Functions of several variables briefly theory. A function of two or more variables. Its domain of definition. Geometric representation of a function of two variables
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. Many 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.
Example.
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) equation u(x, y, z) = c defines a surface in three-dimensional space, which is called level surface.
Example.
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, one should distinguish the limit of a function of several variables in the 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, in which 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, 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.
) we have already repeatedly encountered partial derivatives of complex functions like and more difficult examples. So what else can you talk about?! ...And everything is like in life - there is no complexity that cannot be complicated =) But mathematics is what mathematics is for, to fit the diversity of our world into a strict framework. And sometimes this can be done with one single sentence:
In general, the complex function has the form 
, Where, at least one of letters represents function, which may depend on arbitrary number of variables.
The minimum and simplest option is the long-familiar complex function of one variable, whose derivative we learned how to find last semester. You also have the skills to differentiate functions (take a look at the same functions 
)
.
Thus, now we will be interested in just the case. Due to the great variety of complex functions, the general formulas for their derivatives are very cumbersome and difficult to digest. In this regard, I will limit myself to specific examples from which you can understand the general principle of finding these derivatives:
Example 1
Given a complex function where 
. Required:
1) find its derivative and write down the 1st order total differential;
2) calculate the value of the derivative at .
Solution: First, let's look at the function itself. We are offered a function depending on and , which in turn are functions one variable: 
Secondly, let’s pay close attention to the task itself - we are required to find derivative, that is, we are not talking about partial derivatives, which we are used to finding! Since the function 
actually depends on only one variable, then the word “derivative” means total derivative. How to find her?
The first thing that comes to mind is direct substitution and further differentiation. Let's substitute 
to function: 
, after which there are no problems with the desired derivative:
And, accordingly, the total differential: 
This solution is mathematically correct, but a small nuance is that when the problem is formulated the way it is formulated, no one expects such barbarism from you =) But seriously, you can really find fault here. Imagine that a function describes the flight of a bumblebee, and the nested functions change depending on the temperature. Performing a direct substitution 
, we only get private information, which characterizes flight, say, only in hot weather. Moreover, if a person who is not knowledgeable about bumblebees is presented with the finished result and even told what this function is, then he will never learn anything about the fundamental law of flight!
So, completely unexpectedly, our buzzing brother helped us understand the meaning and importance of the universal formula: 
Get used to the “two-story” notation for derivatives - in the task under consideration, they are the ones in use. In this case, one should be very neat in the entry: derivatives with direct symbols “de” are complete derivatives, and derivatives with rounded icons are partial derivatives. Let's start with the last ones: 
Well, with the “tails” everything is generally elementary:
Let's substitute the found derivatives into our formula:
When a function is initially proposed in an intricate way, it will be logical (and this is explained above!) leave the results as they are: 
At the same time, in “sophisticated” answers it is better to refrain from even minimal simplifications (here, for example, it begs to be removed 3 minuses)- and you have less work, and your furry friend is happy to review the task easier.
However, a rough check will not be superfluous. Let's substitute 
into the found derivative and carry out simplifications: 
(at the last step we used trigonometric formulas 
, 
)
As a result, the same result was obtained as with the “barbaric” solution method.
Let's calculate the derivative at the point. First it is convenient to find out the “transit” values (function values 
)
:
Now we draw up the final calculations, which in this case can be performed in different ways. I use an interesting technique in which the 3rd and 4th “floors” are simplified not according to the usual rules, but are transformed as the quotient of two numbers:
And, of course, it’s a sin not to check using a more compact notation 
:
Answer: 
It happens that the problem is proposed in a “semi-general” form:
"Find the derivative of the function where 
»
That is, the “main” function is not given, but its “inserts” are quite specific. The answer should be given in the same style:
Moreover, the condition can be slightly encrypted:
"Find the derivative of the function 
»
In this case you need on one's own designate nested functions with some suitable letters, for example, through 
and use the same formula:
By the way, about letter designations. I have repeatedly urged not to “cling to letters” as a life preserver, and now this is especially relevant! Analyzing various sources on the topic, I generally got the impression that the authors “went crazy” and began to mercilessly throw students into the stormy abyss of mathematics =) So forgive me :))
Example 2
Find the derivative of a function 
, If 
Other designations should not be confusing! Every time you encounter a task like this, you need to answer two simple questions:
1) What does the “main” function depend on? In this case, the function “zet” depends on two functions (“y” and “ve”).
