Construct level lines of the function z x y. A simple class for plotting level lines of a 2D grid function

FUNCTIONS OF SEVERAL VARIABLES

1. BASIC CONCEPTS

Let: z - a variable value with a range of changes R; R - number line; D - area on the coordinate plane R2.

Any mapping D->R is called a function of two variables with domain D and written z = f(x;y).

In other words:

If each pair (x; y) of two independent variables from the domain D, according to some rule, is associated with one specific value z from R, then the variable z is called a function of two independent variables x and y with domain D and written

http://pandia.ru/text/78/481/images/image002_44.jpg" width="215" height="32 src=">

Example 1.

http://pandia.ru/text/78/481/images/image005_28.jpg" width="157" height="29 src=">

http://pandia.ru/text/78/481/images/image007_16.jpg" align="left" width="110" height="89">

The domain of definition is a part of the plane lying inside a circle of radius r = 3, with the center at the origin, see figure.

Example 3. Find and draw the domain of a function

http://pandia.ru/text/78/481/images/image009_11.jpg" width="86" height="32 src=">

http://pandia.ru/text/78/481/images/image011_10.jpg" width="147" height="30 src=">

2. GEOMETRICAL INTERPRETATION OF THE FUNCTION OF TWO

VARIABLES

2.1.Graph of a function of two variables

Let us consider a rectangular coordinate system in space and a region D on the xOy plane. At each point M(x;y) from this area we restore a perpendicular to the xOy plane and plot the value z = f(x;y) on it. Geometric location of the obtained points

http://pandia.ru/text/78/481/images/image013_10.jpg" width="106" height="23 src=">

http://pandia.ru/text/78/481/images/image015_6.jpg" width="159" height="23 src=">

These are circles centered at the origin, radius R = C1/2 and the equation

x2 + y2 = R2, see figure.

Level lines allow us to represent the surface under consideration, which gives concentric circles when sectioned by planes z = C.

http://pandia.ru/text/78/481/images/image017_16.gif" width="88" height="29"> and find .

Solution. Let's use the section method.

http://pandia.ru/text/78/481/images/image020_11.gif" width="184 height=60" height="60">– in the plane – a parabola.

– in the plane – parabola.

http://pandia.ru/text/78/481/images/image025_5.gif" width="43" height="24 src="> – circle.

The required surface is a paraboloid of revolution.

Distance between two arbitrary points and (Euclidean) space is called a number

http://pandia.ru/text/78/481/images/image030_5.gif" width="153 height=24" height="24"> is called open circle radius centered at point r.

An open circle of radius ε with center at point A is called - ε - surroundings point A.

3task

Find and graphically depict the domain of definition of the function:

Draw function level lines:

3. LIMIT OF A FUNCTION OF TWO VARIABLES

Basic Concepts mathematical analysis, introduced for a function of one variable, extend to functions of several variables.

Definition:

A constant number A is called the limit of a function of two variables z = f(x;y) for x -> x0, y -> y0, if for any

ε >0 there is δ >0 such that |f(x; y) - A|< ε , как только

|x - x0|< δ и |у – у0| < δ.

This fact is indicated as follows:

http://pandia.ru/text/78/481/images/image042_2.jpg" width="160" height="39 src=">

http://pandia.ru/text/78/481/images/image044_2.gif" width="20" height="25 src=">. For a function of two variables, the tendency to a limit point on the plane can occur according to infinite number directions (and not necessarily in a straight line), and therefore the requirement for the existence of a limit for a function of two (or several) variables is “tighter” compared to a function of one variable.

Example 1. Find .

Solution. Let the desire to reach the limiting point http://pandia.ru/text/78/481/images/image048_2.gif" width="55 height=24" height="24">. Then

http://pandia.ru/text/78/481/images/image050_2.gif" width="72 height=48" height="48"> depends on.

Example 2. Find .

Solution. For any straight line the limit is the same:

http://pandia.ru/text/78/481/images/image054_2.gif" width="57" height="29">. Then

http://pandia.ru/text/78/481/images/image056_1.gif" width="64" height="21">, (the rest is by analogy).

Definition. The number is called limit functions for and , if for such that the inequalities and imply the inequality . This fact is briefly written as follows:

http://pandia.ru/text/78/481/images/image065_1.gif" width="124" height="48">.gif" width="236" height="48 src=">;

http://pandia.ru/text/78/481/images/image069_1.gif" width="247" height="60 src=">,

where is the limit point http://pandia.ru/text/78/481/images/image070_1.gif" width="85" height="24 src="> with the domain of definition and let – limit point of the set, i.e. the point to which the arguments tend X And at.

Definition 1. They say that the function is continuous at a point if:

1) ;

2) , i.e. .

