The concept of a neighborhood of a point of a function of several variables. Functions of several variables: basic concepts. Domain of trigonometric functions of two variables
(lecture 1)
Functions of 2 variables.
A variable z is called a function of 2 variables f (x, y) if for any pair of values \u200b\u200b(x, y) G a certain value of the variable z is associated.
Def. The neighborhood of the point p 0 is called a circle with a center at point p 0 and a radius. = (x-x 0 ) 2 + (ooh 0 ) 2
an arbitrarily small number, you can specify a number ()\u003e 0 such that for all values \u200b\u200bof x and y, for which the distance from m. 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 radius, the value of the function differs from A by less than in absolute value. And this means that when the point p approaches the point p 0 along any
Continuity of function.
Let the function z \u003d 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: \u003d f (x 0, y 0);
Lim f (x, y) \u003d f (x 0 , y 0 );
pp 0
Partial derivative.
Give argument x an increment of x; x + x, we get the point p 1 (x + x, y), calculate the difference between the values \u200b\u200bof the function at the point p:
x z \u003d f (p1) -f (p) \u003d f (x + x, y) - f (x, y) is the partial increment of the function corresponding to the increment of the argument x.
z \u003d Lim x z
z \u003d Lim f (x + x, y) - f (x, y)
X x0 X
Defining a function of several variables
When considering many questions from various fields of knowledge, one has to study such relationships between variables, when the numerical values \u200b\u200bof one of them are completely determined by the values \u200b\u200bof several others.
for instanceWhen studying the physical state of a body, one has to observe a change 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 \u200b\u200bof four variables: x, y, z, t.
Another example: the cost of production for the manufacture of a unit of a certain type of product is studied. Let be:
x is the cost of materials,
y - expenses for the payment of wages to employees,
z - depreciation charges.
It is obvious that production costs depend on the values \u200b\u200bof the named parameters x, y, z.
Definition 1.1 If each set of values \u200b\u200bof "n" variables
from some set D of these collections corresponds its unique value to the variable z, then they say that the set D is given a function
"n" variables.
The set D indicated in Definition 1.1 is called the domain of definition or the domain of existence of this function.
If a function of two variables is considered, then the collections of numbers
are denoted, as a rule, (x, y) and are interpreted as points of the coordinate plane Oxy, and the domain of definition of the function z \u003d f (x, y) of two variables is represented as a set of points on the plane Oxy.
So, for example, the scope of the function
is the set of points of the plane Oxy whose coordinates satisfy the relation
that is, it is a circle of radius r centered at the origin.
For function
the domain of definition are points that satisfy the condition
that is, external to a given circle.
Often functions of two variables are given implicitly, i.e., as the equation
linking three variables. In this case, each of the values \u200b\u200bx, y, z can be considered as an implicit function of the other two.
The geometric image (graph) of a function of two variables z \u003d f (x, y) is the set of points P (x, y, z) in the three-dimensional space Oxyz, the coordinates of which satisfy the equation z \u003d f (x, y).
The graph of a function of continuous arguments, as a rule, is some surface in the Oxyz space, which is projected onto the Oxy coordinate plane into the domain of the function z \u003d f (x, y).
So, for example, (Fig.1.1) the graph of the function
is the top half of the sphere, and the graph of the function
The lower half of the sphere.
The graph of the linear function z \u003d ax + by + с is the plane in the Oxyz space, and the graph of the function z \u003d сonst is the plane parallel to the coordinate plane Oxyz.
Note that it is impossible to visualize the function of three or more variables in the form of a graph in three-dimensional space.
In what follows, we will mainly restrict ourselves to considering functions of two or three variables, since the case of a larger (but finite) number of variables is considered in a similar way.
Defining a function of several variables.
(lecture 1)
A variable u is called f (x, y, z, .., t) if for any set of values \u200b\u200b(x, y, z, .., t) a well-defined value of the variable u is associated.
The set of aggregates of the value of the variable is called the domain of definition of the function.
G - set (x, y, z, .., t) - domain of definition.
Functions of 2 variables.
A variable z is called a function of 2 variables f (x, y) if for any pair of values \u200b\u200b(x, y) Î G a certain value of the variable z is associated.
Limit of a function of 2 variables.
Let the function z \u003d 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 neighborhood of point p 0 is called a circle with center at point p 0 and radius r. r= Ö (x-x 0 ) 2 + (ooh 0 ) 2 Ø
The number A is called the limit of the function | at the point p 0 if for any
of an arbitrarily small number e, one can specify a number r (e)\u003e 0 such that for all values \u200b\u200bof x and y for which the distance from m. p to p0 is less than r, the following inequality holds: ½f (x, y) - A10, with radius r , the value of the function differs from A by less than e in absolute value. And this means that when the point p approaches the point p 0 along any path, the value of the function approaches the number A.
