What are functions of several variables. Derivatives of complex functions of several variables. Domain of definition of a logarithmic function of two variables

So far we have considered the simplest functional model, in which function depends on the only thing argument. But when studying various phenomena of the surrounding world, we often encounter simultaneous changes in more than two quantities, and many processes can be effectively formalized function of several variables, Where - arguments or independent variables. Let's start developing the topic with the most common one in practice. functions of two variables .

Function of two variables called law, according to which each pair of values independent variables(arguments) from domain of definition corresponds to the value of the dependent variable (function).

This function denoted as follows:

Either , or another standard letter:

Since the ordered pair of values ​​"x" and "y" determines point on a plane, then the function is also written through , where is a point on the plane with coordinates . This notation is widely used in some practical tasks.

Geometric meaning of a function of two variables very simple. If a function of one variable corresponds to a certain line on a plane (for example, the familiar school parabola), then the graph of a function of two variables is located in three-dimensional space. In practice, most often we have to deal with surface, but sometimes the graph of a function can be, for example, a spatial line(s) or even a single point.

We are well familiar with the elementary example of a surface from the course analytical geometry- This plane. Assuming that , the equation can be easily rewritten in functional form:

The most important attribute of a function of 2 variables is the already stated domain.

Domain of a function of two variables called a set everyone pairs for which the value exists.

Graphically, the domain of definition is the entire plane or part of it. Thus, the domain of definition of the function is the entire coordinate plane - for the reason that for any point exists value .

But such an idle arrangement does not always happen, of course:

Like two variables?

When considering various concepts of a function of several variables, it is useful to draw analogies with the corresponding concepts of a function of one variable. In particular, when figuring out domain of definition we paid Special attention for those functions that contain fractions, even roots, logarithms, etc. Everything is exactly the same here!

The task of finding the domain of definition of a function of two variables with almost 100% probability will be encountered in your thematic work, so I will analyze a decent number of examples:

Example 1

Find the domain of a function

Solution: since the denominator cannot go to zero, then:

Answer: the entire coordinate plane except points belonging to the line

Yes, yes, it is better to write the answer in this style. The domain of definition of a function of two variables is rarely denoted by any symbol; it is much more often used verbal description and/or drawing.

If by condition required make a drawing, then it would be necessary to depict the coordinate plane and dotted line make a straight line. The dotted line indicates that the line Excluded into the domain of definition.

As we will see a little later, in more difficult examples you cannot do without a drawing at all.

Example 2

Find the domain of a function

Solution: the radical expression must be non-negative:

Answer: half-plane

Graphic image here it’s also primitive: we draw a Cartesian coordinate system, solid draw a straight line and shade the top half-plane. The solid line indicates the fact that it included into the domain of definition.

Attention! If you don’t understand ANYTHING from the second example, please study/repeat the lesson in detail Linear inequalities- It will be very difficult without him!

Thumbnail for self-solution:

Example 3

Find the domain of a function

Two line solution and answer at the end of the lesson.

Let's continue to warm up:

Example 4

And depict it on the drawing

Solution: it is easy to understand that this is the formulation of the problem requires execution of the drawing (even if the domain of definition is very simple). But first, analytics: the radical of the expression must be non-negative: and, given that the denominator cannot go to zero, the inequality becomes strict:

How to determine the area that the inequality defines? I recommend the same algorithm of actions as in the solution linear inequalities.

First we draw line, which is set corresponding equality. The equation determines circle centered at the origin of a radius that divides the coordinate plane into two parts - “inside” and “exterior” of the circle. Since we have inequality strict, then the circle itself will certainly not be included in the domain of definition and therefore it must be drawn dotted line.

Now let's take it arbitrary plane point, not belonging to circle, and substitute its coordinates into the inequality. The easiest way, of course, is to choose the origin:

Received false inequality, thus, point does not satisfy inequality Moreover, this inequality is not satisfied by any point lying inside the circle, and, therefore, the desired domain of definition is its outer part. The definition area is traditionally hatched:

Anyone can take any point belonging to the shaded area and make sure that its coordinates satisfy the inequality. By the way, the opposite inequality gives circle centered at the origin, radius .

Answer: outer part of the circle

Let's return to the geometric meaning of the problem: we have found the domain of definition and shaded it, what does this mean? This means that at each point of the shaded area there is a value “zet” and graphically the function is the following surface:

The schematic drawing clearly shows that this surface is located in places above plane (near and far octants from us), in some places – under plane (left and right octants relative to us). The surface also passes through the axes. But the behavior of the function as such is not very interesting to us now - what is important is that all this happens exclusively in the field of definition. If we take any point belonging to the circle, then there will be no surface there (since there is no “zet”), as evidenced by the round space in the middle of the picture.

Please thoroughly understand the analyzed example, since in it I explained in detail the very essence of the problem.

The following task is for you to solve on your own:

Example 5

A short solution and drawing at the end of the lesson. In general, in the topic under consideration among 2nd order lines the most popular is the circle, but, as an option, they can “push” into the problem ellipse, hyperbole or parabola.

Let's move up:

Example 6

Find the domain of a function

Solution: the radical expression must be non-negative: and the denominator cannot be equal to zero: . Thus, the domain of definition is specified by the system.