Let us formulate the definition of continuity in an equivalent form..gif" width="89" height="25 src=">.gif" width="85 height=24" height="24"> is continuous at a point if the equality holds

http://pandia.ru/text/78/481/images/image079_0.gif" width="16" height="20 src=">.gif" width="15 height=16" height="16"> let's give an arbitrary increment. The function will receive a partial increment by X

http://pandia.ru/text/78/481/images/image084_0.gif" width="35" height="25 src="> is a function of one variable. Similarly,

http://pandia.ru/text/78/481/images/image058_1.gif" width="85" height="24"> is called continuous at a point over a variable (over a variable) if

http://pandia.ru/text/78/481/images/image087.gif" width="101" height="36">).

Theorem.If the functionis defined in a certain neighborhood of a point and is continuous at this point, then it is continuous at this point in each of the variables.

The reverse statement is not true.

EXAMPLE Let us prove that the function

continuous at the point http://pandia.ru/text/78/481/images/image081_0.gif" width="15 height=16" height="16">.gif" width="57" height="24" > at point corresponding to the increment http://pandia.ru/text/78/481/images/image081_0.gif" width="15" height="16 src=">:

http://pandia.ru/text/78/481/images/image092_0.gif" width="99" height="36 src=">, which means that it is continuous at a point in the variable.

Similarly, one can prove continuity at a point with respect to a variable.

Let us show that there is no limit. Let a point approach a point along a straight line passing through the point. Then we get

Thus, approaching the point http://pandia.ru/text/78/481/images/image051_1.gif" width="15" height="20">, we obtain different limit values. It follows that the limit of this function does not exist at the point, which means the function http://pandia.ru/text/78/481/images/image097.jpg" width="351" height="48 src=">

Other designations

http://pandia.ru/text/78/481/images/image099.jpg" width="389" height="55 src=">

Other designations

http://pandia.ru/text/78/481/images/image101_0.gif" width="60" height="28 src=">.

Solution. We have:

Example 2.

http://pandia.ru/text/78/481/images/image105.jpg" width="411" height="51 src=">

Example 3. Find partial derivatives of a function

http://pandia.ru/text/78/481/images/image107.jpg" width="477" height="58 src=">

Example 4. Find partial derivatives of a function

http://pandia.ru/text/78/481/images/image109.jpg" width="321" height="54 src=">

5.2. First order differentials of a function of two variables

The partial differentials of the function z = f(x, y) with respect to the variables x and y are determined, respectively, by the formulas x(x;y) and f"y(x;y) exist at the point (x0;y0) and in some of its neighborhood and are continuous at this point, then, by analogy with a function of one variable, a formula is established for the complete increment of a function of two variables

http://pandia.ru/text/78/481/images/image112_0.gif" width="364" height="57 src=">

where http://pandia.ru/text/78/481/images/image114_0.gif" width="154" height="39 src=">

In other words, the function z = f(x, y) is differentiable at the point (x, y) if its increment Δz is equivalent to the function:

Expression

http://pandia.ru/text/78/481/images/image116.jpg" width="192" height="57 src=">

Taking into account the fact that Δх = dx, Δy=dy:

http://pandia.ru/text/78/481/images/image090_0.gif" width="57" height="24 src="> is differentiable at the point, then it is continuous at this point.

The converse statement is false, i.e., continuity is only a necessary, but not a sufficient condition for the differentiability of a function. Let's show it.

EXAMPLE Let's find the partial derivatives of the function http://pandia.ru/text/78/481/images/image120.gif" width="253" height="57 src=">.

The resulting formulas lose their meaning at the point http://pandia.ru/text/78/481/images/image121.gif" width="147" height="33 src="> has no partial derivatives at the point. In fact, . This function of one variable, as is known, does not have a derivative at the point http://pandia.ru/text/78/481/images/image124.gif" width="25" height="48"> does not exist at the point. Similarly , there is no partial derivative. , is obviously continuous at the point .

So, we have shown that a continuous function may not have partial derivatives. It remains to establish the connection between differentiability and the existence of partial derivatives.

5.4. Relationship between differentiability and the existence of partial derivatives.

Theorem 1. A necessary condition for differentiability.

If the function z = f(x, y) is differentiable at the point M(x, y), then it has partial derivatives with respect to each variable and at the point M.

The converse theorem is not true, i.e. the existence of partial derivatives is necessary, but not a sufficient condition for the differentiability of a function.

Theorem 2. Sufficient condition differentiability. If the function z = f(x, y) has continuous partial derivatives at the point , then it is differentiable at the point (and its total differential at this point is expressed by the formula http://pandia.ru/text/78/481/images/image130 .gif" width="101 height=29" height="29">

Example 2. Calculate 3,021.97

3task

Calculate approximately using differential:

5.6. Rules for differentiating complex and implicit functions. Full derivative.

Case 1.

z=f(u, v); u=φ(x, y), v=ψ(x, y)

The functions u and v are continuous functions of the arguments x, y.

Thus, the function z is a complex function of the arguments x and y: z=f(φ(x, y),ψ(x, y))

Let us assume that the functions f(u, v), φ(x, y), ψ(x, y) have continuous partial derivatives with respect to all their arguments.

Let's set the task to calculate http://pandia.ru/text/78/481/images/image140.gif" width="23" height="44 src=">.

