Lecture 1. Functions of Multiple Variables
In this lecture, we begin the section "Functions of Multiple Variables". We have already discussed the concept of a function of a single real variable, but our life is arranged in such a way that little depends only on the change of a single variable. Our life is so diverse, so many factors affect changes in quantities that it is logical to generalize this concept – the concept of functions of multiple variables.
Today we will focus in more detail on the concept of a function of two variables. First, let's focus on the notation. R is a numeric line.
If we give R an upper index, this shows the dimension of the space. R^2 is a coordinate plane, R^3 is a three-dimensional space, and R^n is an arithmetic n-dimensional space.
Elements are n real numbers: numbers are in R, pairs of real numbers of points in the plane are in R^2, triples of real numbers are in R^3, these are points of three-dimensional space.
Let's consider a three-dimensional space R^3, it is defined by three mutually perpendicular coordinate axes: Ox is the abscissa axis, Oy is the ordinate, and Oz is the applicate.
How are the coordinate planes set, there are also three of them, and how are the coordinate axes set? When we move on to solving problems, we will need this knowledge.
The XOY coordinate plane is set by the condition z=0.
Any point has three coordinates.
If it lies in the XOY plane, the third coordinate is z=0, in XOZ y=0, and in YOZ x=0. How can we memorize it easily? In the designation of the plane XOY z is missing, so it is 0, in XOZ y is missing, so y=0, and the third is clear.
Coordinate axis. The Ox axis is set by a system of two conditions (see the video). There is neither z nor y in the axis designation. Thus, these coordinates are 0 on the Ox axis at any point. The OY axis, respectively, is set by the condition z=0, x=0, and on the Oz axis we have x=0, y=0.
Now let's define the function of two variables.
There is a certain set of points G in the plane R^2. If each pair of numbers, a point belonging to the set G, according to some law f corresponds to a single number z, then we say that a function of two variables z=f(x, y) is given.
x and y are independent variables, there are two of them, and z is dependent, its value depends on what values x and y take.
The set G, which is the set of points in the plane R^2, is the domain of definition. The value is real numbers.
Let's consider the tasks that we should be able to solve.
Number 1 (watch the video): find the domain of the function of two variables. The function is defined analytically by the formula. What is its domain?
The domain is a set of the pairs (x, y), they are in the plane; for them the z value can be calculated using this formula (watch the video), there is such a language.
To solve this problem, we use the generalized interval method.
If you remember the method of intervals, then note that our reasoning is very similar to the points of the method of intervals.
Where do we start? We write down the inequation, it shows for which x, y z exist.
Logically we decompose it into multipliers, and we get the inequation (watch the video). We don't need to solve it yet.
We proceed to the first point of the generalized interval method. We set the boundaries of the domain. How can we do this? In the inequation, we pass to the equation and we have the following zeros: where x-y=0 and where x+y=0. If we obtain the points of a numerical line in the interval method, here we get curves on the plane; these are the lines y=x and y=-x.
Let's look again at the inequation we are solving. It is strict, so the points of a straight line are not a solution, we exclude them and mark them with dotted lines.
As a result, the plane is divided into four parts. We take a point in each part and check whether the inequation sign is followed or not.
For example, there are the points (1, 0) and (-1, 0) in the marked areas.
If we substitute (1, 0) or (-1, 0) in the inequation, then the sign of the inequation is greater than 0, plus, everything is fulfilled, so these areas are stroked. However, in the points (0, 1), (0, -1) the product is less than 0, and we don't stroke it. Here is the solution.
While finding the domain of the function of one variable, we write out the answer and note that the domain is a set. The most significant difference here: we do not write the answer in this form; the answer is the construction of a domain.
This is the answer to this problem.
Let's take another example (watch the video).
The function is set analytically. Where do we start? We write an inequation that allows us to determine for which values of x, y the function value exists.
The first stage: we set the boundaries of the domain, these are the zeros of the numerator and denominator.