We deal with the first condition using the standard scheme discussed in the lesson. Linear inequalities: draw a straight line and determine the half-plane that corresponds to the inequality. Because inequality non-strict, then the straight line itself will also be a solution.

With the second condition of the system, everything is also simple: the equation specifies the ordinate axis, and since , then it should be excluded from the domain of definition.

Let's draw the drawing, not forgetting that the solid line indicates its entry into the definition area, and the dotted line indicates its exclusion from this area:

It should be noted that here we are already forced make a drawing. And this situation is typical - in many tasks, a verbal description of the area is difficult, and even if you describe it, you will most likely be poorly understood and forced to depict the area.

Answer: domain:

By the way, such an answer without a drawing really looks damp.

Let us repeat once again the geometric meaning of the result obtained: in the shaded area there is a graph of the function , which represents surface of three-dimensional space. This surface can be located above/below the plane, or can intersect the plane - in this case, all this is parallel to us. The very fact of the existence of the surface is important, and it is important to correctly find the region in which it exists.

Example 7

Find the domain of a function

This is an example for you to solve on your own. An approximate example of a final task at the end of the lesson.

It’s not uncommon for seemingly simple functions to produce a long-term solution:

Example 8

Find the domain of a function

Solution: using square difference formula, let us factorize the radical expression: .

The product of two factors is non-negative , When both multipliers are non-negative: OR When both non-positive: . This is a typical feature. Thus, we need to solve two systems of linear inequalities And COMBINE received areas. In a similar situation, instead of the standard algorithm, the method of scientific, or more precisely, practical poking works much faster =)

We draw straight lines that divide the coordinate plane into 4 “corners”. We take some point belonging to the upper “corner”, for example, a point and substitute its coordinates into the equations of the 1st system: . The correct inequalities are obtained, which means that the solution to the system is all top "corner". Shading.

Now we take the point belonging to the right “corner”. The 2nd system remains, into which we substitute the coordinates of this point: . The second inequality is not true, therefore and all the right "corner" is not a solution to the system.

A similar story is with the left “corner”, which is also not included in the scope of the definition.

And finally, we substitute the coordinates of the experimental point of the lower “corner” into the 2nd system: . Both inequalities are true, which means that the solution to the system is and all the lower “corner”, which should also be shaded.

In reality, of course, there is no need to describe it in such detail - all the commented actions are easily performed orally!

Answer: the domain of definition is Union system solutions .

As you might guess, such an answer is unlikely to work without a drawing, and this circumstance forces you to pick up a ruler and pencil, even though the condition did not require it.

And this is your nut:

Example 9

Find the domain of a function

A good student always misses logarithms:

Example 10

Find the domain of a function

Solution: the argument of the logarithm is strictly positive, so the domain of definition is given by the system.

The inequality indicates the right half-plane and excludes the axis.

With the second condition the situation is more intricate, but also transparent. Let's remember sinusoid. The argument is “Igrek”, but this should not confuse me – Igrek, so Igrek, Zyu, so Zyu. Where is sine greater than zero? Sine is greater than zero, for example, on the interval. Since the function is periodic, there are infinitely many such intervals and in collapsed form the solution to the inequality will be written as follows:
, where is an arbitrary integer.

An infinite number of intervals, of course, cannot be depicted, so we will limit ourselves to the interval and its neighbors:

Let's complete the drawing, not forgetting that according to the first condition, our field of activity is limited strictly to the right half-plane:

hmm...it turned out to be some kind of ghost drawing...a good representation of higher mathematics...

Answer:

The next logarithm is yours:

Example 11

Find the domain of a function

During the solution, you will have to build parabola, which will divide the plane into 2 parts - the “inside” located between the branches, and the outer part. The method of finding the required part has appeared repeatedly in the article Linear inequalities and previous examples in this lesson.

Solution, drawing and answer at the end of the lesson.

The final nuts of the paragraph are devoted to “arches”:

Example 12

Find the domain of a function

Solution: The arcsine argument must be within the following limits:

Then there are two technical possibilities: more prepared readers, similar to the last examples of the lesson Domain of a function of one variable they can “roll” the double inequality and leave the “Y” in the middle. For dummies, I recommend converting the “locomotive” into an equivalent system of inequalities:

The system is solved as usual - we construct straight lines and find the necessary half-planes. As a result:

Please note that here the boundaries are included in the definition area and straight lines are drawn as solid lines. This must always be carefully monitored to avoid a serious mistake.

Answer: the domain of definition represents the solution of the system

Example 13

Find the domain of a function

The sample solution uses an advanced technique - converting double inequalities.

In practice, we also sometimes encounter problems involving finding the domain of definition of a function of three variables. The domain of definition of a function of three variables can be All three-dimensional space, or part of it. In the first case the function is defined for any points in space, in the second - only for those points that belong to some spatial object, most often - body. It can be a rectangular parallelepiped, ellipsoid, "inside" parabolic cylinder etc. The task of finding the domain of definition of a function of three variables usually consists of finding this body and making a three-dimensional drawing. However, such examples are quite rare. (I only found a couple of pieces), and therefore I will limit myself to just this overview paragraph.