Let's give the argument x an increment Δx, fixing the value of the argument y. Then functions of two variables u= φ(x, y) and

v= φ(x, y) will receive partial increments Δxu and Δxv. Consequently, z=f(u, v) will receive the full increment defined in paragraph 5.2 (first-order differentials of a function of two variables):

http://pandia.ru/text/78/481/images/image142.gif" width="293" height="43 src=">

If xu→ 0, then Δxu → 0 and Δxv → 0 (due to the continuity of the functions u and v). Passing to the limit at Δx→ 0, we obtain:

http://pandia.ru/text/78/481/images/image144.gif" width="147" height="44 src="> (*)

EXAMPLE

Z=ln(u2+v), u=ex+y² , v=x2 + y;

http://pandia.ru/text/78/481/images/image146.gif" width="81" height="41 src=">.

http://pandia.ru/text/78/481/images/image148.gif" width="97" height="44 src=">.gif" width="45" height="44 src=">.

Then using formula (*) we get:

http://pandia.ru/text/78/481/images/image152.gif" width="219" height="44 src=">.

To obtain the final result, in the last two formulas, instead of u and v, it is necessary to substitute еx+y² and x2+y, respectively.

Case 2.

The functions x and y are continuous functions.

Thus, the function z=f(x, y) depends through x and y on one independent variable t, i.e. let’s assume that x and y are not independent variables, but functions of the independent variable t, and define the derivative http: //pandia.ru/text/78/481/images/image155.gif" width="235" height="44 src=">

Let's divide both sides of this equality by Δt:

http://pandia.ru/text/78/481/images/image157.gif" width="145" height="44 src="> (**)

Case 3.

Let us now assume that the role of the independent variable t is played by the variable x, that is, that the function z = f(x, y) depends on the independent variable x both directly and through the variable y, which is a continuous function of x.

Taking into account that http://pandia.ru/text/78/481/images/image160.gif" width="120" height="44 src="> (***)

Derivative x(x, y)=http://pandia.ru/text/78/481/images/image162.gif" width="27" height="27 src=">, y=sin x.

Finding partial derivatives

http://pandia.ru/text/78/481/images/image164.gif" width="72" height="48 src=">.gif" width="383" height="48 src=">

The proven rule for differentiating complex functions is applied to find the derivative of an implicit function.

Derivative of a function specified implicitly.

Let us assume that the equation

defines y as an implicit function of x having derivative

y' = φ'(x)_

Substituting y = φ(x) into the equation F(x, y) = 0, we would have to obtain the identity 0 = 0, since y = φ(x) is a solution to this equation. We see, therefore, that the constant zero can be considered as complex function on x, which depends on x both directly and through y =φ(x).

The derivative with respect to x of this constant must be zero; applying rule (***), we get

F’x(x, y) + F’y(x, y) y’ = 0,

http://pandia.ru/text/78/481/images/image168.gif" width="64" height="41 src=">

Hence,

http://pandia.ru/text/78/481/images/image171.gif" width="20" height="24"> is true for both one and the other function.

5.7. First order total differential. Invariance of the form of a first order differential

Let's substitute the expressions for http://pandia.ru/text/78/481/images/image173.gif" width="23" height="41 src="> defined by equalities (*) (see case 1 in clause 5.6 "Rules for differentiation of complex and implicit functions. Total derivative") into the total differential formula.

Gif" width="33" height="19 src=">.gif" width="33" height="19 src=">.gif" width="140" height="44 src=">

Then the formula for the first order total differential of a function of two variables has the form

http://pandia.ru/text/78/481/images/image180.gif" width="139" height="41 src=">

Comparing the last equality with the formula for the first differential of a function of two independent variables, we can say that the expression for the complete first-order differential of a function of several variables has the same form as it would have if u and v were independent variables.

In other words, the form of the first differential is invariant, that is, it does not depend on whether the variables u and v are independent variables or depend on other variables.

EXAMPLE

Find the first order total differential of a complex function

z=u2v3, u=x2 sin y, v=x3·ey.

Solution. Using the formula for the first order total differential, we have

dz = 2uv3 du+3u2v2 dv =

2uv3 (2x sin y·dx+x2·cos y·dy)+3u2v2·(3x2·ey·dx+x3·ey·dy).

This expression can be rewritten like this

dz=(2uv3 2x siny+3u2v2 3x2 ey) dx+(2uv3x2 cozy+3u2v2x3 ey) dy=

The invariance property of a differential allows us to extend the rule for finding the differential of a sum, product, and quotient to the case of a function of several variables:

http://pandia.ru/text/78/481/images/image183.jpg" width="409" height="46 src=">

http://pandia.ru/text/78/481/images/image185.gif" width="60" height="41 src=">. This

the function will be homogeneous of the third degree for all real x, y and t. The same function will be any homogeneous polynomial in x and y of the third degree, i.e. such a polynomial in each term of which the sum of the exponents xn is equal to three:

http://pandia.ru/text/78/481/images/image187.jpg" width="229" height="47 src=">

are homogeneous functions of degrees 1, 0 and (- 1) respectively..jpg" width="36" height="15">. Indeed,

http://pandia.ru/text/78/481/images/image191.jpg" width="363" height="29 src=">

Assuming t=1, we find

http://pandia.ru/text/78/481/images/image193.jpg" width="95" height="22 src=">

Partial derivatives http://pandia.ru/text/78/481/images/image195.jpg" width="77" height="30 src=">), in general

In other words, they are functions of the variables x and y. Therefore, partial derivatives can again be found from them. Consequently, there are four second-order partial derivatives of a function of two variables, since each of the functions and can be differentiated with respect to both x and y.

