of the curve x = h And y = k, we get that k = 2h – h 2, therefore, the angular coefficient of the secant is equal to
At very small h meaning m close to 2. Moreover, choosing h close enough to 0 we can do m arbitrarily close to 2. We can say that m"tends to the limit" equal to 2 when h tends to zero, or whatever the limit m equals 2 at h tending to zero. Symbolically it is written like this:
Then the tangent to the graph at the point ABOUT is defined as a straight line passing through a point ABOUT, with a slope equal to this limit. This definition of a tangent is applicable in the general case.
Let's show the advantages of this approach with one more example: let's find the slope of the tangent to the graph of the curve y = 2x – x 2 at any point P = (x,y), not limited to the simplest case when P = (0,0).
Let Q = (x + h, y + k) – the second point on the graph, located at a distance h to the right of R(Fig. 6). We need to find the slope k/h secant PQ. Dot Q is at a distance
above the axis X.
Opening the brackets, we find:
Subtracting from this equation y = 2x – x 2, find the vertical distance from the point R to the point Q:
Therefore, the slope m secant PQ equals
Now that h tends to zero, m tends to 2 – 2 x; We will take the last value as the angular coefficient of the tangent P.T.. (The same result will occur if h takes negative values, which corresponds to the selection of a point Q on the left of P.) Note that when x= 0 the result obtained coincides with the previous one.
Expression 2 – 2 x called the derivative of 2 x – x 2. In the old days, the derivative was also called "differential ratio" and "differential coefficient". If by expression 2 x – x 2 designate f(x), i.e.
then the derivative can be denoted
In order to find out the slope of the tangent to the graph of the function y = f(x) at some point, it is necessary to substitute in fў ( x) value corresponding to this point X. Thus, the slope fў (0) = 2 at X = 0, fў (0) = 0 at X= 1 and fў (2) = –2 at X = 2.
The derivative is also denoted atў , dy/dx, D x y And Du.
The fact that the curve y = 2x – x 2 near a given point is practically indistinguishable from its tangent at this point, allows us to speak of the angular coefficient of the tangent as the “angular coefficient of the curve” at the point of tangency. Thus, we can say that the slope of the curve we are considering has a slope of 2 at the point (0,0). We can also say that when x= 0 rate of change y relatively x is equal to 2. At point (2,0) the slope of the tangent (and the curve) is –2. (The minus sign means that as we increase x variable y decreases.) At the point (1,1) the tangent is horizontal. We say it's a curve y = 2x – x 2 has a stationary value at this point.
Highs and lows.
We have just shown that the curve f(x) = 2x – x 2 is stationary at point (1,1). Because fў ( x) = 2 – 2x = 2(1 – x), it is clear that when x, less than 1, fў ( x) is positive, and therefore y increases; at x, large 1, fў ( x) is negative, and therefore y decreases. Thus, in the vicinity of the point (1,1), indicated in Fig. 6 letter M, meaning at grows to a point M, stationary at point M and decreases after the point M. This point is called “maximum” because the value at at this point exceeds any of its values in a sufficiently small neighborhood. Similarly, the “minimum” is defined as the point in the vicinity of which all values y exceed the value at at this very point. It may also happen that although the derivative of f(x) at a certain point and vanishes; its sign in the vicinity of this point does not change. Such a point, which is neither a maximum nor a minimum, is called an inflection point.
As an example, let's find the stationary point of the curve
The derivative of this function is equal to
and goes to zero at x = 0, X= 1 and X= –1; those. at points (0,0), (1, –2/15) and (–1, 2/15). If X a little less than –1, then fў ( x) is negative; If X a little more than –1, then fў ( x) is positive. Therefore, the point (–1, 2/15) is the maximum. Similarly, it can be shown that the point (1, –2/15) is a minimum. But the derivative fў ( x) is negative both before the point (0,0) and after it. Therefore, (0,0) is the inflection point.
The study of the shape of the curve, as well as the fact that the curve intersects the axis X at f(x) = 0 (i.e. when X= 0 or ) allow us to present its graph approximately as shown in Fig. 7.
In general, if we exclude unusual cases (curves containing straight segments or an infinite number of bends), there are four options for the relative position of the curve and the tangent in the vicinity of the tangent point R. (Cm. rice. 8, on which the tangent has a positive slope.)
1) On both sides of the point R the curve lies above the tangent (Fig. 8, A). In this case they say that the curve at the point R convex down or concave.
2) On both sides of the point R the curve is located below the tangent (Fig. 8, b). In this case, the curve is said to be convex upward or simply convex.
3) and 4) The curve is located above the tangent on one side of the point R and below - on the other. In this case R– inflection point.
Comparing values fў ( x) on both sides of R with its value at the point R, one can determine which of these four cases one has to deal with in a particular problem.
Applications.
All of the above has important applications in various fields. For example, if a body is thrown vertically upward with an initial speed of 200 feet per second, then the height s, on which they will be located through t seconds compared to the starting point will be
Proceeding in the same way as in the examples we considered, we find
this quantity goes to zero at c. Derivative fў ( x) is positive up to the value c and negative after this time. Hence, s increases to , then becomes stationary, and then decreases. This is a general description of the movement of a body thrown upward. From it we know when the body reaches its highest point. Next, substituting t= 25/4 V f(t), we get 625 feet, the maximum lift height. In this problem fў ( t) has a physical meaning. This derivative shows the speed at which the body is moving at an instant t.
Let us now consider an application of another type (Fig. 9). From a sheet of cardboard with an area of 75 cm2, you need to make a box with a square bottom. What should be the dimensions of this box in order for it to have maximum volume? If X– side of the base of the box and h is its height, then the volume of the box is V = x 2 h, and the surface area is 75 = x 2 + 4xh. Transforming the equation, we get:
Derivative of V turns out to be equal
and goes to zero at X= 5. Then
And V= 125/2. Graph of a function V = (75x – x 3)/4 is shown in Fig. 10 (negative values X omitted as having no physical meaning in this problem).
Derivatives.
An important task of differential calculus is the creation of methods that allow you to quickly and conveniently find derivatives. For example, it is easy to calculate that
(The derivative of a constant is, of course, zero.) It is not difficult to derive a general rule:
Where n– any whole number or fraction. For example,
(This example shows how useful fractional exponents are.)
Here are some of the most important formulas:
There are also the following rules: 1) if each of the two functions g(x) And f(x) has derivatives, then the derivative of their sum is equal to the sum of the derivatives of these functions, and the derivative of the difference is equal to the difference of the derivatives, i.e.
2) the derivative of the product of two functions is calculated by the formula:
3) the derivative of the ratio of two functions has the form
4) the derivative of a function multiplied by a constant is equal to the constant multiplied by the derivative of this function, i.e.