Level lines

To better understand this term, we will compare the axis with height: the higher the “Z” value, the greater the height, the lower the “Z” value, the lower the height. The height can also be negative.

A function in its domain of definition is a spatial graph; for definiteness and greater clarity, we will assume that this is a trivial surface. What are level lines? Figuratively speaking, level lines are horizontal “slices” of the surface at various heights. These “slices” or, more correctly, sections carried out by planes, after which they are projected onto the plane .

Definition: a function level line is a line on the plane at each point of which the function maintains a constant value: .

Thus, level lines help to figure out what a particular surface looks like - and they help without constructing a three-dimensional drawing! Let's consider a specific task:

Example 14

Find and plot several level lines of a function graph

Solution: We examine the shape of a given surface using level lines. For convenience, let’s expand the entry “back to front”:

Obviously, in this case “zet” (height) obviously cannot take negative values (since the sum of squares is non-negative). Thus, the surface is located in the upper half-space (above the plane).

Since the condition does not say at what specific heights the level lines need to be “cut off,” we are free to choose several “Z” values ​​at our discretion.

We examine the surface at zero height, to do this we put the value in the equality :

The solution to this equation is the point. That is, when the level line represents a point.

We rise to a unit height and “cut” our surface plane (substitute into the surface equation):

Thus, for height, the level line is a circle centered at a point of unit radius.

I remind you that all “slices” are projected onto the plane, and that’s why I write down two, not three, coordinates for points!

Now we take, for example, a plane and “cut” the surface under study with it (substituteinto the surface equation):

Thus, for heightthe level line is a circle centered at the radius point.

And, let's build another level line, say for :

–circle centered at a point of radius 3.

The level lines, as I have already emphasized, are located on the plane, but each line is signed - what height it corresponds to:

It is not difficult to understand that other level lines of the surface under consideration are also circles, and the higher we go up (we increase the “Z” value), the larger the radius becomes. Thus, the surface itself It is an endless bowl with an ovoid bottom, the top of which is located on a plane. This “bowl”, together with the axis, “comes right out at you” from the monitor screen, that is, you are looking at its bottom =) And this is not without reason! Only I pour it on the road so lethally =) =)

Answer: the level lines of a given surface are concentric circles of the form

Note : when a degenerate circle of zero radius (point) is obtained

The very concept of a level line comes from cartography. To paraphrase the established mathematical expression, we can say that level line is a geographical location of points of the same height. Consider a certain mountain with level lines of 1000, 3000 and 5000 meters:

The figure clearly shows that the upper left slope of the mountain is much steeper than the lower right slope. Thus, level lines allow you to reflect the terrain on a “flat” map. By the way, here negative altitude values ​​also acquire a very specific meaning - after all, some areas of the Earth’s surface are located below the zero level of the world’s oceans.

(Lecture 1)

Functions of 2 variables.

The variable z is called a function of 2 variables f(x,y), if for any pair of values ​​(x,y) G a certain value of the variable z is associated.

Def. A neighborhood of the point p 0 is a circle with a center at the point p 0 and a radius. = (x-x 0 ) 2 +(oooh 0 ) 2

of an arbitrarily small number, one can specify a number ()>0 such that for all values ​​of x and y, for which the distance from t.p to p0 is less, the following inequality holds: f(x,y) A, i.e. for all points p falling in the vicinity of point p 0, with a radius, the value of the function differs from A by less than in absolute value. And this means that when point p approaches point p 0 by anyone

Continuity of function.

Let the function z=f(x,y) be given, p(x,y) is the current point, p 0 (x 0 ,y 0) is the point under consideration.

Def.

3) The limit is equal to the value of the function at this point: = f(x 0 ,y 0);

Lim f(x,y) = f(x 0 ,y 0 );

pp 0

Partial derivative.

Let's give the argument x an increment of x; x+x, we get point p 1 (x+x,y), calculate the difference between the values ​​of the function at point p:

x z = f(p1)-f(p) = f(x+x,y) - f(x,y) partial increment of the function corresponding to the increment of the argument x.

z= Lim x z

z = Lim f(x+x,y) - f(x,y)

X x0 X

Defining a function of several variables

When considering many issues from various fields of knowledge, it is necessary to study such dependencies between variable quantities when the numerical values ​​of one of them are completely determined by the values ​​of several others.

For example When studying the physical state of a body, one has to observe changes in its properties from point to point. Each point of the body is specified by three coordinates: x, y, z. Therefore, studying, say, the density distribution, we conclude that the density of a body depends on three variables: x, y, z. If the physical state of the body also changes over time t, then the same density will depend on the values ​​of four variables: x, y, z, t.

Another example: the production costs of producing a unit of a certain type of product are studied. Let be:

x - costs of materials,

y - payment costs wages employees,

z - depreciation charges.

It is obvious that production costs depend on the values ​​of the named parameters x, y, z.

Definition 1.1 If for each set of values ​​"n" variables

from some set D of these collections corresponds to its unique value of the variable z, then they say that the function is given on the set D

"n" variables.

The set D specified in Definition 1.1 is called the domain of definition or domain of existence of this function.

If a function of two variables is considered, then the collection of numbers

are denoted, as a rule, (x, y) and are interpreted as points of the Oxy coordinate plane, and the domain of definition of the function z = f (x, y) of two variables is depicted as a certain set of points on the Oxy plane.