The second partial derivatives are denoted as follows:

is the nth order derivative; here the function z was first differentiated p times with respect to x, and then n - p times with respect to y.

For a function of any number of variables, partial derivatives of higher orders are determined similarly.

P R And m e r 1. Calculate second order partial derivatives of a function

http://pandia.ru/text/78/481/images/image209.jpg" width="600" height="87 src=">

Example 2. Calculate and http://pandia.ru/text/78/481/images/image212.jpg" width="520" height="97 src=">

Example 3. Calculate if

http://pandia.ru/text/78/481/images/image215.jpg" width="129" height="36 src=">

x, f"y, f"xy and f"yx are defined and continuous at the point M(x, y) and in some of its neighborhood, then at this point

http://pandia.ru/text/78/481/images/image218.jpg" width="50 height=28" height="28">.jpg" width="523" height="128 src=">

Hence,

http://pandia.ru/text/78/481/images/image222.jpg" width="130" height="30 src=">

Solution.

Mixed derivatives are equal.

5.10. Higher order differentials of a functionnvariables.

Total differential d u functions of several variables are in turn a function of the same variables, and we can determine the total differential of this last function. Thus, we will obtain a second-order differential d2u of the original function and, which will also be a function of the same variables, and its complete differential will lead us to a third-order differential d3u of the original function, etc.

Let us consider in more detail the case of the function u=f(x, y) of two variables x and y and assume that the variables x and y are independent variables. A-priory

http://pandia.ru/text/78/481/images/image230.jpg" width="463" height="186 src=">

Calculating d3u in exactly the same way, we get

http://pandia.ru/text/78/481/images/image232.jpg" width="347" height="61 src="> (*)-

Moreover, this formula should be understood as follows: the amount worth parentheses, must be raised to the power n, using Newton's Binomial Formula, after which the exponents of y and http://pandia.ru/text/78/481/images/image235.jpg" width="22" height="21 src=" >.gif" width="22" height="27"> with direction cosines cos α, cos β (α + β = 90°). On the vector, consider the point M1(x + Δx; y + Δy). When moving from point M to point M1, the function z = f(x; y) will receive a full increment

http://pandia.ru/text/78/481/images/image239.jpg" width="133 height=27" height="27"> tending to zero (see figure).

http://pandia.ru/text/78/481/images/image241.jpg" width="324" height="54 src=">

where http://pandia.ru/text/78/481/images/image243.gif" width="76" height="41 src=">, and therefore we get:

http://pandia.ru/text/78/481/images/image245.gif" width="24" height="41 src="> for Δs->0 is called the product

water function z = f(x; y) at the point (x; y) in the direction of the vector and is denoted

http://pandia.ru/text/78/481/images/image247.jpg" width="227" height="51 src="> (*)

Thus, knowing the partial derivatives of the function

z = f(x; y) you can find the derivative of this function in any direction, and each partial derivative is a special case of the directional derivative.

EXAMPLE Find the derivative of a function

http://pandia.ru/text/78/481/images/image249.jpg" width="287" height="56 src=">

http://pandia.ru/text/78/481/images/image251.jpg" width="227" height="59 src=">

http://pandia.ru/text/78/481/images/image253.gif" width="253 height=62" height="62">

Consequently, the function z = f(x;y) increases in a given direction.

5. 12 . Gradient

The gradient of a function z = f(x; y) is a vector whose coordinates are the corresponding partial derivatives of this function

http://pandia.ru/text/78/481/images/image256.jpg" width="205" height="56 src=">

i.e..jpg" width="89" height="33 src=">

at point M(3;4).

Solution.

http://pandia.ru/text/78/481/images/image259.jpg" width="213" height="56 src=">

several functions

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Graphing a function online

instantly.

Online service instantly draws a graph

Absolutely supported All mathematical functions

Trigonometric functions
		Cosecant	Cotangent	arcsine	arc cosine	Arctangent	Arcsecant	Arccosecant	Arccotangent

Hyperbolic functions

Other
Natural logarithm	Logarithm	Square root				Round down		Round up

Minimum				Maximum
min(expression1,expression2,…)				max(expression1,expression2,…)

Graph the function

Construction of a 3D surface

Enter the equation

Let us construct a surface defined by the equation f(x, y, z) = 0, where a< x < b, c < y < d, m < z < n.