It often happens that the values of a function have to be calculated step by step. For example, to calculate sin x 2, we need to first find u = x 2, and then calculate the sine of the number u. We find the derivative of such complex functions using the so-called “chain rule”:
In our example f(u) = sin u, fў ( u) = cos u, hence,
These and other similar rules allow you to immediately write down derivatives of many functions.
Linear approximations.
The fact that, knowing the derivative, we can in many cases replace the graph of a function near a certain point with its tangent at this point is of great importance, since it is easier to work with straight lines.
This idea finds direct application in calculating approximate values of functions. For example, it is quite difficult to calculate the value when x= 1.033. But you can use the fact that the number 1.033 is close to 1 and that . Up close x= 1 we can replace the graph with a tangent curve without making any serious mistakes. The angular coefficient of such a tangent is equal to the value of the derivative ( x 1/3)ў = (1/3) x–2/3 at x = 1, i.e. 1/3. Since point (1,1) lies on the curve and the angular coefficient of the tangent to the curve at this point is equal to 1/3, the tangent equation has the form
On this straight line X = 1,033
Received value y should be very close to the true value y; and, indeed, it is only 0.00012 more than the true one. In mathematical analysis, methods have been developed that make it possible to increase the accuracy of this kind of linear approximations. These methods ensure the reliability of our approximate calculations.
The procedure just described suggests one useful notation. Let P– point corresponding to the function graph f variable X, and let the function f(x) is differentiable. Let's replace the graph of the curve near the point R tangent to it drawn at this point. If X change by value h, then the ordinate of the tangent will change by the amount h H f ў ( x). If h is very small, then the latter value serves as a good approximation to the true change in the ordinate y graphic arts. If instead h we will write a symbol dx(this is not a product!), but a change in ordinate y let's denote dy, then we get dy = f ў ( x)dx, or dy/dx = f ў ( x) (cm. rice. eleven). Therefore, instead of Dy or f ў ( x) the symbol is often used to denote a derivative dy/dx. The convenience of this notation depends mainly on the explicit appearance of the chain rule (differentiation of a complex function); in the new notation this formula looks like this:
where it is implied that at depends on u, A u in turn depends on X.
Magnitude dy called differential at; in reality it depends on two variables, namely: from X and increments dx. When the increment dx very small size dy is close to the corresponding change in value y. But assume that the increment dx little, no need.
Derivative of a function y = f(x) we designated f ў ( x) or dy/dx. It is often possible to take the derivative of the derivative. The result is called the second derivative of f (x) and is denoted f ўў ( x) or d 2 y/dx 2. For example, if f(x) = x 3 – 3x 2, then f ў ( x) = 3x 2 – 6x And f ўў ( x) = 6x– 6. Similar notation is used for higher order derivatives. However, to avoid a large number of strokes (equal to the order of the derivative), the fourth derivative (for example) can be written as f (4) (x), and the derivative n-th order as f (n) (x).
It can be shown that the curve at a point is convex downward if the second derivative is positive, and convex upward if the second derivative is negative.
If a function has a second derivative, then the change in value y, corresponding to the increment dx variable X, can be approximately calculated using the formula
This approximation is usually better than that given by the differential fў ( x)dx. It corresponds to replacing part of the curve not with a straight line, but with a parabola.
If the function f(x) there are derivatives of higher orders, then
The remainder term has the form
Where x- some number between x And x + dx. The above result is called Taylor's formula with remainder term. If f(x) has derivatives of all orders, then usually Rn® 0 at n ® Ґ .
INTEGRAL CALCULUS
Squares.
When studying the areas of curvilinear plane figures, new aspects of mathematical analysis are revealed. The ancient Greeks tried to solve problems of this kind, for whom determining, for example, the area of a circle was one of the most difficult tasks. Archimedes achieved great success in solving this problem, who also managed to find the area of a parabolic segment (Fig. 12). Using very complex reasoning, Archimedes proved that the area of a parabolic segment is 2/3 of the area of the circumscribed rectangle and, therefore, in this case is equal to (2/3)(16) = 32/3. As we will see later, this result can be easily obtained by methods of mathematical analysis.
The predecessors of Newton and Leibniz, mainly Kepler and Cavalieri, solved problems of calculating the areas of curvilinear figures using a method that can hardly be called logically sound, but which turned out to be extremely fruitful. When Wallis in 1655 combined the methods of Kepler and Cavalieri with the methods of Descartes (analytic geometry) and took advantage of the newly emerging algebra, the stage was fully prepared for the appearance of Newton.
Wallis divided the figure, the area of which needed to be calculated, into very narrow strips, each of which he approximately considered a rectangle. Then he added up the areas of the approximating rectangles and in the simplest cases obtained the value to which the sum of the areas of the rectangles tended when the number of strips tended to infinity. In Fig. Figure 13 shows rectangles corresponding to some division into strips of the area under the curve y = x 2 .
Main theorem.
The great discovery of Newton and Leibniz made it possible to eliminate the laborious process of going to the limit of the sum of areas. This was done thanks to a new look at the concept of area. The point is that we must imagine the area under the curve as generated by an ordinate moving from left to right and ask at what rate the area swept by the ordinates changes. We will get the key to answering this question if we consider two special cases in which the area is known in advance.
Let's start with the area under the graph of a linear function y = 1 + x, since in this case the area can be calculated using elementary geometry.
Let A(x) – part of the plane enclosed between the straight line y = 1 + x and a segment OQ(Fig. 14). When driving QP right area A(x) increases. At what speed? It is not difficult to answer this question, since we know that the area of a trapezoid is equal to the product of its height and half the sum of its bases. Hence,
Rate of area change A(x) is determined by its derivative
We see that Aў ( x) coincides with the ordinate at points R. Is this a coincidence? Let's try to check on the parabola shown in Fig. 15. Area A (x) under the parabola at = X 2 in the range from 0 to X equal to A(x) = (1 / 3)(x)(x 2) = x 3/3. The rate of change of this area is determined by the expression
which exactly coincides with the ordinate at moving point R.
If we assume that this rule holds in the general case such that
is the rate of change of the area under the graph of the function y = f(x), then this can be used for calculations and other areas. In fact, the ratio Aў ( x) = f(x) expresses a fundamental theorem that could be formulated as follows: the derivative, or rate of change of area as a function of X, equal to the function value f (x) at point X.
For example, to find the area under the graph of a function y = x 3 from 0 to X(Fig. 16), let's put
A possible answer reads:
since the derivative of X 4 /4 is really equal X 3. Besides, A(x) is equal to zero at X= 0, as it should be if A(x) is indeed an area.
Mathematical analysis proves that there is no other answer other than the above expression for A(x), does not exist. Let us show that this statement is plausible using the following heuristic (non-rigorous) reasoning. Suppose there is some second solution IN(x). If A(x) And IN(x) “start” simultaneously from zero value at X= 0 and change at the same rate all the time, then their values cannot be X cannot become different. They must coincide everywhere; therefore, there is a unique solution.