So, for example, the domain of definition of the function

is the set of points of the Oxy plane whose coordinates satisfy the relation

i.e., it is a circle of radius r with its center at the origin.

For function

the domain of definition is the points that satisfy the condition

i.e. external with respect to a given circle.

Often functions of two variables are specified implicitly, i.e., as an equation

connecting three variables. In this case, each of the quantities x, y, z can be considered as an implicit function of the other two.

The geometric image (graph) of a function of two variables z = f (x, y) is the set of points P (x, y, z) in three-dimensional space Oxyz, whose coordinates satisfy the equation z = f (x, y).

The graph of a function of continuous arguments, as a rule, is a certain surface in the Oxyz space, which is projected onto the coordinate plane Oxy into the domain of definition of the function z= f (x, y).

So, for example, (Fig. 1.1) the graph of the function

is the upper half of the sphere, and the graph of the function

Bottom half of the sphere.

The graph of the linear function z = ax + by + с is a plane in the Oxyz space, and the graph of the function z = const is a plane parallel to the coordinate plane Oxyz.

Note that it is impossible to visually depict a function of three or more variables in the form of a graph in three-dimensional space.

In what follows, we will mainly limit ourselves to the consideration of functions of two or three variables, since the consideration of the case of a larger (but finite) number of variables is carried out similarly.

Definition of a function of several variables.

(Lecture 1)

The variable u is called f(x,y,z,..,t) if for any set of values ​​(x,y,z,..,t) a well-defined value of the variable u is associated.

The set of collections of the value of a variable is called the domain of definition of a function.

G - set (x,y,z,..,t) - domain of definition.

Functions of 2 variables.

The variable z is called a function of 2 variables f(x,y), if for any pair of values ​​(x,y) О G a certain value of the variable z is associated.

Limit of a function of 2 variables.

Let the function z=f(x,y) be given, p(x,y) is the current point, p 0 (x 0 ,y 0) is the point under consideration.

Def. A neighborhood of the point p 0 is a circle with a center at the point p 0 and radius r. r= Ö (x-x 0 ) 2 +(oooh 0 ) 2 Ø

The number A is called the limit of the function | at the point p 0 if for any

for an arbitrarily small number e, one can specify a number r (e)>0 such that for all values ​​of x and y, for which the distance from t. p to p0 is less than r, the following inequality holds: ½f(x,y) - A½0, with radius r , the value of the function differs from A by less than e in absolute value. And this means that when point p approaches point p 0 by anyone path, the value of the function indefinitely approaches the number A.

Continuity of function.

Let the function z=f(x,y) be given, p(x,y) is the current point, p 0 (x 0 ,y 0) is the point under consideration.

Def. The function z=f(x,y) is called continuous at t. p 0 if 3 conditions are met:

1) the function is defined at this point. f(p 0) = f(x,y);

2)f-i has a limit at this point.

3) The limit is equal to the value of the function at this point: b = f(x 0 ,y 0);

Lim f(x,y)= f(x 0 ,y 0 ) ;

pà p 0

If at least 1 of the continuity conditions is violated, then point p is called a break point. For functions of 2 variables, there can be separate break points and entire break lines.

The concept of limit and continuity for functions of a larger number of variables is defined similarly.

A function of three variables cannot be depicted graphically, unlike a function of 2 variables.

For a 3-variable function, there can be discontinuity points, discontinuity lines, and discontinuity surfaces.

Partial derivative.

Let's consider the function z=f(x,y), p(x,y) is the point under consideration.

Let's give the argument x the increment Dx; x+Dx, we get point p 1 (x+Dx,y), calculate the difference between the values ​​of the function at point p:

D x z = f(p1)-f(p) = f(x+Dx,y) - f(x,y) - partial increment of the function corresponding to the increment of the argument x.

Def. The quotient of the derivative of a function z=f(x,y) with respect to the variable x is called the limit of the ratio of the partial increment of this function with respect to the variable x to this increment when the latter tends to zero.

¶ z= Lim D x z

à ¶ z = Lim f(x+ D x,y) - f(x,y)

¶ x Dx® 0 Dx

Similarly, we determine the quotient of the derivative with respect to the variable y.

Finding partial derivatives.

When determining partial derivatives, only one variable changes each time; the remaining variables are treated as constants. As a result, each time we consider a function of only one variable and the partial derivative coincides with the usual derivative of this function of one variable. Hence the rule for finding partial derivatives: the partial derivative with respect to the variable under consideration is sought as the ordinary derivative of a function of this one variable, the remaining variables are treated as constants. In this case, all formulas for differentiating a function of one variable (derivative of a sum, product, quotient) turn out to be valid.

Concept of a function of several variables

If each point X = (x 1, x 2, ... x n) from the set (X) of points of n-dimensional space is associated with one well-defined value of the variable z, then they say that the given function of n variables z = f(x 1, x 2, ...x n) = f (X).

In this case, the variables x 1, x 2, ... x n are called independent variables or arguments functions, z - dependent variable, and the symbol f denotes law of correspondence. The set (X) is called domain of definition functions (this is a certain subset of n-dimensional space).