Continuity of function.
Let the function z \u003d 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 \u003d f (x, y) is called continuous in m. P \u200b\u200b0 if 3 conditions are satisfied:
1) the function is defined at this point. f (p 0) \u003d f (x, y);
2) ph-i has a limit at this point.
3) The limit is equal to the value of the function at this point: b \u003d f (x 0, y 0);
Lim f (x, y)= f (x 0 , y 0 ) ;
pà p 0
If at least one of the continuity conditions is violated, then the point p is called a discontinuity point. For functions of 2 variables, separate break points and whole break lines may exist.
The notions of limit and continuity for functions of a larger number of variables are defined similarly.
The function of three variables cannot be represented graphically, in contrast to the function of two variables.
For a 3-variable function, break points, lines and break surfaces can exist.
Partial derivative.
Consider the function z \u003d f (x, y), p (x, y) is the point under consideration.
Let's give the argument x an increment Dx; x + Dx, we get the point p 1 (x + Dx, y), calculate the difference between the values \u200b\u200bof the function at the point p:
D x z \u003d f (p1) -f (p) \u003d f (x + Dx, y) - f (x, y) is the partial increment of the function corresponding to the increment of the argument x.
Def. The partial derivative of the function z \u003d f (x, y) with respect to the variable x is 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 \u003d Lim D x z
à ¶ z \u003d Lim f (x + D x, y) - f (x, y)
¶ x Dx® 0 Dx
Similarly, we define the partial derivative with respect to the variable y.
Finding partial derivatives.
When determining partial derivatives, only one variable is changed each time, the rest of the variables are considered constant. 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 usual derivative of the function of this one variable, the remaining variables are considered as constant values. In this case, all formulas for differentiating a function of one variable (derivative of a sum, product, quotient) are valid.
The concept of a function of several variables
If each point X \u003d (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 function of n variables z \u003d f (x 1, x 2, ... x n) \u003d f (X).
In this case, the variables x 1, x 2, ... x n are called independent variables or arguments functions, z - dependent variableand the symbol f stands for conformity law... The set (X) is called scope functions (this is a subset of n-dimensional space).
For example, the function z \u003d 1 / (x 1 x 2) is a function of two variables. Its arguments are variables x 1 and x 2, and z is the dependent variable. The domain of definition is the entire coordinate plane, except for the straight lines x 1 \u003d 0 and x 2 \u003d 0, i.e. without abscissa and ordinate axes. Substituting into the function any point from the domain of definition, according to the law of correspondence, we get a certain number. For example, taking point (2; 5), i.e. x 1 \u003d 2, x 2 \u003d 5, we get
z \u003d 1 / (2 * 5) \u003d 0.1 (i.e. z (2; 5) \u003d 0.1).
A function of the form z \u003d a 1 x 1 + a 2 x 2 + ... + a n x n + b, where a 1, a 2, ..., and n, b are constant numbers, are called linear... It can be viewed as the sum of n linear functions of variables x 1, x 2, ... x n. All other functions are called nonlinear.
For example, the function z \u003d 1 / (x 1 x 2) is nonlinear, and the function z \u003d
\u003d x 1 + 7x 2 - 5 - linear.
Any function z \u003d f (X) \u003d f (x 1, x 2, ... x n) can be associated with n functions of one variable, if we fix the values \u200b\u200bof all variables except one.
For example, functions of three variables z \u003d 1 / (x 1 x 2 x 3) can be associated with three functions of one variable. If you fix x 2 \u003d a and x 3 \u003d b then the function will take the form z \u003d 1 / (abx 1); if we fix x 1 \u003d a and x 3 \u003d b, then it will take the form z \u003d 1 / (abx 2); if we fix x 1 \u003d a and x 2 \u003d b, then it will take the form z \u003d 1 / (abx 3). In this case, all three functions have the same look. It is not always so. For example, if x 2 \u003d a is fixed for a function of two variables, then it will take the form z \u003d 5x 1 a, i.e. a power function, and if we fix x 1 \u003d a, then it will take the form, i.e. exponential function.
Schedule a function of two variables z \u003d f (x, y) is the set of points of the three-dimensional space (x, y, z), the applicate z of which is associated with the abscissa x and the ordinate y by the functional relation
z \u003d f (x, y). This graph represents some surface in three-dimensional space (for example, as in Figure 5.3).
It can be proved that if a function is linear (i.e. z \u003d ax + by + c), then its graph is a plane in three-dimensional space. It is recommended to study other examples of three-dimensional graphs on your own using Kremer's textbook (pp. 405-406).
If there are more than two variables (n variables), then schedulefunction is a set of points of the (n + 1) -dimensional space, for which the coordinate x n + 1 is calculated in accordance with a given functional law. Such a schedule is called hypersurface (for a linear function - hyperplane), and it is also a scientific abstraction (it is impossible to depict it).