How can you justify the relationship? Aў ( x) = f(x) in general? This question can only be answered by studying the rate of change of area as a function of X in general. Let m– the smallest value of the function f (x) in the range from X before ( x + h), A M– the largest value of this function in the same interval. Then the increase in area when going from X To ( x + h) must be enclosed between the areas of two rectangles (Fig. 17). The bases of both rectangles are equal h. The smaller rectangle has a height m and area mh, larger, respectively, M And Mh. On the graph of area versus X(Fig. 18) it is clear that when the abscissa changes to h, the ordinate value (i.e. area) increases by the amount between mh And Mh. The secant slope on this graph is between m And M. what happens when h tends to zero? If the graph of a function y = f(x) is continuous (i.e. does not contain discontinuities), then M, And m tend to f(x). Therefore, the slope Aў ( x) graph of area as a function of X equals f(x). This is precisely the conclusion that needed to be reached.
Leibniz proposed for the area under a curve y = f(x) from 0 to A designation
In a rigorous approach, this so-called definite integral should be defined as the limit of certain sums in the manner of Wallis. Considering the result obtained above, it is clear that this integral is calculated provided that we can find such a function A(x), which vanishes when X= 0 and has a derivative Aў ( x), equal to f (x). Finding such a function is usually called integration, although it would be more appropriate to call this operation anti-differentiation, meaning that it is in some sense the inverse of differentiation. In the case of a polynomial, integration is simple. For example, if
which is easy to verify by differentiating A(x).
To calculate the area A 1 under the curve y = 1 + x + x 2 /2, enclosed between ordinates 0 and 1, we simply write
and, substituting X= 1, we get A 1 = 1 + 1/2 + 1/6 = 5/3. Square A(x) from 0 to 2 is equal to A 2 = 2 + 4/2 + 8/6 = 16/3. As can be seen from Fig. 19, the area enclosed between ordinates 1 and 2 is equal to A 2 – A 1 = 11/3. It is usually written as a definite integral
Volumes.
Similar reasoning makes it surprisingly easy to calculate the volumes of bodies of rotation. Let's demonstrate this with the example of calculating the volume of a sphere, another classical problem that the ancient Greeks, using the methods known to them, managed to solve with great difficulty.
Let's rotate part of the plane contained inside a quarter circle of radius r, at an angle of 360° around the axis X. As a result, we get a hemisphere (Fig. 20), the volume of which we denote V(x). We need to determine the rate at which it increases V(x) with increasing x. Moving from X To X + h, it is easy to verify that the increment in volume is less than the volume p(r 2 – x 2)h circular cylinder with radius and height h, and more than volume p[r 2 – (x + h) 2 ]h cylinder radius and height h. Therefore, on the graph of the function V(x) the angular coefficient of the secant is between p(r 2 – x 2) and p[r 2 – (x + h) 2 ]. When h tends to zero, the slope tends to
At x = r we get
for the volume of the hemisphere, and therefore 4 p r 3/3 for the volume of the entire ball.
A similar method allows one to find the lengths of curves and the areas of curved surfaces. For example, if a(x) – arc length PR in Fig. 21, then our task is to calculate aў( x). At the heuristic level, we will use a technique that allows us not to resort to the usual passage to the limit, which is necessary for a rigorous proof of the result. Let us assume that the rate of change of the function A(x) at point R the same as it would be if the curve were replaced by its tangent P.T. at the point P. But from Fig. 21 is directly visible when stepping h to the right or left of the point X along RT meaning A(x) changes to
Therefore, the rate of change of the function a(x) is
To find the function itself a(x), you just need to integrate the expression on the right side of the equality. It turns out that integration is quite difficult for most functions. Therefore, the development of methods of integral calculus constitutes a large part of mathematical analysis.
Antiderivatives.
Every function whose derivative is equal to the given function f(x), is called antiderivative (or primitive) for f(x). For example, X 3 /3 – antiderivative for the function X 2 since ( x 3 /3)ў = x 2. Of course X 3 /3 is not the only antiderivative of the function X 2 because x 3 /3 + C is also a derivative for X 2 for any constant WITH. However, in what follows we agree to omit such additive constants. In general
Where n is a positive integer, since ( x n + 1/(n+ 1))ў = x n. Relation (1) is satisfied in an even more general sense if n replace with any rational number k, except –1.
An arbitrary antiderivative function for a given function f(x) is usually called the indefinite integral of f(x) and denote it in the form
For example, since (sin x)ў = cos x, the formula is valid
In many cases where there is a formula for the indefinite integral of a given function, it can be found in numerous widely published tables of indefinite integrals. Integrals from elementary functions are tabular (they include powers, logarithms, exponential functions, trigonometric functions, inverse trigonometric functions, as well as their finite combinations obtained using the operations of addition, subtraction, multiplication and division). Using table integrals you can calculate integrals of more complex functions. There are many ways to calculate indefinite integrals; The most common of these is the variable substitution or substitution method. It consists in the fact that if we want to replace in the indefinite integral (2) x to some differentiable function x = g(u), then for the integral to remain unchanged, it is necessary x replaced by gў ( u)du. In other words, the equality
(substitution 2 x = u, from where 2 dx = du).
Let us present another integration method - the method of integration by parts. It is based on the already known formula
By integrating the left and right sides, and taking into account that
This formula is called the integration by parts formula.
Example 2. You need to find . Since cos x= (sin x)ў , we can write that
From (5), assuming u = x And v= sin x, we get
And since (–cos x)ў = sin x we find that
It should be emphasized that we have limited ourselves to only a very brief introduction to a very vast subject in which numerous ingenious techniques have been accumulated.
Functions of two variables.
Due to the curve y = f(x) we considered two problems.
1) Find the angular coefficient of the tangent to the curve at a given point. This problem is solved by calculating the value of the derivative fў ( x) at the specified point.
2) Find the area under the curve above the axis segment X, bounded by vertical lines X = A And X = b. This problem is solved by calculating a definite integral.
Each of these problems has an analogue in the case of a surface z = f(x,y).
1) Find the tangent plane to the surface at a given point.
2) Find the volume under the surface above the part of the plane xy, bounded by a curve WITH, and from the side – perpendicular to the plane xy passing through the points of the boundary curve WITH (cm. rice. 22).
The following examples show how these problems are solved.
Example 4. Find the tangent plane to the surface
at point (0,0,2).
A plane is defined if two intersecting lines lying in it are given. One of these straight lines ( l 1) we get in the plane xz (at= 0), second ( l 2) – in the plane yz (x = 0) (cm. rice. 23).