Other examples:

y = x^2
z = x^2 + y^2
0.3 * z^2 + x^2 + y^2 = 1
z = sin((x^2 + y^2)^(1/2))
x^4+y^4+z^4-5.0*(x^2+y^2+z^2)+11.8=0

Canonical view of curve and surface

You can determine the type of curve and 2nd order surface online with a detailed solution:

Rules for entering expressions and functions

Expressions can consist of functions (notations are given in alphabetical order):

absolute(x) Absolute value x
(module x or |x|) arccos(x) Function - arc cosine of xarccosh(x) Arc cosine hyperbolic from xarcsin(x) Arcsine from xarcsinh(x) Arcsine hyperbolic from xarctan(x) Function - arctangent of xarctgh(x) Arctangent hyperbolic from xee a number that is approximately equal to 2.7 exp(x) Function - exponent of x(which is e^x) log(x) or ln(x) Natural logarithm of x
(To obtain log7(x), you need to enter log(x)/log(7) (or, for example, for log10(x)=log(x)/log(10)) pi The number is "Pi", which is approximately equal to 3.14 sin(x) Function - Sine of xcos(x) Function - Cosine of xsinh(x) Function - Hyperbolic sine of xcosh(x) Function — Hyperbolic cosine of xsqrt(x) Function - Square root from xsqr(x) or x^2 Function - Square xtan(x) Function - Tangent from xtgh(x) Function — Tangent hyperbolic from xcbrt(x) Function - cube root of xfloor(x) Function - rounding x downward (example floor(4.5)==4.0) sign(x) Function - Sign xerf(x) Error function (Laplace or probability integral)

The following operations can be used in expressions:

Real numbers enter as 7.5 , Not 7,5 2*x- multiplication 3/x- division x^3- exponentiation x+7- addition x - 6- subtraction

How to graph a function online on this site?

To plot a function online, you just need to enter your function in a special field and click somewhere outside it. After this, the graph of the entered function will be drawn automatically. Let's say you want to build a classic graph of the "x squared" function. Accordingly, you need to enter “x^2” in the field.

If you need to plot several functions at the same time, then click on the blue “Add more” button. After this, another field will open in which you will need to enter the second function. Its schedule will also be built automatically.

You can adjust the color of the graph lines by clicking on the square located to the right of the function input field. The remaining settings are located directly above the graph area. With their help, you can set the background color, the presence and color of the grid, the presence and color of the axes, the presence of marks, as well as the presence and color of the numbering of graph segments. If necessary, you can scale the function graph using the mouse wheel or special icons in the lower right corner of the drawing area.

After plotting and entering necessary changes in settings, you can download chart by using big green"Download" buttons at the very bottom. You will be prompted to save the function graph as a PNG image.

Why do you need to graph a function?

On this page you can build interactive chart online functions.

Graph a function online

Plotting a graph of a function allows you to see the geometric image of a particular mathematical function. To make it more convenient for you to build such a graph, we have created a special online application. It's completely free, doesn't require registration, and can be used directly in your browser without any hassle. additional settings and manipulation. Build graphs for various functions Most often it is required for middle and high school students studying algebra and geometry, as well as first and second year students as part of higher mathematics courses. Usually, this process It takes a lot of time and requires a lot of office supplies to draw the graph axes on paper, put down coordinate points, connect them with a straight line, etc. Using this online service you can calculate and create graphic image functions instantly.

How does a graphing calculator work for graphing functions?

Online service It works very simply. The function (i.e. the equation itself, the graph of which needs to be plotted) is entered into the field at the very top. Immediately after entering the application instantly draws a graph in the area below this field. Everything happens without refreshing the page. Next, you can enter various color settings, as well as hide/show some elements of the function graph. After that, ready schedule can be downloaded by clicking on the appropriate button at the very bottom of the application. The drawing will be downloaded to your computer in .png format, which you can print or transfer to a paper notebook.

What features does the graph builder support?

Absolutely supported all mathematical functions, which can be useful when plotting graphs. It is important to emphasize here that, in contrast to the classical language of mathematics adopted in schools and universities, the degree sign within the application is designated international sign"^". This is due to the lack of the ability to write a degree in the usual format on a computer keyboard. Below is a table with full list supported functions.

The application supports the following functions:

Trigonometric functions
		Cosecant	Cotangent	arcsine	arc cosine	Arctangent	Arcsecant	Arccosecant	Arccotangent

Hyperbolic functions

Other
Natural logarithm	Logarithm	Square root				Round down		Round up

Minimum				Maximum
min(expression1,expression2,…)				max(expression1,expression2,…)

Examples. Construct function level lines corresponding to the values

Construct function level lines corresponding to the values .

Assuming , we obtain the equations of the corresponding level lines:

By constructing these lines in the Cartesian coordinate system xOy, we obtain straight lines parallel to the bisector of the second and fourth coordinate angles (Fig. 1)

Let's write the equations of the level lines:

, , , And .

By constructing them in the xOy plane, we obtain concentric circles with the center at the origin of coordinates (Fig. 2)

The level lines of this function , , , and are parabolas symmetrical with respect to Oy with a common vertex at the origin (Fig. 3).

2. Directional derivative

An important characteristic of a scalar field is the rate of change of the field in a given direction.