For example, the function z = 1/(x 1 x 2) is a function of two variables. Its arguments are the variables x 1 and x 2, and z is the dependent variable. The domain of definition is the entire coordinate plane, with the exception of the straight lines x 1 = 0 and x 2 = 0, i.e. without x- and ordinate-axes. Substituting any point from the domain of definition into the function, according to the correspondence law we obtain a certain number. For example, taking the point (2; 5), i.e. x 1 = 2, x 2 = 5, we get
z = 1/(2*5) = 0.1 (i.e. z(2; 5) = 0.1).

A function of the form z = a 1 x 1 + a 2 x 2 + ... + a n x n + b, where a 1, a 2,..., and n, b are constant numbers, is called linear. It can be considered as the sum of n linear functions of the variables x 1, x 2, ... x n. All other functions are called nonlinear.

For example, the function z = 1/(x 1 x 2) is nonlinear, and the function z =
= x 1 + 7x 2 - 5 – linear.

Any function z = f (X) = f(x 1, x 2, ... x n) can be associated with n functions of one variable if we fix the values ​​of all variables except one.

For example, functions of three variables z = 1/(x 1 x 2 x 3) can be associated with three functions of one variable. If we fix x 2 = a and x 3 = b, then the function will take the form z = 1/(abx 1); if we fix x 1 = a and x 3 = b, then it will take the form z = 1/(abx 2); if we fix x 1 = a and x 2 = b, then it will take the form z = 1/(abx 3). In this case, all three functions have the same form. It is not always so. For example, if for a function of two variables we fix x 2 = a, then it will take the form z = 5x 1 a, i.e. power function, and if we fix x 1 = a, then it will take the form, i.e. exponential function.

Schedule function of two variables z = f(x, y) is the set of points in three-dimensional space (x, y, z), the applicate z of which is related to the abscissa x and ordinate y by a functional relation
z = f (x, y). This graph represents some surface in three-dimensional space (for example, as in Figure 5.3).

It can be proven that if a function is linear (i.e. z = ax + by + c), then its graph is a plane in three-dimensional space. Other examples 3D graphs It is recommended to study independently using Kremer's textbook (pp. 405-406).

If there are more than two variables (n variables), then schedule function is a set of points in (n+1)-dimensional space for which the x coordinate n+1 is calculated in accordance with a given functional law. Such a graph is called hypersurface(for a linear function – hyperplane), and it also represents a scientific abstraction (it is impossible to depict it).

Figure 5.3 – Graph of a function of two variables in three-dimensional space

Level surface a function of n variables is a set of points in n-dimensional space such that at all these points the value of the function is the same and equal to C. The number C itself in this case is called level.

Usually, for the same function, it is possible to construct an infinite number of level surfaces (corresponding to different levels).

For a function of two variables, the level surface takes the form level lines.

For example, consider z = 1/(x 1 x 2). Let's take C = 10, i.e. 1/(x 1 x 2) = 10. Then x 2 = 1/(10x 1), i.e. on the plane the level line will take the form shown in Figure 5.4 as a solid line. Taking another level, for example, C = 5, we obtain the level line in the form of a graph of the function x 2 = 1/(5x 1) (shown with a dotted line in Figure 5.4).

Figure 5.4 - Function level lines z = 1/(x 1 x 2)

Let's look at another example. Let z = 2x 1 + x 2. Let's take C = 2, i.e. 2x 1 + x 2 = 2. Then x 2 = 2 - 2x 1, i.e. on the plane the level line will take the form of a straight line, represented in Figure 5.5 by a solid line. Taking another level, for example, C = 4, we obtain a level line in the form of a straight line x 2 = 4 - 2x 1 (shown with a dotted line in Figure 5.5). The level line for 2x 1 + x 2 = 3 is shown in Figure 5.5 as a dotted line.

It is easy to verify that for a linear function of two variables, any level line will be a straight line on the plane, and all level lines will be parallel to each other.

Figure 5.5 - Function level lines z = 2x 1 + x 2

Functions of many variables

§1. The concept of a function of many variables.

Let there be n variable quantities. Each set
denotes a point n- dimensional set
(P-dimensional vector).

Let given sets
And
.

ODA. If each point
matches the singular number
, then we say that a numerical function is given n variables:

.

is called the domain of definition,
- a set of values ​​of a given function.

When n=2 instead
usually write x, y, z. Then the function of two variables has the form:

z= f(x, y).

For example,
- function of two variables;

- function of three variables;

Linear function n variables.

ODA. Function graph n variables are called n- dimensional hypersurface in space
, each point of which is specified by coordinates

For example, a graph of a function of two variables z= f(x, y) is a surface in three-dimensional space, each point of which is specified by coordinates ( x, y, z) , Where
, And
.

Since it is not possible to depict a graph of a function of three or more variables, we will mainly (for clarity) consider functions of two variables.

Plotting functions of two variables is quite a difficult task. The construction of so-called level lines can provide significant assistance in solving this problem.

ODA. Level line of a function of two variables z= f(x, y) is called the set of points on the plane HOU, which are the projection of the section of the graph of the function by a plane parallel HOU. At each point on the level line the function has the same value. Level lines are described by the equation f(x, y)=c, Where With– a certain number. There are infinitely many level lines, and one of them can be drawn through each point of the definition domain.