First of all, if at= 0, then z = f(x,0) = 2 – 2x – 3x 2. Derivative with respect to X, denoted fў x(x,0) = –2 – 6x, at X= 0 has a value of –2. Straight l 1 given by the equations z = 2 – 2x, at= 0 – tangent to WITH 1, lines of intersection of the surface with the plane at= 0. Similarly, if X= 0, then f(0,y) = 2 – y – y 2 , and the derivative with respect to at looks like
Because fў y(0,0) = –1, curve WITH 2 – line of intersection of the surface with the plane yz– has a tangent l 2 given by the equations z = 2 – y, X= 0. The desired tangent plane contains both lines l 1 and l 2 and is written by the equation
This is the equation of a plane. In addition, we receive direct l 1 and l 2, assuming, accordingly, at= 0 and X = 0.
The fact that equation (7) really defines a tangent plane can be verified at a heuristic level by noting that this equation contains first-order terms included in equation (6), and that second-order terms can be represented in the form -. Since this expression is negative for all values X And at, except X = at= 0, surface (6) lies below plane (7) everywhere, except for the point R= (0,0,0). We can say that surface (6) is convex upward at the point R.
Example 5. Find the tangent plane to the surface z = f(x,y) = x 2 – y 2 at origin 0.
On surface at= 0 we have: z = f(x,0) = x 2 and fў x(x,0) = 2x. On WITH 1, intersection lines, z = x 2. At the point O the slope is equal to fў x(0,0) = 0. On the plane X= 0 we have: z = f(0,y) = –y 2 and fў y(0,y) = –2y. On WITH 2, intersection lines, z = –y 2. At the point O curve slope WITH 2 is equal fў y(0,0) = 0. Since the tangents to WITH 1 and WITH 2 are axes X And at, the tangent plane containing them is the plane z = 0.
However, in the neighborhood of the origin, our surface is not on the same side of the tangent plane. Indeed, a curve WITH 1 everywhere, except point 0, lies above the tangent plane, and the curve WITH 2 – respectively below it. Surface intersects tangent plane z= 0 in straight lines at = X And at = –X. Such a surface is said to have a saddle point at the origin (Fig. 24).
Partial derivatives.
In previous examples we used derivatives of f (x,y) By X and by at. Let us now consider such derivatives in a more general sense. If we have a function of two variables, for example, F(x,y) = x 2 – xy, then we can determine at each point its two “partial derivatives”, one by differentiating the function with respect to X and fixing at, the other – differentiating by at and fixing X. The first of these derivatives is denoted as fў x(x,y) or ¶ f/¶ x; second - how f f ў y. If both mixed derivatives (by X And at, By at And X) are continuous, then ¶ 2 f/¶ x¶ y= ¶ 2 f/¶ y¶ x; in our example ¶ 2 f/¶ x¶ y= ¶ 2 f/¶ y¶ x = –1.
Partial derivative fў x(x,y) indicates the rate of change of the function f at point ( x,y) in the direction of increasing X, A fў y(x,y) – rate of change of function f in the direction of increasing at. Rate of change of function f at point ( X,at) in the direction of a straight line making an angle q with positive axis direction X, is called the derivative of the function f towards; its value is a combination of two partial derivatives of the function f in the tangent plane is almost equal (at small dx And dy) true change z on the surface, but calculating the differential is usually easier.
The formula from the variable change method that we have already considered, known as the derivative of a complex function or the chain rule, in the one-dimensional case when at depends on X, A X depends on t, has the form:
For functions of two variables, a similar formula has the form:
The concepts and notations of partial differentiation are easy to generalize to higher dimensions. In particular, if the surface is specified implicitly by the equation f(x,y,z) = 0, the equation of the tangent plane to the surface can be given a more symmetrical form: the equation of the tangent plane at the point ( x(x 2 /4)], then integrated over X from 0 to 1. The final result is 3/4.
Formula (10) can also be interpreted as a so-called double integral, i.e. as the limit of the sum of the volumes of elementary “cells”. Each such cell has a base D x D y and a height equal to the height of the surface above some point of the rectangular base ( cm. rice. 26). It can be shown that both points of view on formula (10) are equivalent. Double integrals are used to find centers of gravity and numerous moments encountered in mechanics.
A more rigorous justification of the mathematical apparatus.
So far we have presented the concepts and methods of mathematical analysis on an intuitive level and did not hesitate to resort to geometric figures. It remains for us to briefly consider the more rigorous methods that emerged in the 19th and 20th centuries.
At the beginning of the 19th century, when the era of storm and pressure in the “creation of mathematical analysis” ended, questions of its justification came to the fore. In the works of Abel, Cauchy and a number of other outstanding mathematicians, the concepts of “limit”, “continuous function”, “convergent series” were precisely defined. This was necessary in order to introduce logical order into the basis of mathematical analysis in order to make it a reliable research tool. The need for a thorough justification became even more obvious after the discovery in 1872 by Weierstrass of functions that were everywhere continuous but nowhere differentiable (the graph of such functions has a kink at each point). This result had a stunning effect on mathematicians, since it clearly contradicted their geometric intuition. An even more striking example of the unreliability of geometric intuition was the continuous curve constructed by D. Peano, which completely fills a certain square, i.e. passing through all its points. These and other discoveries gave rise to the program of “arithmetization” of mathematics, i.e. making it more reliable by grounding all mathematical concepts using the concept of number. The almost puritanical abstinence from clarity in works on the foundations of mathematics had its historical justification.
According to modern canons of logical rigor, it is unacceptable to talk about the area under the curve y = f(x) and above the axis segment X, even f- a continuous function, without first defining the exact meaning of the term “area” and without establishing that the area thus defined actually exists. This problem was successfully solved in 1854 by B. Riemann, who gave a precise definition of the concept of a definite integral. Since then, the idea of summation behind the concept of a definite integral has been the subject of many in-depth studies and generalizations. As a result, today it is possible to give meaning to the definite integral, even if the integrand is discontinuous everywhere. New concepts of integration, to the creation of which A. Lebesgue (1875–1941) and other mathematicians made a great contribution, increased the power and beauty of modern mathematical analysis.
It would hardly be appropriate to go into detail about all these and other concepts. We will limit ourselves only to giving strict definitions of the limit and the definite integral.
In conclusion, let us say that mathematical analysis, being an extremely valuable tool in the hands of a scientist and engineer, still attracts the attention of mathematicians today as a source of fruitful ideas. At the same time, modern development seems to indicate that mathematical analysis is increasingly being absorbed by those dominant in the 20th century. branches of mathematics such as abstract algebra and topology.
Compiled by Yu.V. Obrubov
Kaluga - 2012
Introduction to mathematical analysis.
Real numbers. Variables and constants.
One of the basic concepts of mathematics is number.
Positive numbers 1,2,3, ..., which are obtained when counting, are called natural.
The numbers... -3,-2,-1,0,1,2,3,... are called integers. Numbers that can be expressed as a finite ratio of two integers ( 
) are called rational.
These include integers and fractions, positive and negative numbers. Numbers that are represented by infinite non-periodic fractions are called irrational.