Figure 5.3 - Graph of a function of two variables in three-dimensional space
Surface level a function of n variables is the set of points in an n-dimensional space such that at all these points the value of the function is the same and is equal to C. The number C itself in this case is called level.
Typically, infinitely many level surfaces (corresponding to different levels) can be plotted for the same function.
For a function of two variables, the level surface takes the form level lines.
For example, consider z \u003d 1 / (x 1 x 2). Take C \u003d 10, i.e. 1 / (x 1 x 2) \u003d 10. Then x 2 \u003d 1 / (10 x 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 \u003d 5, we get a level line in the form of a graph of the function x 2 \u003d 1 / (5x 1) (shown by a dotted line in Figure 5.4).
Figure 5.4 - Level lines of the function z \u003d 1 / (x 1 x 2)
Let's take another example. Let z \u003d 2x 1 + x 2. Take C \u003d 2, i.e. 2x 1 + x 2 \u003d 2. Then x 2 \u003d 2 - 2x 1, i.e. on the plane, the level line will take the form of a straight line, shown in Figure 5.5 as a solid line. Taking another level, for example, C \u003d 4, we get a level line in the form of a straight line x 2 \u003d 4 - 2x 1 (shown by a dotted line in Figure 5.5). The level line for 2x 1 + x 2 \u003d 3 is shown in Figure 5.5 with a dotted line.
It is easy to make sure that for a linear function of two variables, any level line will be a straight line on the plane, and all the level lines will be parallel to each other.
Figure 5.5 - Level lines of the function z \u003d 2x 1 + x 2
) we have repeatedly encountered partial derivatives of complex functions like and more difficult examples. So what else can you tell about ?! ... And everything is like in life - there is no such complexity that could not be complicated \u003d) But mathematics - that's what mathematics is for, to fit the diversity of our world into strict frames. And sometimes it can be done in one single sentence:
In general, the complex function has the form 
where, at least one of letters is functionwhich may depend on arbitrary the number of variables.
The smallest and easiest option is a long-familiar complex function of one variable, whose derivative we learned to find in the 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 poorly assimilated. In this regard, I will limit myself to specific examples, from which you can understand the general principle of finding these derivatives:
Example 1
A complex function is given, where 
... Required:
1) find its derivative and write down the total differential of the 1st order;
2) calculate the value of the derivative at.
Decision: First, let's deal with the function itself. We are offered a function that depends 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 full derivative ... How to find it?
The first thing that comes to mind is direct substitution and further differentiation. Substitute 
into a function: 
, after which there are no problems with the desired derivative:
And, accordingly, the full differential: 
This solution is mathematically correct, but a small nuance is that when the problem is formulated as it is, no one expects such barbarity from you \u003d) But seriously, you can really find fault here. Imagine that the function describes the flight of the bumblebee, and the nested functions change with temperature. By performing forward substitution 
, we only get private information , which characterizes flight, say, only in hot weather. Moreover, if a person who is not versed in bumblebees is presented with a ready-made result and even said what this function is, then he will never learn anything about the fundamental law of flight!
So, quite unexpectedly, our buzzing brother helped to realize the meaning and importance of the universal formula: 
Get used to the "two-story" designations of derivatives - they are used in this task. In this case, one should be very neat in the notation: derivatives with direct signs "de" are full derivativesand derivatives with rounded symbols are partial derivatives... Let's start with the latter: 
Well, with "tails" in general, everything is 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 in the same form: 
At the same time, in the "fancy" answers, it is better to refrain from even minimal simplifications (here, for example, it begs to remove 3 minuses) - and you have less work, and a furry friend is happy to review the task easier.
However, a rough check will not be superfluous. Substitute 
into the found derivative and simplify: 
(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. At 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 a quotient of two numbers:
And, of course, it's a sin not to check it using a more compact record. 
:
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 by yourselfdenote nested functions with some suitable letters, for example, through 
and use the same formula:
By the way, about the letter designations. I have repeatedly urged not to "cling to the letters" as a lifeline, and now this is especially important! Analyzing various sources on the topic, I generally got the impression that the authors "went wild" and began to ruthlessly throw students into the stormy abyss of mathematics \u003d) So forgive :))
Example 2
Find the derivative of a function 
, if 
Other designations should not be confusing! Every time you come across such a task, you need to answer two simple questions:
1) What does the "main" function depend on? In this case, the "z" function depends on two functions ("y" and "ve").
2) What variables do the nested functions depend on? In this case, both "inserts" depend only on the "x".
Thus, you should have no difficulty in adapting the formula to this task!
A short solution and answer at the end of the tutorial.
Additional examples for 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
A function is given where.