ODA. Surface level function n variables y= f (
) is called a hypersurface in space
, at each point of which the value of the function is constant and equal to a certain value With. Level surface equation: f (
)=s.

Example. Graph a function of two variables

.

.

When c=1:
;
.

With c=4:
;
.

At c=9:
;
.

Level lines are concentric circles, the radius of which decreases with increasing z.

§2. Limit and continuity of a function of several variables.

For functions of many variables, the same concepts are defined as for functions of one variable. For example, one can give definitions of the limit and continuity of a function.

ODA. z= f(x, y) The number A is called the limit of a function of two variables
,
at
and is designated , if for any positive number there is a positive number
, such that if the point
away from the point less distance f(x, y) , then the quantities .

ODA and A differ by less than z= f(x, y) . If the function
defined at point
and has a limit at this point equal to the value of the function

.

, then it is called continuous at a given point.

§3. Partial derivatives of functions of several variables.
.

Consider a function of two variables Let's fix the value of one of its arguments, for example
, putting
. Then the function there is a function of one variable :

.

. Let it have a derivative at the point
This derivative is called the partial derivative (or first order partial derivative) of the function By
at the point
;
;
;
.

and is designated: at
:

The difference is called the partial increment

.

Taking into account the above notations, we can write

.

Defined similarly Partial derivative

functions of several variables in one of these variables is called the limit of the ratio of the partial increment of a function to the increment of the corresponding independent variable, when this increment tends to zero.

When finding the partial derivative with respect to any argument, the other arguments are considered constant. All rules and formulas for differentiating functions of one variable are valid for partial derivatives of functions of many variables. Note that the partial derivatives of a function are functions of the same variables. These functions, in turn, can have partial derivatives, which are called second partial derivatives

(or second order partial derivatives) of the original function.
For example, the function

;
;

;
.

And
has four second-order partial derivatives, which are denoted as follows:

- mixed partial derivatives. Example.

.

Find second order partial derivatives for a function
,
.

,
.

,
.

Solution..

1. Find second-order partial derivatives for functions

,
;

2. For function
prove that
.

Full differential functions of many variables.

With simultaneous changes in values X And at function
will change by an amount called the total increment of the function z at the point
. Just as in the case of a function of one variable, the problem arises of approximate replacement of the increment
to a linear function of
And
. The role of linear approximation is performed by full differential Features:

Second order total differential:

=
.

=
.

IN general view full differential P-th order has the form:

Directional derivative. Gradient.

Let the function z= f(x, y) is defined in some neighborhood of the point M( x, y) And - some direction specified by the unit vector
. The coordinates of a unit vector are expressed through the cosines of the angles formed by the vector and the coordinate axes and called direction cosines:

,

.

When moving point M( x, y) in this direction l exactly
function z will receive an increment

called the increment of the function in a given direction l.

If MM 1 =∆ l, That

T

ABOUT

The directional derivative characterizes the rate of change of a function in a given direction. It is obvious that partial derivatives And represent derivatives in directions parallel to the axes Ox And Oy.

Example It is easy to show that
. Calculate the derivative of a function
.

ODA. at point (1;1) in the direction functions z= f(x, y) Gradient

.

is a vector with coordinates equal to partial derivatives:
And
:

Consider the scalar product of vectors
It's easy to see that .

, i.e. the directional derivative is equal to the scalar product of the gradient and the unit direction vector
Because the

, then the scalar product is maximum when the vectors have the same directions. Thus, the gradient of a function at a point specifies the direction of the fastest increase in the function at this point, and the modulus of the gradient is equal to the maximum growth rate of the function.

Knowing the gradient of a function, one can locally construct function level lines. Theorem z= f(x, y) . Let a differentiable function be given
and at the point
the gradient of the function is not zero:

. Then the gradient is perpendicular to the level line passing through the given point.

Thus, if, starting from a certain point, we construct the gradient of the function and a small part of the level line perpendicular to it at nearby points, then we can (with some error) construct level lines.

Let the function
Local extremum of a function of two variables
.

ODA defined and continuous in some neighborhood of the point
is called the local maximum point of the function
, if there is such a neighborhood of the point , in which for any point
inequality holds:

.

The concept of a local minimum is introduced similarly.

Theorem (necessary condition for local extremum).

In order for a differentiable function
had a local extremum at the point
, it is necessary that all its first-order partial derivatives at this point be equal to zero:

So, the points of possible presence of an extremum are those points at which the function is differentiable and its gradient is equal to 0:
. As in the case of a function of one variable, such points are called stationary.

Definition. Variable z(with change area Z) called function of two independent variables x,y in abundance M, if each pair ( x,y) from many M z from Z.

Definition. A bunch of M, in which the variables are specified x,y, called domain of the function, set Z – function range, and themselves x,y- her arguments.

Designations: z = f(x,y), z = z(x,y).

Examples.

Definition . Variable z(with change area Z) called function of several independent variables in abundance M, if each set of numbers from the set M according to some rule or law, one specific value is assigned z from Z. The concepts of arguments, domain of definition, and domain of value are introduced in the same way as for a function of two variables.

Designations: z = f, z = z.