Examples of irrational numbers are 
,
. In the set of irrational numbers there are transcendental
numbers. These are numbers that are the result of non-algebraic operations. The most famous of them are the number 
and Neperovo number 
. Rational and irrational numbers are called valid
. Real numbers are represented by dots on the number line. Each point on the number line corresponds to one single real number and, conversely, each real number corresponds to a single point on the number line. Thus, a one-to-one correspondence is established between real numbers and points on the number line. This makes it possible to use the terms “number a” and “point a” equally.
In the process of studying various physical, economic, and social processes, one often has to deal with quantities that represent the numerical values of the parameters of the phenomena being studied. At the same time, some of them change, while others retain their values.
Variable
is a quantity that takes on different numerical values. A quantity whose numerical value does not change in a given problem or experiment is called constant.
Variable quantities are usually denoted in Latin letters 
and constants 
.
Variable value 
is considered given if the set of values it can take is known. This set is called the range of variation of the variable.
There are different types of sets of values of a numeric variable.
Interval
is the set of values of x contained between the numbers a and b, while the numbers a and b do not belong to the set in question. The interval is denoted by: (a,b);a By segment
is the set of values of x contained between the numbers a and b, while the numbers a and b belong to the set in question. The segment is denoted by ,a≤x≤b. The set of all real numbers is an open interval. Denoted by: (- ∞,+ ∞), -∞<х <+∞, R. Neighborhood of point x
0
is an arbitrary interval (a,b) containing the point x 0, all points of this interval satisfy the inequalitya ε
- neighborhood of point a
is an interval with center at point a that satisfies the inequality a–ε Function is one of the basic concepts of mathematical analysis. Let X and Y be arbitrary sets of real numbers. If each number x X, according to some rule or law, is associated with a single well-defined real number yY, then they say that the given function
with the domain of definition of X and the set of values of Y. Denoted by y = f (x). The variable x is called argument
functions. In defining a function, two points are essential: indicating the domain of definition and establishing the law of correspondence. Domain of definition
or area of existence
A function is the set of argument values for which the function exists, that is, it makes sense. Change area
A function is the set of values y that it takes given acceptable values of x. Methods for specifying a function.
Analytical method of specifying a function. With this method of specifying a function, the correspondence law is written in the form of a formula (an analytical expression) indicating through what mathematical transformations the corresponding value of y can be found from a known value of the argument x. A function can be specified by one analytical expression throughout its entire domain of definition or represent a collection of several analytical expressions. For example: y = sin (x 2 + 1) 2. Tabular method of specifying a function As a result of direct observation or experimental study of any phenomenon or process, the values of the argument x and the corresponding values of y are written out in a certain order. This table defines the function y of x. An example of a tabular method of specifying a function can be tables of trigonometric functions, tables of logarithms, dates and exchange rates, air temperature and humidity, etc. 3. Graphical method of specifying a function. The graphical method of specifying a function consists of depicting points (x, y) on the coordinate plane using technical devices. The graphical method of specifying a function is not used in mathematical analysis, but graphical illustration of analytically defined functions is always used. According to the Russian language dictionary analysis is a method of scientific research by considering individual aspects, properties, and components of something. One of the most important branches of mathematics is called mathematical analysis, and often even just analysis. The question immediately arises: what exactly is analyzed by mathematical analysis? The answer is clear - functions are analyzed. Function(from the Latin “functionio” - implementation) represents the relationship between variable numerical values. Since analysis is a research method, a second question arises: what is this method? The answer is given by the second name of mathematical analysis - differential and integral calculus. Calculus is the branch of mathematics that sets out the rules of calculation. Word " differential" comes from the Latin word "differentiation", i.e. difference. Word " integral“does not have such a clear origin (“integer” - whole; “integra” - restore), but it has the meaning of combining parts into a whole, restoring what has been broken into differences. This recovery is achieved using summation. Let's summarize the first results: · Main objects, studied in mathematical analysis are functions. · Functions are dependencies of various types between variable numerical values. · The method of mathematical analysis is differentiation– working with differences in function values, and integration– calculation of amounts. Thus, to master mathematical analysis, first of all, you need to understand the concept of function. Function is an important mathematical concept because functions are a mathematical way of describing motion and change. Function is a process. The most important type of movement is mechanical movement in a straight line. When moving, the distances traveled by an object are measured, but this is clearly not enough to fully describe the movement. Both Achilles and the tortoise can move the same distance from the starting point, but their movement differs in speed, and speed cannot be measured without measuring time. Already from considering this example, it becomes clear that one variable is not enough to describe movement and change. It is intuitively clear that time changes uniformly, but distance can change either faster or slower. The movement is fully described if at each moment of time it is known how far the object has moved from the starting point. So, with mechanical movement, a correspondence arises between the values of two variable quantities - time, which changes regardless of anything, and distance, which depends on time. This fact forms the basis for the definition of a function. In this case, the two variables are no longer called time and distance. Function Definition: function – is this a rule or law, assigning to each value of the independent variable X
specific value of the dependent variable at
. Independent variable X
is called an argument, and the dependent at
– function. It is sometimes said that a function is a relationship between two variables. How to visualize what a variable is? A variable is a number line (ruler or scale) along which a point (thermometer or knitting needle with a bead) moves. A function is a mechanism of gears with two windows x and y. This mechanism allows you to install in a window X
any value, and in the window at
The function value will automatically appear using the gears. Problem 1. The patient's temperature is measured every hour. There is a function - the dependence of temperature on time. How to present this function? Answer: table and graph. A function is continuous, just as movement is continuous, but in practice it is impossible to fix this continuity. You can only catch individual argument and function values. However, theoretically it is still possible to describe continuity. Problem 2. Galileo Galilei discovered that a freely falling body travels one unit of distance in the first second, 3 units in the second, 5 units in the third, etc. Determine the dependence of time on distance. Note: derive a general formula for the dependence of the distance traveled on the distance number. Methods for specifying functions. Problems of mathematical analysis. Transition from one representation of a function to another (calculating function values, constructing approximate analytical functions from experimental numerical and graphical data, studying functions and constructing graphs). Mathematical study of the properties of a function as a process. Example 1: searching for speed using a known function of path versus time (differentiation). Example 2: finding a path using a known function of speed versus time (integration). MATHEMATICAL ANALYSIS part of mathematics, in which functions and their generalizations are studied by the method limits. The concept of a limit is closely related to the concept of an infinitesimal quantity, so we