Calculate the derivative at a point
The formula for the derivative of a complex function, as many people guess, has a related form: 
Decide, once you guessed it \u003d)
Just in case, I will also give a general formula for the function:
, although in practice you are unlikely to come across anything longer than Example 3.
In addition, sometimes it is necessary to differentiate the "stripped down" option - as a rule, a function of the form either. I leave this question for you to study on your own - think of some simple examples, think, experiment and derive shortened formulas for derivatives.
If something remains misunderstood, please slowly re-read and comprehend the first part of the lesson, as now the task will become more difficult:
Example 4
Find the partial derivatives of a complex function, where 
Decision: this function has the form, and after direct substitution we get the usual function of two variables: 
But such fear is not something that is not accepted, but no longer wants to differentiate \u003d) Therefore, we will use ready-made formulas. To help you catch the pattern more quickly, I'll make some notes: 
Look carefully from top to bottom and from left to right….
First, let's find the partial derivatives of the "main" function:
Now we find the "x" derivatives of the "inserts":
and write down the final "x" derivative:
Likewise with the "game":
and
You can also stick to another style - immediately find all the "tails" 
and then write down both derivatives.
Answer:
About substitution 
somehow I don't think at all \u003d) \u003d), but you can comb the results a little bit. Although, again, why? - just complicate the test for the teacher.
If required, then full differential here it is written according to the usual formula, and, by the way, just at this step light makeup becomes appropriate: 
So ... .... a coffin on wheels.
Due to the popularity of the considered type of complex function, a couple of tasks for an independent solution. A simpler example in a "semi-general" form - for understanding the formula itself ;-):
Example 5
Find the partial derivatives of the function, where 
And more difficult - with the connection of the differentiation technique:
Example 6
Find the Total Differential of a Function 
where 
No, I'm not trying to "send you to the bottom" at all - all examples are taken from real works, and "on the high seas" you may come across any letters. In any case, you need to analyze the function (by answering 2 questions - see above), present it in general form and carefully modify the formulas for the partial derivatives. Perhaps now you will be a little confused, but you will understand the very principle of their construction! For the real tasks are just beginning :)))
Example 7
Find partial derivatives and complete differential of complex function
where
Decision: The "main" function has the form and still depends on two variables - "x" and "game". But compared to Example 4, one more nested function has been added, and therefore the PD formulas are also lengthened. As in the example, for a better visualization of the pattern, I will highlight the "main" partial derivatives in different colors: 
Again, study the entry carefully from top to bottom and left to right.
Since the problem is formulated in a "semi-general" form, all our works are essentially limited to finding the partial derivatives of embedded functions: 
The first grader will cope: 
And even the full differential turned out to be quite nice:
I deliberately did not begin to offer you any specific function - so that unnecessary heaps would not interfere with a good understanding of the concept of the problem.
Answer: 
Quite often you can find "different-sized" attachments, for example:
Here the "main" function, although it has the form, still depends on both "x" and "game". 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 "insert" depends on one variable.
A similar situation occurs in the two final examples of the lesson:
Example 8
Find the total differential of a compound function at a point
Decision: the condition is formulated in a "budgetary" way, and we ourselves must designate the nested functions. Not a bad option in my opinion:
The "inserts" contain ( ATTENTION!) THREE letters are the good old "X-Y-Y-Z", which means that the "main" function actually depends on three variables. It can be formally rewritten in the form, and the partial derivatives in this case are determined by the following formulas: 
We scan, penetrate, catch….
In our task: 
So far, we have considered the simplest functional model, in which function depends on the only argument... But when studying various phenomena of the surrounding world, we often encounter a simultaneous change 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 a theme with the most common in practice functions of two variables .
The function of two variables called law, according to which each pair of values independent variables (arguments) from areas of definition corresponds to the value of the dependent variable (function).
This function is denoted as follows:
Either, or another standard letter:
Since an ordered pair of values \u200b\u200b"x" and "game" determines point on the plane, then the function is also written through, where is the point of the plane with coordinates. This designation is widely used in some practical exercises.
The geometric meaning of a function of two variables very simple. If the function of one variable corresponds to a certain line on the plane (for example, the familiar school parabola), then the graph of the function of two variables is located in three-dimensional space. In practice, most often you have to deal with surface, but sometimes the graph of a function can represent, for example, a spatial line (s) or even a single point.
We are well acquainted with an elementary example of a surface from the course analytic geometry - this is plane ... Assuming the equation is easy to rewrite in functional form:
The most important attribute of the function of 2 variables is the already voiced domain.
The domain of the function of two variables called the set of all pairs for which there is a value.
Graphically, the domain is the whole plane or part of it... So, the domain of the function 
is the entire coordinate plane - for the reason that for anypoint there is a value.