To characterize the rate of change of the field in the direction of the vector, the concept of the derivative of the field in direction is introduced.

Consider the function at point and point.

Let's draw through the points and the vector. The angles of inclination of this vector to the direction of the coordinate axes x, y, z let's denote a, b, g, respectively. The cosines of these angles are called direction cosines vector

LECTURE NOTES ON MATANALYSIS

Functions of several variables. Geometric representation of a function of two variables. Level lines and surfaces. Limit and continuity of functions of several variables, their properties. Partial derivatives, their properties and geometric meaning.

Definition 1.1. 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 1.2. A bunch of M, in which the variables are specified x,y, called domain of the function, and themselves x,y- her arguments.

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

Examples.

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
, which are arguments to a function of several variables.

Definition 1.3. . Variable z (with change area Z) called function of several independent variables
in abundance M, if each set of numbers
from many M according to some rule or law, one specific value is assigned z from Z. The concepts of arguments and domain are introduced in the same way as for a function of two variables.

Designations: z = f
,z = z
.

Geometric representation of a function of two variables.

Consider the function

z = f(x, y) , (1.1)

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 (1.1) defines a certain surface in three-dimensional space, it will be the geometric image of the function under consideration.

z = f(x,y)

M y

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 (1.1), 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 are x² + y² = 4 – c (c=const) – equations of concentric circles with a center at the origin and with radii
. For example, when With=0 we get a circle x² + y² = 4.

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

Example.

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

3x + 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 with center 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
- point coordinates M 0 . Sometimes this set is called a “ball” in n-dimensional space.

Definition 1.4. 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 1.5. Function f
called continuous at the point M 0
, If
(1.2)

If we introduce the notation

That condition (1.2) can be rewritten in the form

(1.3)

Definition 1.6. Inner point M 0 function domain z = f (M) called break point function if conditions (1.2), (1.3) are not satisfied at this point.

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

When processing data in subject areas related to scientific activities, there is often a need to construct and visualize a function of two independent variables. A typical example is a necessity visual representation results of solving two-dimensional differential equations in partial derivatives, obtained in the form of so-called grid functions.

A simple class is proposed for constructing level lines (isolines) of the function: Z=F(X,Y) in the form of lines on X-Y plane, satisfying the equations Z=const (where const is a set of given values).

It is assumed that the function Z is specified as an array z on an arbitrary grid with quadrangular cells. The grid is specified by two arrays x, y, where J and K are the grid sizes.

The function values are defined in the corners of the quadrilateral cell. In each cell, the passage of the calculated level line through its faces is checked and, provided that the line passes through the cell, the coordinates of the intersection of the level line with the faces are calculated. Inside the cell, the line is drawn as a straight segment.

The source text is provided with detailed comments.

File LinesLevels.cs:

Using System.Collections.Generic; using System.Linq; using System.Windows; namespace WpfLinesLevels ( public class LinesOfLevels ( private int J, K; private double[,] X; private double[,] Y; private double[,] Z; // List of isolines public List Lines ( get; set; ) ///

/// Preparation ///

/// Array of levels /// X area coordinates /// Y coordinates of the area /// Grid function public LinesOfLevels(double _levels, double[,] _x, double[,] _y, double[,] _z) ( Lines = new List (_levels.Count());

foreach (double l in _levels) ( Lines.Add(new LineLevel(l)); ) X = _x;

Y = _y;< J - 1; j++) for (int k = 0; k < K - 1; k++) { Ceil ir = new Ceil(j, k, X, Y, Z); for (int l = 0; l < Lines.Count(); l++) ir.AddIntoLineLevel(Lines[l]); } } } ///

Z = _z;

J = X.GetLength(0); K = X.GetLength(1); (); } } ///

) ///

/// Calculation of isolines. /// (); } } ///

public void Calculate() ( for (int j = 0; j

/// One isoline ///

public class LineLevel ( // List of isoline points in the form of pairs of points // belonging to the same quadrangular cell public List

Pairs ( get; set; ) // Isoline level public double Level ( get; set; ) public LineLevel(double _level) ( Level = _level; Pairs = new List

/// A pair of isoline points belonging to the same cell /// public class PairOfPoints ( public List Points ( get; set; ) public PairOfPoints() ( Points = new List

/// /// Cell angle. /// /// Indices for defining one corner of a quadrilateral cell /// /// internal struct Dot ( internal int j ( get; set; ) internal int k ( get; set; ) internal Dot(int _j, int _k) ( j = _j; k = _k; ) ) /// /// /// Quadrangular grid cell. Determines the current cell. /// /// Calculates isoline segments in a cell /// internal class Ceil ( // Cell corners private Dot d = new Dot; // Coordinate points of corners private Point r = new Point; // Arrays of coordinates of the entire area private double[,] X; private double[,] Y; // Array grid function private double[,] Z; ///

/// Cell definition /// Defined by the lower left corner. Index iteration cycles should be 1 less dimensions J,K arrays ///