can also say that M. a. studies functions and their generalizations using the infinitesimal method. The name "M. a." - an abbreviated modification of the old name of this part of mathematics - “Analysis of infinitesimals”; the latter reveals the content more fully, but it is also abbreviated (the title “Analysis by means of infinitesimals” would characterize the subject more accurately). In classical M. a. the objects of study (analysis) are primarily functions. “First of all” because the development of M. a. led to the possibility of studying with its methods more complex formations than , - functionals, operators, etc. In nature and technology, movements and processes are found everywhere, which are described by functions; the laws of natural phenomena are also usually described by functions. Hence the objective importance of M. a. as a means of studying functions. M. a. in the broad sense of the term, it covers a very large part of mathematics. It includes differential, integral calculus, theory of functions of a complex variable, theory ordinary differential equations, theory partial differential equations, theory integral equations, calculus of variations, functional analysis and some other mathematical disciplines. Modern number theory And probability theory apply and develop methods of MA. Still, the term M. a. often used to name only the foundations of mathematical analysis, which combine the theory real number theory of limits, theory rows, differential and integral calculus and their direct applications, such as the theory of maxima and minima, the theory implicit functions, Fourier series, Fourier integrals. Function. In M. a. start from the definition of a function according to Lobachevsky and Dirichlet. If each number xy of a certain set of Fnumbers, by virtue of k.-l. law is included in the number y, then this defines the function from one variable X. The function is defined similarly from variables, where x=(x 1 , ..., x p) -
point in n-dimensional space; also consider functions from points x=(x 1 , X 2 ,
...) of a certain infinite-dimensional space, which, however, are more often called functionals. Elementary functions. Fundamental importance in M. a. play elementary functions. In practice, they mainly operate with elementary functions; they are used to approximate functions of a more complex nature. Elementary functions can be considered not only for real, but also for complex x; then the ideas about these functions become, in a certain sense, complete. In connection with this, an important branch of M. arose, called. theory of functions of a complex variable, or theory analytical functions. Real number. The concept of a function is essentially based on the concept of a real (rational and irrational) number. It was finally formed only at the end of the 19th century. In particular, a logically flawless connection has been established between numbers and geometric points. straight line, which led to the formal substantiation of the ideas of R. Descartes (R. Descartes, mid-17th century), who introduced rectangular coordinate systems into mathematics and the representation of functions in them by graphs. Limit. In M. a. method of studying functions is . A distinction is made between the limit of a sequence and the limit of a function. These concepts were finally formed only in the 19th century, although ancient Greek had an idea about them. scientists. Suffice it to say that Archimedes (3rd century BC) was able to calculate a segment of a parabola using a process that we would call the passage to the limit (see. Exhaustion method). Continuous functions. Important functions studied in MA are formed continuous functions. One possible definition of this concept: function y=f(x).from one variable X, given on the interval ( a, b),
called continuous at a point X, If The function is continuous on the interval ( a, b),
if it is continuous at all its points; then it is a curve, continuous in the everyday understanding of the word. Derivative and . Among continuous functions, we should highlight functions that have derivative. Derivative of a function at a point is the rate of change at this point, i.e. the limit If you have the coordinate of a point moving along the ordinate axis in time X, then f" (x). is the instantaneous speed of the point at the moment of time X. By the sign of the derivative f" (x) .
judge the nature of the change in f(x): if f"(z)>0 ( f"(x) <0
).
on the interval ( s, d), then the function / increases (decreases) on this interval. If the function / at a point x reaches a local extremum (maximum or minimum) and has a derivative at this point, then the latter is equal to zero at this point f "(x 0) = 0. Equality (1) can be replaced by the equivalent equality where is infinitesimal, when i.e. if the function f has a derivative at the point X, then its increment at this point is decomposed into two terms. Of these, the first is from (proportional), the second - tends to zero faster than Value (2) called. differential functions corresponding to the increment At small can be considered approximately equal dy: The above considerations about the differential are typical for MA. They extend to functions of many variables and to functionals. For example, if the function from variables has continuous partial derivatives at the point x=(x 1 , ... , x n), then its increment
corresponding to the increments of independent variables can be written in the form where when that is if all Here the first term on the right side of (3) is the differential dz functions f. It depends linearly on and the second term tends to zero at faster than Let be given (see art. Calculus of variations) extended to function classes x(t) ,
having a continuous derivative on the segment and satisfying the boundary conditions x( t 0)=x 0, x( t 1)=x l , Where x 0, x 1 - data numbers; let, further, be the class of the function h(t) ,
having a continuous derivative on and such that h( t 0)=h(t 1)=0. Obviously, if In the calculus of variations it is proven that, under certain conditions on L, the increment of the functional J(x) can be written in the form at where and, thus, the second term on the right side of (4) tends to zero faster than ||h||, and the first term depends linearly on the first term in (4) called. variation of the functional and is denoted by dJ( x,h). Integral. Along with the derivative, it is of fundamental importance in mathematics. There are indefinite and definite integrals. The indefinite integral is closely related to the antiderivative function. The function F(x).is called. antiderivative of the function f on the interval ( a, b), if on this interval F"(x) =f(x). Definite integral (Riemann) of the function / on the interval [ a, b]there is a limit If the function f is positive and continuous on the interval [ a, b],
then the integral of it on this segment is equal to the area of the figure bounded by the curve y=f(x), axis Oh and straight x=a, x=b. The class of Riemann integrable functions contains all continuous functions on [ a, b]functions and certain discontinuous functions. But they are all necessarily limited. For unbounded functions that do not grow very quickly, as well as for certain functions defined on infinite intervals, the so-called improper integrals, requiring a double passage to the limit for their definition. The concept of the Riemann integral for a function of one variable extends to functions of many variables (see Multiple integral). On the other hand, the needs of M. a. led to a generalization of the integral in a completely different direction, meaning Lebesgue integral or more general Lebesgue-Stieltjes integral. Essential in the definition of these integrals is the introduction for certain sets, called measurable, of the concept of their measure and, on this basis, of the concept of a measurable function. For measurable functions, the Lebesgue - Stieltjes integral is introduced. In this case, a wide range of different measures and the corresponding classes of measurable sets and functions are considered. This makes it possible to adapt one or another integral to a specific specific problem. Newton-Leibniz formula. There is a connection between the derivative and the integral, expressed by the Newton-Leibniz formula (theorem) Here f(x).continuous on [ a, b]function, a F(x) -