But such an idle alignment does not always happen, of course:
How are two variables?
Considering the 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 clarifying areas 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 topical work, so I will analyze a decent number of examples:
Example 1
Find the domain of a function 
Decision: since the denominator cannot vanish, then:
Answer: the whole coordinate plane except points belonging to a straight line
Yes, yes, the answer is best written in this style. The domain of a function of two variables is rarely denoted by any symbol, much more often they use verbal description and / or drawing.
If by condition required to complete the drawing, then the coordinate plane should be drawn and dotted line draw a straight line. The dotted line signals that the line excluded into the domain of definition.
As we will see a little later, in the more difficult examples, a drawing is not necessary at all.
Example 2
Find the domain of a function 
Decision: 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 with a line and hatch the upper half-plane... The solid line indicates the fact that she enters into the domain of definition.
Attention! If ANYTHING is not clear to you from the second example, please study / repeat the lesson in detail Linear inequalities - it will be very difficult without it!
Thumbnail for self-solution:
Example 3
Find the domain of a function
Two-line solution and answer at the end of the lesson.
We continue to warm up:
Example 4
And depict her in the drawing 
Decision: it is easy to understand that such a problem statement requires execution of the drawing (even if the definition area is very simple). But analytics first: the radical expression must be non-negative: and, given that the denominator cannot vanish, the inequality becomes strict:
How to determine the area that inequality defines? I recommend the same algorithm of actions as when solving linear inequalities.
Draw first linegiven by corresponding equality... The equation defines circle centered at the origin of the radius that divides the coordinate plane into two parts - "inside" and "outside" of the circle. Since the inequality we have strict, then the circle itself will certainly not be included in the domain of definition and therefore it must be drawn dotted line.
Now we take arbitrary point of the plane, not owned circle, and substitute its coordinates in the inequality. The easiest way, of course, is to select the origin:
Received wrong inequalitythus point does not satisfy inequality. Moreover, this inequality is not satisfied by any point lying inside the circle, and, therefore, the sought domain is its outer part. The definition area is traditionally hatched: 
Those interested 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 a circle centered at the origin, radius.
Answer: outside of the circle
Let's go back to the geometric meaning of the problem: we found the domain and shaded it, what does this mean? This means that at each point of the shaded area there is a value "z" and graphically the function 
represents the following surface:
The schematic drawing clearly shows that this surface is located in places over plane (near and far from us octants), in places - under plane (left and right octants relative to us)... Also, the surface passes through the axes. But the behavior of the function as such is not very interesting to us now - it is important that all this happens exclusively in the field of definition... If we take any point belonging to a circle, then no surface will be there (because there is no "z"), which is what the round space in the middle of the picture says.
Please, understand well the analyzed example, because in it I explained in detail the very essence of the problem.
The next task for independent solution:
Example 5
A short solution and a 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 task ellipse, hyperbole or parabola.
Going for a promotion:
Example 6
Find the domain of a function
Decision: the radical expression must be non-negative: and the denominator cannot be zero:. Thus, the domain is defined by the system.
We deal with the first condition according to the standard scheme discussed in the lesson Linear inequalities: draw a straight line and define a half-plane that corresponds to the inequality. Since the inequality lax, then the straight line itself will also be a solution.
With the second condition of the system, everything is also simple: the equation sets the ordinate axis, and as soon as it should be excluded from the domain of definition.
Let's execute the drawing, not forgetting that the solid line denotes its entry into the definition area, and the dotted line - the exclusion from this area: 
It should be noted that here we are actually forced make a drawing. And this situation is typical - in many problems the verbal description of the area is difficult, and even if you describe it, then, most likely, you will be poorly understood and forced to depict the area.
Answer: domain:
By the way, such an answer without a drawing really looks damp.
Once again, we repeat the geometric meaning of the result obtained: in the shaded area, there is a graph of a function, which is three-dimensional surface... This surface can be located above / below the plane, it can intersect the plane - in this case, everything is parallel to us. The very fact of the existence of a surface is important, and it is important to correctly find the area in which it exists.
Example 7
Find the domain of a function 
This is an example for a stand alone solution. An approximate example of finishing the task at the end of the lesson.
It is not uncommon when seemingly simple-looking functions cause a far from hasty decision:
Example 8
Find the domain of a function
Decision: using difference of squares formula, we factor out the radical expression: 
.
The product of two factors is non-negative 
when both multipliers are non-negative: ORwhen both non-positive:. This is a typical feature. Thus, two systems of linear inequalities and COMBINE received areas. In a similar situation, instead of the standard algorithm, the scientific, or rather, practical, poke method works much faster \u003d)
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, and therefore the solution to the system is whole upper "corner". Shading.
Now we take a point belonging to the right "corner". There remains the 2nd system, into which we substitute the coordinates of this point: 
... The second inequality is not true, therefore, and all the right corner is not a system solution. 