Comment. Since a couple of numbers ( x,y) can be considered the coordinates of a certain point on the plane, we will subsequently use the term “point” for a pair of arguments to a function of two variables, as well as for an ordered set of numbers that are arguments to a function of several variables.

Geometric representation of a function of two variables

Consider the function

z = f(x,y), (15.1)

defined in some area M on the O plane xy. Then the set of points in three-dimensional space with coordinates ( x,y,z), where , is the graph of a function of two variables. Since equation (15.1) defines a certain surface in three-dimensional space, it will be the geometric image of the function under consideration.

Function Domain z = f(x,y) in the simplest cases, it is either a part of the plane bounded by a closed curve, and the points of this curve (the boundaries of the region) may or may not belong to the domain of definition, or the entire plane, or, finally, a set of several parts of the xOy plane.

z = f(x,y)

Examples include the equations of the plane z = ax + by + c

and second order surfaces: z = x² + y² (paraboloid of revolution),

(cone), etc.

Comment. For a function of three or more variables we will use the term “surface in n-dimensional space,” although it is impossible to depict such a surface.

Level lines and surfaces

For a function of two variables given by equation (15.1), we can consider a set of points ( x,y) O plane xy, for which z takes on the same constant value, that is z= const. These points form a line on the plane called level line.

- mixed partial derivatives.

Find the level lines for the surface z = 4 – x² - y². Their equations look like x² + y² = 4 – c(c=const) – equations of concentric circles with a center at the origin and with radii . For example, when With=0 we get a circle x² + y² = 4 .

For a function of three variables u = u (x, y, z) the equation u(x, y, z) = c defines a surface in three-dimensional space, which is called level surface.

- mixed partial derivatives.

For function u = 3x + 5y – 7z–12 level surfaces will be a family of parallel planes given by equations 3 x + 5y – 7z –12 + With = 0.

Limit and continuity of a function of several variables

Let's introduce the concept δ-neighborhoods points M 0 (x 0, y 0) on the O plane xy as a circle of radius δ with center at a given point. Similarly, we can define a δ-neighborhood in three-dimensional space as a ball of radius δ centered at the point M 0 (x 0, y 0, z 0). For n-dimensional space we will call the δ-neighborhood of a point M 0 set of points M with coordinates satisfying the condition

where are the coordinates of the point M 0 . Sometimes this set is called a “ball” in n-dimensional space.

Definition. The number A is called limit functions of several variables f at the point M 0 if such that | f(M) – A| < ε для любой точки M from δ-neighborhood M 0 .

Designations: .

It must be taken into account that in this case the point M may be approaching M 0, relatively speaking, along any trajectory inside the δ-neighborhood of the point M 0 . Therefore, it is necessary to distinguish the limit of a function of several variables in in a general sense from the so-called repeated limits obtained by successive passages to the limit for each argument separately.

Examples.

Comment. It can be proven that from the existence of a limit at a given point in the usual sense and the existence at this point of limits on individual arguments, the existence and equality of repeated limits follows. The reverse statement is not true.

Definition Function f called continuous at the point M 0 if (15.2)

If we introduce the notation , then condition (15.2) can be rewritten in the form (15.3)

Definition . Inner point M 0 function domain z = f(M) called break point function if conditions (15.2), (15.3) are not satisfied at this point.

Comment. Many discontinuity points can form on a plane or in space lines or fracture surface.

Examples.

Properties of limits and continuous functions

Since the definitions of limit and continuity for a function of several variables practically coincide with the corresponding definitions for a function of one variable, then for functions of several variables all the properties of limits and continuous functions proven in the first part of the course are preserved, namely:

1) If they exist, then they exist and (if).

2) If a and for any i there are limits and there is where M 0, then there is a limit of a complex function at , where are the coordinates of the point R 0 .

3) If the functions f(M) And g(M) continuous at a point M 0, then at this point the functions are also continuous f(M) + g(M), kf(M), f(M) g(M), f(M)/g(M)(If g(M 0) ≠ 0).

4) If the functions are continuous at the point P 0, and the function is continuous at the point M 0, where , then the complex function is continuous at the point R 0 .

5) The function is continuous in a closed limited area D, takes its largest and smallest values ​​in this region.

6) If the function is continuous in a closed limited area D, takes values ​​in this region A And IN, then she takes in the area D and any intermediate value lying between A And IN.

7) If the function is continuous in a closed limited area D, takes values ​​of different signs in this region, then there is at least one point from the region D, wherein f = 0.

Partial derivatives

Let's consider changing a function when specifying an increment to only one of its arguments - x i, and let's call it .

Definition . Partial derivative functions by argument x i called .

Designations: .

Thus, the partial derivative of a function of several variables is actually defined as the derivative of the function one variable – x i. Therefore, all the properties of derivatives proven for a function of one variable are valid for it.

Comment. In the practical calculation of partial derivatives, we use the usual rules for differentiating a function of one variable, assuming that the argument by which differentiation is carried out is variable, and the remaining arguments are constant.

Examples .