its prototype. Formula and Taylor. Along with the derivative and integral, the most important concept (research tool) in mathematical mathematics. are Taylor n Taylor row. If the function f(x) , a called by its Taylor polynomial (degree n).by powers x-x 0: (Taylor formula); in this case the approximation error tends to zero at faster than Thus, the function f(x).in the neighborhood of the point x 0 can be approximated with any degree of accuracy by a very simple function (polynomial), which requires only arithmetic for its calculation. operations - addition, subtraction and multiplication. Particularly important are the so-called. functions that are analytical in a certain neighborhood of x 0 and have an infinite number of derivatives, such that for them in this neighborhood at they can be represented in the form of an infinite Taylor power series: Under certain conditions, Taylor expansions are also possible for functions of many variables, as well as functionals and operators. Historical reference. Until the 17th century M. a. was a set of solutions to isolated particular problems; for example, in integral calculus, these are problems of calculating the areas of figures, volumes of bodies with curved boundaries, the work of a variable force, etc. Each problem or particular problem was solved by its own method, sometimes complex and cumbersome (for the prehistory of mathematics, see article Infinitesimal calculus),
M. a. as a single and systematic the whole was formed in the works of I. Newton, G. Leibniz, L. Euler, J. Lagrange and other scientists of the 17th -18th centuries, and his theory of limits was developed by O. Komi (A. Cauchy) in the beginning. 19th century An in-depth analysis of the initial concepts of MA. was associated with the development in the 19th and 20th centuries. set theory, measure theory, theory of functions of a real variable and led to various generalizations. Lit.: La Valle - P u s e n Sh.-J. d e, Course of analysis of infinitesimals, trans. from French, vol. 1-2, M., 1933; Ilyin V. A., Poznyak E. G., Fundamentals of mathematical analysis, 3rd ed., part 1, M., 1971; 2nd ed., part 2, M., 1980; Il and N V. A., Sadovnichy V. A., Seidov B. X., Mathematical Analysis, M., 1979; K u d r i v c e v L. D., Mathematical analysis, 2nd ed., vol. 1-2, M., 1973; Nikolsky S. M., Course of mathematical analysis, 2nd ed., vol. 1-2, M., 1975; U i t t e k e r E. T., V a t s o n D J. N., Course of modern analysis, trans. from English, parts 1-2, 2nd ed., M., 1962-63; F ikhtengolts G.M., Course of differential and integral calculus, 7th ed., vol. 1-2, M., 1970; 5th ed., vol. 3, M., 1970. S. M. Nikolsky. Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985. MATHEMATICAL ANALYSIS, a set of branches of mathematics devoted to the study of functions by methods of differential calculus and integral calculus... Modern encyclopedia A set of branches of mathematics devoted to the study of functions by methods of differential and integral calculus. The term is more pedagogical than scientific: courses in mathematical analysis are taught in universities and technical schools... Big Encyclopedic Dictionary English mathematical analysis German mathematische Analysis. A branch of mathematics devoted to the study of functions by methods of differential and integral calculus. Antinazi. Encyclopedia of Sociology, 2009 ... Encyclopedia of Sociology Exist., number of synonyms: 2 matan (2) mathematical analysis (2) Dictionary of synonyms ASIS. V.N. Trishin. 2013… Synonym dictionary MATHEMATICAL ANALYSIS- MATHEMATICAL ANALYSIS. A set of branches of mathematics devoted to the study of mathematical functions by methods of differential and integral calculus. Use of M. a. methods. is an effective means of solving the most important... ... New dictionary of methodological terms and concepts (theory and practice of language teaching) mathematical analysis- — EN mathematical analysis The branch of mathematics most explicitly concerned with the limit process or the concept of convergence; includes the theories of differentiation,… … Technical Translator's Guide Mathematical analysis- MATHEMATICAL ANALYSIS, a set of branches of mathematics devoted to the study of functions by methods of differential calculus and integral calculus. ... Illustrated Encyclopedic Dictionary In the history of mathematics, we can roughly distinguish two main periods: elementary and modern mathematics. The milestone from which it is customary to count the era of new (sometimes called higher) mathematics was the 17th century - the century of the appearance of mathematical analysis. By the end of the 17th century. I. Newton, G. Leibniz and their predecessors created the apparatus of a new differential calculus and integral calculus, which forms the basis of mathematical analysis and even, perhaps, the mathematical basis of all modern natural science. Mathematical analysis is a broad field of mathematics with a characteristic object of study (variable quantity), a unique research method (analysis by means of infinitesimals or by means of passages to limits), a certain system of basic concepts (function, limit, derivative, differential, integral, series) and constantly improving and a developing apparatus, the basis of which is differential and integral calculus. Let's try to give an idea of what kind of mathematical revolution occurred in the 17th century, what characterizes the transition associated with the birth of mathematical analysis from elementary mathematics to what is now the subject of research in mathematical analysis, and what explains its fundamental role in the entire modern system of theoretical and applied knowledge . Imagine that in front of you is a beautifully executed color photograph of a stormy ocean wave rushing onto the shore: a powerful stooped back, a steep but slightly sunken chest, a head already tilted forward and ready to fall with a gray mane tormented by the wind. You stopped the moment, you managed to catch the wave, and you can now carefully study it in every detail without haste. A wave can be measured, and using the tools of elementary mathematics, you can draw many important conclusions about this wave, and therefore all its ocean sisters. But by stopping the wave, you deprived it of movement and life. Its origin, development, running, the force with which it hits the shore - all this turned out to be outside your field of vision, because you do not yet have either a language or a mathematical apparatus suitable for describing and studying not static, but developing, dynamic processes, variables and their relationships. “Mathematical analysis is no less comprehensive than nature itself: it determines all tangible relationships, measures times, spaces, forces, temperatures.” J. Fourier Movement, variables and their relationships surround us everywhere. Various types of motion and their patterns constitute the main object of study of specific sciences: physics, geology, biology, sociology, etc. Therefore, precise language and corresponding mathematical methods for describing and studying variable quantities turned out to be necessary in all areas of knowledge to approximately the same extent as numbers and arithmetic are necessary when describing quantitative relationships. So, mathematical analysis forms the basis of the language and mathematical methods for describing variables and their relationships. Nowadays, without mathematical analysis, it is impossible not only to calculate space trajectories, the operation of nuclear reactors, the movement of ocean waves and patterns of cyclone development, but also to economically manage production, resource distribution, organization of technological processes, predict the course of chemical reactions or changes in the number of various species interconnected in nature animals and plants, because all of these are dynamic processes. Elementary mathematics was mainly the mathematics of constant quantities, it studied mainly the relationships between the elements of geometric figures, the arithmetic properties of numbers and algebraic equations. Its attitude to reality can to some extent be compared with an attentive, even thorough and complete study of each fixed frame of a film that captures the changing, developing living world in its movement, which, however, is not visible in a separate frame and which can only be observed by looking the tape as a whole. But just as cinema is unthinkable without photography, so modern mathematics is impossible without that part of it that we conventionally call elementary, without the ideas and achievements of many outstanding scientists, sometimes separated by tens of centuries. Mathematics is united, and the “higher” part of it is connected with the “elementary” part in much the same way as the next floor of a house under construction is connected with the previous one, and the width of the horizons that mathematics opens to us in the world around us depends on which floor of this building we managed to reach rise. Born in the 17th century. mathematical analysis has opened up opportunities for us to scientifically describe, quantitatively and qualitatively study variables and motion in the broad sense of the word. What are the prerequisites for the emergence of mathematical analysis? By the end of the 17th century. The following situation has arisen. Firstly, within the framework of mathematics itself over many years, some important classes of problems of the same type have accumulated (for example, problems of measuring areas and volumes of non-standard figures, problems of drawing tangents to curves) and methods for solving them in various special cases have appeared. Secondly, it turned out that these problems are closely related to the problems of describing arbitrary (not necessarily uniform) mechanical motion, and in particular with the calculation of its instantaneous characteristics (speed, acceleration at any time), as well as with finding the distance traveled for movement occurring at a given variable speed. The solution to these problems was necessary for the development of physics, astronomy, and technology. Finally, thirdly, by the middle of the 17th century. the works of R. Descartes and P. Fermat laid the foundations of the analytical method of coordinates (the so-called analytical geometry), which made it possible to formulate geometric and physical problems of heterogeneous origin in the general (analytical) language of numbers and numerical dependencies, or, as we now say, numerical functions. NIKOLAY NIKOLAEVICH LUZIN