A similar story with the left "corner", which also will not be included in the definition area.
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, it is naturally not necessary to describe in such detail - all commented-out actions are easily performed orally!
Answer: the scope is union systems solutions 
.
As you might guess, such an answer is unlikely to pass without a drawing, and this circumstance forces you to pick up a ruler and a pencil, although this was not required by the condition.
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 
Decision: The argument of the logarithm is strictly positive, so the domain is given by the system.
The inequality points to the right half-plane and excludes the axis.
With the second condition, the situation is more intricate, but also transparent. Remember sinusoid... The argument is "yer", but this should not confuse - yer, so yer, siu, siu. Where is the sine greater than zero? The sine is greater than zero, for example, in the interval. Since the function is periodic, there are infinitely many such intervals, and in a reduced form the solution to the inequality is written as follows: 
, where is an arbitrary integer.
An infinite number of intervals, of course, cannot be depicted, so we will limit ourselves to an interval 
and its neighbors: 
Let's execute the drawing, not forgetting that according to the first condition, our field of activity is limited to the strictly right half-plane: 
hmm ... some kind of ghost drawing turned out ... a good cast 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 outside. The method for finding the desired part has repeatedly appeared in the article. Linear inequalities and the previous examples in this tutorial.
Solution, drawing and answer at the end of the lesson.
The final nuts of this paragraph are about "arches":
Example 12
Find the domain of a function 
Decision: arcsine argument must be within the following ranges: 
Then there are two technical possibilities: more prepared readers by analogy with the last examples of the lesson Domain of a function of one variable can "roll over" double inequality and leave "games" in the middle. For dummies, I recommend converting the "train" into an equivalent system of inequalities:
The system is solved as usual - we build straight lines and find the necessary half-planes. As a result: 
Note that here the boundaries are included in the definition area and the straight lines are drawn with solid lines. This should always be carefully monitored in order to avoid a gross error.
Answer: the domain is the solution of the system
Example 13
Find the domain of a function
The sample solution uses an advanced technique - double inequality transforms.
In practice, there are also sometimes problems of finding the domain of a function of three variables. The domain 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 in finding this body and performing a three-dimensional drawing. However, such examples are quite rare. (I found only a couple of them), and therefore I will limit myself to just this overview paragraph.
Level lines
For a better understanding of this term, we will compare the axis with height: the higher the value of "z" - the greater the height, the lower the value of "z" - 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 “cuts” of the surface at different heights. These "slices" or, more correctly, cross-sections carried by planes then projected onto the plane .
Definition: the level line of a function is a line on the plane, at each point of which the function remains constant:.
Thus, the level lines help to figure out what a particular surface looks like - and they help without building a three-dimensional drawing! Let's consider a specific task:
Example 14
Find and plot multiple level lines of a function graph 
Decision: Examine the shape of this surface using level lines. For convenience, let's expand the record "backwards": 
It is obvious that in this case "z" (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 you need to "cut" the level lines, then we are free to choose several values \u200b\u200bof "z" at our discretion.
Examine the surface at zero height, for this we put the value in equality 
:
The solution to this equation is a point. That is, for the level line represents a point.
We rise to a unit height and "cut" our surface 
plane (substituted into the surface equation):
In this way, 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 therefore I write two, not three coordinates for points!
Now we take, for example, a plane and "cut it" the investigated surface 
(substitute into the surface equation):
In this way, for height the level line is a circle centered on the radius point.
And let's build another level line, say, for :
– circle centered at radius 3.
The level lines, as I have already emphasized, are located on the plane, but each line is signed - what height does it correspond to: 
It is easy to understand that other level lines of the surface under consideration are also circles, and the higher we go up (increase the value of "z"), the larger the radius becomes. In this way, the surface itself is an endless bowl with an egg-shaped bottom, the top of which is located on a plane. This "bowl" together with the axis "goes straight to you" from the monitor screen, that is, you look at its bottom \u003d) And this is no accident! Only I pour so destructively on the road \u003d) \u003d)
Answer: the level lines of this surface are concentric circles.
Note : when we get a degenerate circle of zero radius (point)
The very concept of a level line comes from cartography. To paraphrase the well-established mathematical phrase, we can say that a level line is a geographical location of points of equal height... Consider a certain mountain with 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 values \u200b\u200bof altitude also acquire a very specific meaning - after all, some parts of the Earth's surface are located below the zero mark of the world ocean level.
Functions of many variables
§1. The concept of a function of many variables.
Let there be n variables. Each set 
denotes a point n-
dimensional set 
(p-dimensional vector).
Given sets 
and 
.
Def... If every point 
matches the singular 
, then they say that a numerical function is given n variables:
.
called the domain, 
- the set of values \u200b\u200bof this function.