/// j - index of the lower left corner /// k - index of the lower left corner

Array X Array Y ///

/// j - index of the lower left corner /// Grid function array Z

internal Ceil(int _j, int _k, double[,] _x, double[,] _y, double[,] _z) ( d = new Dot(_j, _k); d = new Dot(_j + 1, _k); d = new Dot(_j + 1, _k + 1); ); r = dotPoint(d); r = dotPoint(d);

/// /// Definition of coordinate /// points< _l) || (dotZ(d) >angle ///<= _l)) { double t = (_l - dotZ(d)) / (dotZ(d) - dotZ(d)); double x = r.X * t + r.X * (1 - t); double y = r.Y * t + r.Y * (1 - t); p.Points.Add(new Point(x, y)); } // Ребро 1 if ((dotZ(d) >Angle defined by Dot structure< _l) || (dotZ(d) >angle ///<= _l)) { double t = (_l - dotZ(d)) / (dotZ(d) - dotZ(d)); double x = r.X * t + r.X * (1 - t); double y = r.Y * t + r.Y * (1 - t); p.Points.Add(new Point(x, y)); if (p.Points.Count == 2) return p; } // Ребро 2 if ((dotZ(d) >Angle defined by Dot structure< _l) || (dotZ(d) >angle ///<= _l)) { double t = (_l - dotZ(d)) / (dotZ(d) - dotZ(d)); double x = r.X * t + r.X * (1 - t); double y = r.Y * t + r.Y * (1 - t); p.Points.Add(new Point(x, y)); if (p.Points.Count == 2) return p; } // Ребро 3 if ((dotZ(d) >Angle defined by Dot structure< _l) || (dotZ(d) >angle ///<= _l)) { double t = (_l - dotZ(d)) / (dotZ(d) - dotZ(d)); double x = r.X * t + r.X * (1 - t); double y = r.Y * t + r.Y * (1 - t); p.Points.Add(new Point(x, y)); } return p; } ///

private Point dotPoint(Dot _d) ( return new Point(X[_d.j, _d.k], Y[_d.j, _d.k]); ) ///

/// /// Function definition in internal void AddIntoLineLevel(LineLevel _lL) ( PairOfPoints lp = ByLevel(_lL.Level); if (lp.Points.Count > 0) _lL.Pairs.Add(lp); ) ) )
To demonstrate how the class works, a small WPF test application is offered that builds level lines for a function of the form: z = x^2 + y^2 on a 10 by 10 grid.

MainWindow.xaml file:

And the MainWindow.xaml.cs code file:

Using System.Linq; using System.Windows; using System.Windows.Controls; using System.Windows.Media; using System.Windows.Shapes; namespace WpfLinesLevels ( ///

/// Interaction logic for MainWindow.xaml ///

public partial class MainWindow: Window ( private double Xmax; private double Xmin; private double Ymax; private double Ymin; private double xSt; private double ySt; public MainWindow() ( InitializeComponent(); // Defining the levels that will be displayed double levels = ( 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 ); double[,] X = new double[,] Y = new double[,] Z = new double; double; // Variables for converting physical coordinates to screen ones Xmin = 0; Ymin = 0; xSt = 525 / (Xmax - Xmin); arrays of coordinates and functions for (int k = 0; k< 10; k++) for (int j = 0; j < 10; j++) { X = j; Y = k; Z = j * j + k * k; } // Создание изолиний LinesOfLevels lol = new LinesOfLevels(levels, X, Y, Z); // Их расчет lol.Calculate(); // Построение DrowLevelLine(lol, X, Y); } ///

/// Method for constructing isolines ///

/// Calculated object with isolines /// array of X coordinates /// array of Y coordinates private void DrowLevelLine(LinesOfLevels lL, double[,] x, double[,] y) ( Canvas can = new Canvas(); foreach (LineLevel l in lL.Lines) ( foreach (PairOfPoints pp in l.Pairs) ( if ( pp.Points.Count() == 2) ( Line pl = new Line(); pl.Stroke = new SolidColorBrush(Colors.BlueViolet); pl.X1 = xCalc(pp.Points.X); pl.X2 = xCalc (pp.Points.X); pl.Y1 = yCalc(pp.Points.Y); pl.Y2 = yCalc(pp.Points.Y); can.Children.Add(pl) ) can.Margin; new Thickness(10, 10, 10, 10); can.VerticalAlignment = VerticalAlignment.Stretch; can.HorizontalAlignment = HorizontalAlignment.Stretch;

/// Converting the physical coordinate X to the screen coordinate ///

/// Physical coordinate X /// Screen X coordinate private double xCalc(double _x) ( return xSt * (_x - Xmin); ) ///

/// Converting the physical Y coordinate to the screen coordinate ///

/// Physical coordinate Y /// Screen Y coordinate private double yCalc(double _y) ( return ySt * (Ymax - _y); ) ) )
The result of the test example is shown in the figure.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Defining a function of several variables

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

Example 1. Square S rectangle with sides whose lengths are equal X And at, is expressed by the formula S = xy. Each pair of values X And at corresponds to a certain area value S; S is a function of two variables.