N. N. Luzin - Soviet mathematician, founder of the Soviet school of function theory, academician (1929). Luzin was born in Tomsk and studied at the Tomsk gymnasium. The formalism of the gymnasium mathematics course alienated the talented young man, and only a capable tutor was able to reveal to him the beauty and greatness of mathematical science. In 1901, Luzin entered the mathematics department of the Faculty of Physics and Mathematics of Moscow University. From the first years of his studies, issues related to infinity fell into his circle of interests. At the end of the 19th century. The German scientist G. Cantor created the general theory of infinite sets, which received numerous applications in the study of discontinuous functions. Luzin began to study this theory, but his studies were interrupted in 1905. The student, who took part in revolutionary activities, had to leave for France for a while. There he listened to lectures by the most prominent French mathematicians of that time. Upon returning to Russia, Luzin graduated from the university and was left to prepare for a professorship. Soon he again left for Paris, and then to Göttingen, where he became close to many scientists and wrote his first scientific works. The main problem that interested the scientist was the question of whether there could be sets containing more elements than the set of natural numbers, but less than the set of points on a segment (the continuum problem). For any infinite set that could be obtained from segments using the operations of union and intersection of countable collections of sets, this hypothesis was satisfied, and in order to solve the problem, it was necessary to find out what other ways there were to construct sets. At the same time, Luzin studied the question of whether it is possible to represent any periodic function, even one with infinitely many discontinuity points, as a sum of a trigonometric series, i.e. the sum of an infinite number of harmonic vibrations. On these issues, Luzin obtained a number of significant results and in 1915 he defended his dissertation “Integral and trigonometric series,” for which he was immediately awarded the academic degree of Doctor of Pure Mathematics, bypassing the intermediate master’s degree that existed at that time. In 1917 Luzin became a professor at Moscow University. A talented teacher, he attracted the most capable students and young mathematicians. Luzin's school reached its peak in the first post-revolutionary years. Luzin’s students formed a creative team, which they jokingly called “Lusitania.” Many of them received first-class scientific results while still a student. For example, P. S. Aleksandrov and M. Ya. Suslin (1894-1919) discovered a new method for constructing sets, which served as the beginning of the development of a new direction - descriptive set theory. Research in this area carried out by Luzin and his students showed that the usual methods of set theory are not enough to solve many of the problems that arise in it. Luzin's scientific predictions were fully confirmed in the 60s. XX century Many of N. N. Luzin’s students later became academicians and corresponding members of the USSR Academy of Sciences. Among them is P. S. Alexandrov. A. N. Kolmogorov. M. A. Lavrentyev, L. A. Lyusternik, D. E. Menshov, P. S. Novikov. L. G. Shnirelman and others. Modern Soviet and foreign mathematicians in their works develop the ideas of N. N. Luzin. The confluence of these circumstances led to the fact that at the end of the 17th century. two scientists - I. Newton and G. Leibniz - independently of each other managed to create a mathematical apparatus for solving these problems, summing up and generalizing individual results of their predecessors, including the ancient scientist Archimedes and contemporaries of Newton and Leibniz - B. Cavalieri, B. Pascal , D. Gregory, I. Barrow. This apparatus formed the basis of mathematical analysis - a new branch of mathematics that studies various developing processes, i.e. relationships between variables, which in mathematics are called functional dependencies or, in other words, functions. By the way, the term “function” itself was required and naturally arose precisely in the 17th century, and by now it has acquired not only general mathematical, but also general scientific significance. Initial information about the basic concepts and mathematical apparatus of analysis is given in the articles “Differential calculus” and “Integral calculus”. In conclusion, I would like to dwell on only one principle of mathematical abstraction, common to all mathematics and characteristic of analysis, and in this regard explain in what form mathematical analysis studies variables and what is the secret of such universality of its methods for studying all kinds of specific developing processes and their interrelations . Let's look at a few illustrative examples and analogies. Sometimes we no longer realize that, for example, a mathematical relation written not for apples, chairs or elephants, but in an abstract form abstracted from specific objects, is an outstanding scientific achievement. This is a mathematical law that, as experience shows, is applicable to various specific objects. This means that by studying in mathematics the general properties of abstract, abstract numbers, we thereby study the quantitative relationships of the real world. For example, from a school mathematics course it is known that, therefore, in a specific situation you could say: “If they don’t give me two six-ton dump trucks to transport 12 tons of soil, then I can ask for three four-ton dump trucks and the work will be done, and if they give me only one four-ton dump truck, then she will have to make three flights.” Thus, the abstract numbers and numerical patterns that are now familiar to us are associated with their specific manifestations and applications. The laws of change in specific variables and developing processes of nature are related in approximately the same way to the abstract, abstract form-function in which they appear and are studied in mathematical analysis. For example, an abstract ratio may reflect the dependence of a cinema's box office on the number of tickets sold, if 20 is 20 kopecks - the price of one ticket. But if we are riding a bicycle on a highway, traveling 20 km per hour, then this same ratio can be interpreted as the relationship between the time (hours) of our cycling trip and the distance covered during this time (kilometers). You can always say that, for example, a change of several times leads to a proportional (i.e., the same number of times) change in the value of , and if , then the opposite conclusion is also true. This means, in particular, to double the box office of a movie theater, you will have to attract twice as many spectators, and in order to travel twice as far on a bicycle at the same speed, you will have to ride twice as long. Mathematics studies both the simplest dependence and other, much more complex dependences in a general, abstract form, abstracted from a particular interpretation. The properties of a function or methods for studying these properties identified in such a study will be of the nature of general mathematical techniques, conclusions, laws and conclusions applicable to each specific phenomenon in which the function studied in abstract form occurs, regardless of what area of knowledge this phenomenon belongs to . So, mathematical analysis as a branch of mathematics took shape at the end of the 17th century. The subject of study in mathematical analysis (as it appears from modern positions) are functions, or, in other words, dependencies between variable quantities. With the advent of mathematical analysis, mathematics became accessible to the study and reflection of developing processes in the real world; mathematics included variables and motion.Function. Basic definitions and concepts.
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