2) What variables do nested functions depend on? In this case, both “inserts” depend only on the “X”.
So you shouldn't have any difficulty adapting the formula to this task!
A short solution and answer at the end of the lesson.
Additional examples of the first type can be found in Ryabushko's problem book (IDZ 10.1), well, we are heading for function of three variables:
Example 3
Given a function where .
Calculate derivative at point
The formula for the derivative of a complex function, as many guess, has a related form: 
Decide once you guessed it =)
Just in case, I will give a general formula for the function:
, although in practice you are unlikely to see anything longer than Example 3.
In addition, sometimes it is necessary to differentiate a “truncated” version - as a rule, a function of the form or. I leave this question for you to study on your own - come up with some simple examples, think, experiment and derive shortened formulas for derivatives.
If anything is still unclear, please slowly re-read and comprehend the first part of the lesson, because now the task will become more complicated:
Example 4
Find the partial derivatives of a complex function, where 
Solution: this function has the form , and after direct substitution and we get the usual function of two variables: 
But such fear is not only not accepted, but one no longer wants to differentiate =) Therefore, we will use ready-made formulas. To help you quickly grasp the pattern, I will make some notes: 
Look carefully at the picture from top to bottom and left to right….
First, let's find the partial derivatives of the “main” function:
Now we find the “X” derivatives of the “liners”:
and write down the final “X” derivative:
Similarly with the “game”:
And
You can stick to another style - find all the “tails” at once 
and then write down both derivatives.
Answer:
About substitution 
somehow I don’t think anything at all =) =), but you can tweak the results a little. Although, again, why? – only make it more difficult for the teacher to check.
If necessary, then full differential here it is written according to the usual formula, and, by the way, it is at this step that light cosmetics become appropriate: 
This is... ...a coffin on wheels.
Due to the popularity of the type of complex function under consideration, there are a couple of tasks for independent solution. A simpler example in a “semi-general” form is for understanding the formula itself;-):
Example 5
Find the partial derivatives of the function, where 
And more complicated - with the inclusion of differentiation techniques:
Example 6
Find the complete differential of a function 
, Where 
No, I’m not trying to “send you to the bottom” at all - all the examples are taken from real works, and “on the high seas” you can come across any letters. In any case, you will need to analyze the function (answering 2 questions – see above), present it in general form and carefully modify the partial derivative formulas. You may be a little confused now, but you will understand the very principle of their construction! Because the real challenges are just beginning :)))
Example 7
Find partial derivatives and create the complete differential of a complex function
, Where
Solution: the “main” function has the form and still depends on two variables – “x” and “y”. But compared to Example 4, another nested function has been added, and therefore the partial derivative formulas are also lengthened. As in that example, for a better visualization of the pattern, I will highlight the “main” partial derivatives in different colors: 
And again, carefully study the record from top to bottom and from left to right.
Since the problem is formulated in a “semi-general” form, all our work is essentially limited to finding partial derivatives of embedded functions: 
A first grader can handle: 
And even the full differential turned out quite nice:
I deliberately did not offer you any specific function - so that unnecessary clutter would not interfere with a good understanding of the concept of the task.
Answer: 
Quite often you can find “mixed-sized” investments, for example:
Here the “main” function, although it has the form , still depends on both “x” and “y”. Therefore, the same formulas work - just some partial derivatives will be equal to zero. Moreover, this is also true for functions like 
, in which each “liner” depends on one variable.