When n\u003d 2 instead of 
usually write x,
y,
z... Then the function of two variables has the form:
z= f(x, y).
For instance, 
- a function of two variables;
- function of three variables;
Linear function n variables.
Def... Function graph n variables 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 the 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 a rather difficult task. The construction of the so-called level lines can provide significant assistance in solving it.
Def... Level line function of two variables z= f(x, y) is the set of points of the plane HOW, which are the projection of the section of the graph of the function by a plane parallel to HOW. At each point on the level line, the function has the same meaning. The level lines are described by the equation f(x, y) \u003d withwhere from - some number. There are infinitely many level lines, and one of them can be drawn through each point of the definition area.
Def... 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 some value from... Level surface equation: f
(
)\u003d s.
Example... Plot a function of two variables
.
.
With c \u003d 1: 
;
.
With c \u003d 4: 
;
.
With c \u003d 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, you can define the limit and continuity of a function.
Def... The number A is called the limit of the function of two variables z=
f(x,
y)
at 
,
and denoted 
if for any positive number 
there is a positive number 
, such that if point 
removed from point 
less distance 
, then the quantities f(x,
y)
and A differ by less than 
.
Def... If the function z=
f(x,
y)
defined at point 
and at this point has a limit equal to the value of the function 
, then it is called continuous at this 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 
is a function of one variable 
... Let it have a derivative at the point 
:
.
This derivative is called the partial derivative (or partial derivative of the first order) of the function 
by 
at the point 
and is indicated by: 
;
;
;
.
The difference is called the partial increment in 
and denoted 
:
Considering the above designations, we can write
.
Similarly,
.
Partial derivative a function of several variables in one of these variables is called the limit of the ratio of the partial increment of the function to the increment of the corresponding independent variable when this increment tends to zero.
When finding a partial derivative with respect to any argument, the other arguments are considered constant. All rules and formulas for differentiation of functions of one variable are valid for partial derivatives of functions of several 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 
has four second-order partial derivatives, which are denoted as follows:
;
;
;
.
and 
- mixed partial derivatives.
Example.Find the partial derivatives of the second order for the function
.
Decision.
,
.
,
.
, 
.
The task.
1. Find the partial derivatives of the second order for the functions
,
;
2. For function 
prove that 
.
Full differential functions of several variables.
While changing the values xand at function 
will change by an amount called the total function increment z
at the point 
... Just as in the case of a function of one variable, the problem arises of an approximate replacement of the increment 
to a linear function of 
and 
... The role of the linear approximation is performed by full differential functions:
Full differential of the second order:
=
.
=
.
In general, the total 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 given by a unit vector 
... The coordinates of a unit vector are expressed in terms of the cosines of the angles formed by the vector and the coordinate axes, called direction cosines:
,
.
When moving point M ( x,
y) in this direction l
exactly 
function z will receive an increment
called incrementing a function in a given direction l.
If MM 1 \u003d ∆ lthen
T
ABOUT
functions z=
f(x,
y)
towards
is the limit of the ratio of the increment of the function in this direction to the value of the displacement ∆ l
when the latter tends to zero:
The directional derivative characterizes the rate of change of a function in a given direction. Obviously, the partial derivatives 
and 
are derivatives in directions parallel to the axes Ox
and Oy... It is easy to show that
Example... Calculate the derivative of a function 
at point (1; 1) in the direction 
.
Def. Gradient functions z= f(x, y) a vector with coordinates equal to partial derivatives is called:
.
Consider the scalar product of vectors 
and 
:
It is easy to see that 
, i.e. the directional derivative is equal to the dot product of the gradient and the unit direction vector 
.
Because the 
, then the dot product is maximal when the vectors are in the same direction. Thus, the gradient of the 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 the function, you can locally build the function level lines.
Theorem... Let a differentiable function be given z=
f(x,
y)
and at the point 
the gradient of the function is not zero: 
... Then the gradient is perpendicular to the level line passing through this point.
Thus, if, starting from a certain point, the gradient of the function and a small part of the level line perpendicular to it are plotted at close points, then it is possible (with some error) to plot the level lines.
Local extremum of a function of two variables
Let the function 
is defined and continuous in some neighborhood of the point 
.
Def... Dot 
is called the point of local maximum 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 the differentiable function 
had a local extremum at the point 
, it is necessary that all its first-order partial derivatives at this point are equal to zero:
So, the points of possible existence of an extremum are those points at which the function is differentiable, and its gradient is 0: 
... As in the case of a function of one variable, such points are called stationary.
Until now, we have been studying the function of one variable, i.e. the study of a variable whose values \u200b\u200bdepend on the values \u200b\u200bof one independent variable.