1. z = 2x² + 3 xy –12y² + 5 x – 4y +2,

2. z = xy,

Geometric interpretation of partial derivatives of a function of two variables

Consider the surface equation z = f(x,y) and draw a plane x = const. Select a point on the line of intersection of the plane and the surface M(x,y). If you give the argument at increment Δ at and consider point T on the curve with coordinates ( x, y+Δ y, z+Δy z), then the tangent of the angle formed by the secant MT with the positive direction of the O axis at, will be equal to . Passing to the limit at , we find that the partial derivative is equal to the tangent of the angle formed by the tangent to the resulting curve at the point M with positive direction of the O axis u. Accordingly, the partial derivative is equal to the tangent of the angle with the O axis X tangent to the curve obtained as a result of sectioning the surface z = f(x,y) plane y= const.

Differentiability of a function of several variables

When studying issues related to differentiability, we will limit ourselves to the case of a function of three variables, since all proofs for a larger number of variables are carried out in the same way.

Definition . Full increment functions u = f(x, y, z) called

Theorem 1. If partial derivatives exist at the point ( x 0, y 0, z 0) and in some of its neighborhoods and are continuous at the point ( x 0 , y 0 , z 0) then are limited (since their modules do not exceed 1).

Then the increment of a function that satisfies the conditions of Theorem 1 can be represented as: , (15.6)

Definition . If the function increment u = f (x, y, z) at point ( x 0 , y 0 , z 0) can be represented in the form (15.6), (15.7), then the function is called differentiable at this point, and the expression is main linear part of the increment or full differential the function in question.

Designations: du, df (x 0 , y 0 , z 0).

Just as in the case of a function of one variable, the differentials of independent variables are considered to be their arbitrary increments, therefore

Note 1. So, the statement “the function is differentiable” is not equivalent to the statement “the function has partial derivatives” - for differentiability, the continuity of these derivatives at the point in question is also required.

Consider the function and choose x 0 = 1, y 0 = 2. Then Δ x = 1.02 – 1 = 0.02; Δ y = 1.97 – 2 = -0.03. Let's find

Therefore, given that f ( 1, 2) = 3, we get.

When considering functions of one variable, we pointed out that when studying many phenomena one has to encounter functions of two or more independent variables. Let's give a few examples.

Example 1. The area S of a rectangle with sides whose lengths are equal to x and y is expressed by the formula Each pair of values ​​x and y corresponds to a certain value of area S; S is a function of two variables.

Example 2. The volume V of a rectangular parallelepiped with edges whose lengths are equal to x is expressed by the formula. Here V is a function of three variables x.

Example 3. Range R of a projectile fired with an initial speed . From a gun whose barrel is inclined to the horizon at an angle , expressed by the formula if air resistance is neglected). Here is the acceleration due to gravity. For each pair of values, this formula gives a specific value of R, i.e. R is a function of two variables

Example 4. and Here is a function of four variables

Definition 1. If each pair of values ​​of two independent variables x and y from a certain region of their variation D corresponds to a certain value of the quantity , then we say that there is a function of two independent variables x and y defined in the region

Symbolically, a function of two variables is denoted as follows:

A function of two variables can be specified, for example, using a table or analytically - using a formula, as was done in the four examples discussed above. Based on the formula, you can create a table of function values ​​for some pairs of values ​​of independent variables. Yes, for

For the first example, you can create the following table:

In this table, at the intersection of a row and a column corresponding to certain values ​​of x and y, the corresponding function value is indicated

If the functional dependence is obtained as a result of measuring the value of z during the experimental study of a phenomenon, then a table is immediately obtained that defines z as a function of two variables. In this case, the function is specified only by the table.

As in the case of one independent variable, a function of two variables does not, generally speaking, exist for any values ​​of x and y.

Definition 2. The set of pairs of values ​​for which a function is defined is called the domain of definition or the domain of existence of this function.

The domain of definition of a function is clearly illustrated geometrically. If we depict each pair of values ​​x and y as a point in the plane, then the domain of definition of the function will be depicted as a certain collection of points on the plane. We will also call this collection of points the domain of definition of the function. In particular, the domain of definition can be the entire plane. In what follows we will mainly deal with such areas, which are parts of the plane bounded by lines. The line limiting this area will be called the boundary of the area. Points of the region that do not lie on the boundary will be called internal points of the region. An area consisting of only internal points is called open or unclosed. If the boundary points also belong to the region, then the region is called closed. A region is called bounded if there is such a constant C that the distance of any point M of the region from the origin of coordinates O is less than C, i.e.

Example 5: Determine the natural domain of a function

The analytical expression makes sense for any values ​​of x and y. Consequently, the natural domain of definition of the function is the entire plane

Example 6. .

In order for it to have a real value, the root must have a non-negative number, that is, x and y must satisfy the inequality or

All points whose coordinates satisfy the specified inequality lie in a circle of radius 1 with a center at the origin and on the boundary of this circle.

Example 7. .

Since logarithms are defined only for positive numbers, the inequality or must be satisfied.

This means that the domain of definition of the function is half of the plane located above the straight line, not including the straight line itself (Fig. 166).

Example 8: The area of ​​triangle 5 is a function of the base and height

The domain of definition of this function is the area like the base of the triangle and its height can be neither negative nor zero). Note that the domain of definition of the function under consideration does not coincide with the natural domain of definition of the analytical expression with which the function is specified, since the natural domain of definition of the expression is, obviously, the entire Oxy plane.