A similar situation occurs in the final two examples of the lesson:
Example 8
Find the total differential of a complex function at a point
Solution: the condition is formulated in a “budgetary” way, and we must label the nested functions ourselves. I think this is a good option:
The “inserts” contain ( ATTENTION!) THREE letters are the good old “X-Y-Z”, which means that the “main” function actually depends on three variables. It can be formally rewritten as , and the partial derivatives in this case are determined by the following formulas: 
We scan, we delve into, we capture….
In our task: 
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 is designated as follows:
Either , or another standard letter:
Since the ordered pair of values "x" and "y" determines point on the 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 of definition.
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 to 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 not included 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
The graphic representation here is 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– without him it will be very difficult!
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 over 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, then most likely you will be poorly understood and will be forced to depict the area.
Answer: scope of definition:
By the way, such an answer without a drawing really looks damp.
Let us once again repeat 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 rather, 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 association 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.
When studying many patterns in natural science and economics, one encounters functions of two (or more) independent variables.
Definition (for a function of two variables).Let X , Y And Z - multitudes. If each couple (x, y) elements from sets respectively X And Y by virtue of some law f matches one and only one element z from many Z , then they say that a function of two variables is given z = f(x, y) .
In general domain of a function of two variables geometrically can be represented by a certain set of points ( x; y) plane xOy .
The basic definitions relating to functions of several variables are a generalization of the corresponding definitions for a function of one variable .
Many D called domain of the function z, and the set E – its many meanings. Variables x And y in relation to function z are called its arguments. Variable z called the dependent variable.
Private values of arguments
corresponds to the private value of the function
Domain of a function of several variables
If function of several variables (for example, two variables) given by the formula z = f(x, y) , That area of its definition is the set of all such points of the plane x0y, for which the expression f(x, y) makes sense and accepts real values. The general rules for the domain of a function of several variables are derived from the general rules for domain of definition of a function of one variable. The difference is that for a function of two variables, the domain of definition is a certain set of points on the plane, and not a straight line, as for a function of one variable. For a function of three variables, the domain of definition is the corresponding set of points in three-dimensional space, and for a function n variables - the corresponding set of points of the abstract n-dimensional space.
Domain of a function of two variables with a root n th degree
In the case where a function of two variables is given by the formula and n - natural number :
If n is an even number, then the domain of definition of the function is the set of points of the plane corresponding to all values of the radical expression that are greater than or equal to zero, that is
If n is an odd number, then the domain of definition of the function is the set of any values, that is, the entire plane x0y .
Domain of a power function of two variables with an integer exponent
:
If a- positive, then the domain of definition of the function is the entire plane x0y ;
If a- negative, then the domain of definition of the function is the set of values different from zero: .
Domain of a power function of two variables with a fractional exponent
In the case when the function is given by the formula 
:
if is positive, then the domain of definition of the function is the set of those points in the plane at which it takes values greater than or equal to zero: ;
if - is negative, then the domain of definition of the function is the set of those points in the plane at which it takes values greater than zero: .
Domain of definition of a logarithmic function of two variables
Logarithmic function of two variables 
is defined provided that its argument is positive, that is, the domain of its definition is the set of those points in the plane at which it takes values greater than zero: .
Domain of definition of trigonometric functions of two variables
Function Domain 
- the whole plane x0y
.
Function Domain 
- the whole plane x0y
.
The domain of definition of the function is the entire plane x0y
Function Domain 
- the whole plane x0y, except for pairs of numbers for which takes values .
Domain of definition of inverse trigonometric functions of two variables
Function Domain 
.
Function Domain 
- the set of points on the plane for which .
Function Domain 
- the whole plane x0y
.
Function Domain 
- the whole plane x0y
.
The domain of definition of a fraction as a function of two variables
If a function is given by the formula, then the domain of definition of the function is all points of the plane in which .
Domain of a linear function of two variables
If the function is given by a formula of the form z = ax + by + c , then the domain of definition of the function is the entire plane x0y .
Example 1.
Solution. According to the rules for the domain of definition, we compose a double inequality
We multiply the entire inequality by and get
The resulting expression specifies the domain of definition of this function of two variables.