Example 2. Volume V rectangular parallelepiped with edges whose lengths are equal X, at, z, is expressed by the formula V= xyz. Here V there is a function of three variables X, at, z.

Example 3. Range R flight of projectiles fired at initial speed v 0 from a gun whose barrel is inclined to the horizontal at an angle  is expressed by the formula
(if we neglect air resistance). Here g– acceleration of gravity. For each pair of values v 0 and  this formula gives a certain value R, i.e. R is a function of two variables v 0 and .

Example 4.
. Here And there is a function of four variables X, at, z, t.

Definition 1. If each pair ( X, at) values of two independent from each other variables X And at from some area of their change D, corresponds to a certain value of the quantity z, then we say that z there is a function two independent variables x And at, defined in the area D.

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

z= f(x, y), z = F(x, y) etc.

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

S = xy

In this table, at the intersection of a row and a column corresponding to certain values X And at, the corresponding function value is entered S. If functional dependence z= f(x, y) is obtained as a result of measurements of the quantity z When experimentally studying any phenomenon, a table is immediately obtained that determines z as a function of two variables. In this case, the function is specified only by the table.

As in the case of one independent variable, a function of two variables does not exist, generally speaking, for any values X And at.

Definition 2. A set of pairs ( X, at) values X And at, at which the function is determined z= f(x, y), called domain of definition or area of existence this function.

The domain of definition of a function is clearly illustrated geometrically. If every pair of values X And at we will represent it with a dot M(X, at) in the plane Ohoo, then the domain of definition of the function will be depicted as a certain collection of points on the plane. We will also call this collection of points the domain of definition of the function. In particular, the domain of definition can be the entire plane. In what follows we will mainly deal with areas such as parts of the plane, bounded by lines. The limiting line this area, we will call border areas. Points of the region that do not lie on the boundary will be called internal points of the area. An area consisting of only interior points is called open or open. If the boundary points also belong to the region, then the region is called closed. An area is called bounded if there is such a constant WITH, that the distance of any point M area from the origin ABOUT less WITH, i.e. | OM| < WITH.

Example 5. Determine the natural domain of a function

z = 2X – at.

Analytical expression 2 X – at makes sense for any value X And at. Consequently, the natural domain of definition of the function is the entire plane Ohoo.

Example 6.
.

In order to z had a real value, it is necessary that there be a non-negative number under the root, i.e. X And at must satisfy the inequality 1 – X 2 – at 2  0, or X 2 + at 2  1.

All points M(X, at), whose coordinates satisfy the indicated inequality, lie in a circle of radius 1 with a center at the origin and on the boundary of this circle.

Example 7.
.

Since logarithms are defined only for positive numbers, the inequality must be satisfied X + at> 0, or at >  X.

This means that the domain of definition of the function z is the half of the plane located above the line at =  X, not including the straight line itself.

Example 8. Area of a triangle S represents the base function X and heights at: S= xy/2.

The domain of definition of this function is the domain X  0, at 0 (since the base of a triangle and its height can be neither negative nor zero). Note that the domain of definition of the function under consideration does not coincide with the natural domain of definition of the analytical expression with which the function is specified, since the natural domain of definition of the expression xy/ 2 is obviously the entire plane Ohoo.

The definition of a function of two variables can easily be generalized to the case of three or more variables.

Definition 3. If each considered set of variable values X, at, z, …, u, t corresponds to a certain variable value w, then we will call w function of independent variables X, at, z, …, u, t and write w= F(X, at, z, …, u, t) or w= f(X, at, z, …, u, t) and so on.

Just as for a function of two variables, we can talk about the domain of definition of a function of three, four or more variables.

So, for example, for the function of three variable area The definition is a certain collection of triples of numbers ( X, at, z). Let us immediately note that each triple of numbers defines a certain point M(X, at, z) in space Ohooz. Consequently, the domain of definition of a function of three variables is a certain set of points in space.

Similarly, we can talk about the domain of definition of a function of four variables u= f(x, y, z, t) as about some collection of quadruples of numbers ( x, y, z, t). However, the domain of definition of a function of four or more variables no longer allows for a simple geometric interpretation.

Example 2 shows a function of three variables defined for all values X, at, z.

Example 4 shows a function of four variables.

Example 9. .

Here w– function of four variables X, at, z, And, defined with values of variables satisfying the relation:

Concept of a function of several variables

Let us introduce the concept of a function of several variables.

Definition 1. Let every point M from a set of points ( M) Euclidean space Em according to some law, a certain number is put into correspondence And from a numerical set U. Then we will say that on the set ( M) function is given and =f(M). Moreover, the sets ( M) And U are called, respectively, the domain of definition (assignment) and the domain of change of the function f(M).

As you know, a function of one variable at = f(x) is depicted on the plane as a line. In the case of two variables, the domain of definition ( M P) functions z = f(x, y) represents a certain set of points on the coordinate plane Ohoo(Fig. 8.1). Coordinate z called applicate, and then the function itself is depicted as a surface in space E3 . Similarly, the function from T variables