In practice, it is often necessary to deal with quantities whose numerical values \u200b\u200bdepend on the values \u200b\u200bof several quantities varying independently of each other. The study of such quantities leads to the concept of a function of several variables. Here are some examples.
Example 1. The area of \u200b\u200ba rectangle is a function of two independently changing variables - the sides of the rectangle and:.
Example 2. The work of the electric current in the section of the circuit depends on the potential difference at the ends of the section, current strength and time:.
Example 3. The temperature measured at various points of a certain body is a function of the coordinates of the point at which it is measured, and from the moment in time:.
Definition 1. Let's call n -measuring point an ordered set of numbers. The numbers are called coordinates -dimensional point. The set of all possible -dimensional points is called n-dimensional space and we will denote it. The point is called origin in -dimensional space, and the number - dimension space.
Special cases:
1. - number line;
2. - plane;
3. - three-dimensional space.
Definition 2. Let there are variable quantities, and each set of their values \u200b\u200bfrom a certain set corresponds to one well-defined value of the variable. Then they say that it is given function of several variables
The variables are called independent variables or arguments , – dependent variable , symbol - conformity law .
As well as a function of one variable, a function of several variables can be set clearly - and implicitly – .
Any explicit function of several variables can be represented as a function of a point in -dimensional space:, where a point is determined by a set of its coordinates.
If one value corresponds to each point from the domain of definition, then the function is called unambiguous , otherwise - ambiguous .
The set is called function scope , it is a subset of -dimensional space. Like a gap, an area can be closed or about open depending on whether it contains its border or not.
Natural domain of definition function (1) is the set of points whose coordinates uniquely provide real and finite values \u200b\u200bof the function. In what follows, if additional restrictions on the change in independent variables are not imposed by the statement of the problem, by the domain of definition of a function we mean its natural domain of definition.
Let us consider in more detail two special cases, which are the simplest and allow a geometric interpretation.
1. A function of two variables ( n = 2)
The function of two variables will be denoted by. The particular value of the function at, or at the point is written in the form,, or.
The domain of a function is a subset of the points of the coordinate plane. In particular, the domain of a function can be the whole plane or part of the plane bounded by lines. The line bounding this area will be called border area. Points of the plane that do not lie on the boundary will be called internal .
Example 4. The function is defined over the entire plane.
Example 5. The function is defined on the entire plane, except for the straight line.
Example 6. The domain of the function is the set of points on the plane whose coordinates satisfy the relation, i.e. a circle of radius 1 and centered at the origin. The domain of this function is closed.
Let's take a closer look at the next example.
Example 7. Find the domain of the function.
Decision.
The logarithm is only defined when the argument is positive, so there is one condition for the arguments:.
To depict the area geometrically, we first find its border:. The resulting equation defines a parabola whose vertex is located at a point, and the axis is directed towards the positive side of the axis.
|
Consequently, the required region consists of the interior points of the parabola. The parabola itself is not included in the region, which means that the region is open.
Definition 3: Neighborhoodpoint is any open circle containing a point.
In particular, an open circle with a center at a point and a radius is called a -neighborhood.
Obviously, a circle on a plane is a two-dimensional analogue of an interval on a straight line.
When studying functions of several variables, the already developed mathematical apparatus for a function of one variable is largely used. Namely, any function can be associated with a pair of functions of one variable: for a fixed value, a function and for a fixed value, a function.
It should be borne in mind that although the functions have the same "origin", their form can differ significantly.
Example 9. Let's consider a function. When the function is exponential, and when the function is exponential.
Geometric representation of a function of two variables.
As you know, a function of one variable can be depicted by some curve on a plane if we consider the values \u200b\u200bof its argument as abscissas, and the values \u200b\u200bof the function as ordinates of the points of the curve.
Likewise, a function of two variables can be represented graphically.
Consider a function defined in a region on a plane and a system of rectangular Cartesian coordinates. To each point of the set we put in correspondence a point in space, the applicate of which is equal to the value of the function at the point:. The collection of all such points represents a certain surface, which is natural to take for a graphical representation of a function.
Definition 4: Graph of a function of two variables is called the set of points in three-dimensional space, the applicate of which is associated with the abscissa and ordinate by a functional relationship.
|
2. Function of three variables (n \u003d 3)
The function of three variables will be denoted, in this case, we will assume that, and are independent variables (or arguments), and is a dependent variable (or function).
The scope of such a function is called the set of all considered triples of numbers. If the function is set analytically, under its natural domain of definition imply the collection of all triples of numbers for which the function takes real values.
Definition 6: neighborhoodpoints are any open sphere containing a point.
In particular, an open sphere with a center at a point and a radius is called a neighborhood.
Depicting triples of numbers as points in space, one can consider a function of three variables as a function of a point in space, and the domain of a function of three variables as a set of points in space.