Example 2. Find the domain of a function of two variables.
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Lectures 1-4
FUNCTIONS OF SEVERAL VARIABLES.
Test questions.
Partial and total increment of a function of several variables (FNP).
Limit of a function of several variables. Properties of FNP limits.
Continuity of the FNP. Properties of continuous functions.
First order partial derivatives.
Definition : if each considered set of variable values corresponds to a certain variable valuew, then we'll callw function of independent variables:
(1)
Definition
: domain of definitionD
(
f
)
function (1) is a collection of such sets of numbers 
, for which function (1) is defined.
Region D ( f ) can be open or closed. For example for a function:
D (f ) there will be all points in space for which the inequality holds (closed ball), and for the function (open ball).
In what follows, we will mainly consider functions of two variables, because firstly, there is no fundamental difference between two or more variables; increasing the number of variables only leads to cumbersome calculations. Secondly, the case of two variables allows for a clear geometric interpretation.
Geometric representation of a function of two variables 
is some surface that can be specified explicitly or implicitly. For example: a ) 
— explicit task (paraboloid of rotation), b) 
— implicit task (sphere).
When constructing a graph, functions are often usedby section method .
Example . Construct a graph of the function. 
Let's use the section method.
– in the plane 
- parabola.
– in the plane 
-parabola.
– in the plane 
– circle.
The required surface is a paraboloid of revolution.
Distance
between two arbitrary points 
And 
(Euclidean) spaces 
called number
The set of points is calledopen circle
radius 
centered at a point 
, –
circumference
radius with center at point .
Open radius circle 
with center at a point is called-surroundings
dots
ABOUT 
determination. The point is calledinternal point
sets 
, if there is a -neighborhood 
point, entirely belonging to the set (i.e. 
).
Definition . The point is calledboundary point of a set if any of its -neighborhoods contains points both belonging to the set and not belonging to it.
The boundary point of a set may or may not belong to this set.
Definition . The set is calledopen , if all its points are internal.
Definition . The set is calledclosed
, if it contains all its boundary points. The set of all boundary points of a set is called itsborder
(and is often indicated by the symbol 
). Note that the set 
is closed and is calledclosure of the set.
Example . If, then. At the same time.
Partial and total increment of a function.
If one independent variable (for example,X ) is incremented X , and the other variable does not change, then the function is incremented:
which is called the partial increment of a function by argumentX .
If all variables are incremented, then the function receives a full increment:
For example, for the function 
we will have:
Limit of a function of several variables.
Definition . We will say that the sequence of points 
converges
at 
to the point 
, if at .
In this case, the point 
calledlimit
the specified sequence and write: 
at 
.
It is easy to show that if and only if both 
, 
(i.e. the convergence of a sequence of points in space is equivalentcoordinate-wise convergence
).
Definition . The number is called limit
functions 
at 
, if for 
such that 
, as soon as.
In this case they write 
or 
at 
.
Despite the apparent complete analogy of the concepts of the limit of functions of one and two variables, there is a deep difference between them. In the case of a function of one variable, for the existence of a limit at a point, the equality of only two numbers is necessary and sufficient - the limits in two directions: to the right and to the left of the limit point 
. For a function of two variables, the tendency to the limit point 
on a plane can occur in an infinite number of directions (and not necessarily along a straight line), and therefore the requirement for the existence of a limit for a function of two (or several) variables is “stricter” compared to a function of one variable.
Example . Find 
.
Let the desire for the limiting point 
happens in a straight line 
. Then 
.
The limit obviously does not exist, since the number 
depends on 
.
Properties of FNP limits:
If there are 
, That:, The partial derivative with respect to 
and its notation is introduced.
It is easy to see that a partial derivative is the derivative of a function of one variable when the value of the other variable is fixed. Therefore, partial derivatives are calculated according to the same rules as the calculation of derivatives of functions of one variable.
Example . Find partial derivatives of a function 
.
We have: 
, 
.