In the first and second cases, we get the equation of a straight line, but the numerator can be zero, so we will build a solid line, a solid straight line, and the denominator does not turn to zero, so we draw a dotted line x=3.
Here is the first stage on the plane: we set the boundaries of the domain.
The second stage: in each of the domains (watch the video); these red lines divide the plane into four Let’s take, for example, the point (0, 0). We can see which part of the domain it is in. We substitute and see that the inequation is fulfilled. This means that the entire area containing the point O will be stroked, but at the point with coordinates (0, 5), this is on the axis Oy, the inequation is not fulfilled.
Therefore, we are not going to stroke this part. We check in two more parts in the same way, and as a result, the hatching gives us the answer. The answer is a stroked set of points in the XOY plane.
The next task that often occurs is to plot a function of two variables.
What is a graph? The graph of a function is a set of points with three coordinates (x, y, z).
They are already in three-dimensional space.
What are these points?
The first pair (x, y) are points from the domain, and the third number in this triple is the corresponding value of the function at this point. This is a point in three-dimensional space, the first two coordinates define a point in the X and Y plane, and the third one is the value of the function at the specified point.
Thus, the function definition area is the projection of the graph on the X and Y plane.
What is the most important property of the graph: any vertical straight line parallel to the Oz axis must intersect the surface, in general, at no more than one point, otherwise it will not be a function.
Let's move on to the task of plotting a graph. In general, it is possible to set it in a different way if we can start on the basis of some ideas. We are going to use the cross-section method.
There are two examples: you need to plot function graphs (watch the video). We are going to have a section of analytical geometry later, but now, looking a little ahead, note that the first case is an equation of the plane, and we will start with this knowledge. Let’s set a plane.
Leet’s solve Example One. Here is the task for you: you need to set a plane, a graph of this function. We set a coordinate system. What should we do to solve it? We determine the points at which this plane intersects the coordinate axes. The Ox axis is set by the condition z=0, z=0.
If we substitute z and y in the equation, we get x=3.
This plane intersects the Ox axis at point 3. the Next axis is Oy, y=0, z=0. Thus, y=2. The third case: the Oz axis appears on the z-axis at point 6.
What should we do next? We know perfectly well that if any two points lie in a plane, then the line connecting these points also lies in this plane, so we connect each pair of points with straight lines.
As a result, we have a three-dimensional image of this plane (watch the video).
The problem is solved.
Try to plot the function of task 2 yourself, and the answer here is a surface called a paraboloid of rotation (watch the video).
Another illustration that allows us to visualize the function of two variables is the concept of level lines. Where do we have them?
In general, this concept is related to maps in cartography, when we want to depict the terrain without having any volumetric capabilities, and the level line shows us the height of the terrain.
What is a level line for functions of two variables? This is the curve given by the equation f(x, y)=c, where c is a real number.
What does this level line show?
It shows which line intersects the surface at the height c, z=c. This is the intersection of the given surface z=f(x, y) and the plane z=c at the height c above the XOY plane. Then this line is projected onto the XOY plane, and the level line appears.
Let's take some examples. We have reviewed the functions, built the graphs, and concluded that (watch the video): the first is a plane; the second case is a paraboloid of rotation.
What are their level lines?
We write the level lines (watch the video): this is an equation connecting the variables x and y, with the number c on the right.
If we express y in terms of x, we see that all these lines have the same angular coefficient, all these lines are parallel, and each c has its own image.
(Watch the video) c=0 means that at the height of 0, this plane intersects the XOY plane in a straight line, we depict it as c=0.
All others are drawn parallel, and at c=6, the level line passes through the point (0, 0). Here is the level line (watch the video).
Let's consider the second surface (watch the video).
It is clear that the level line is a circle. For different c's, we get different circles with the same center, these ar concentric circles.
If c=0, this is the origin coordinate point.
At the height of 4, we intersect the paraboloid of rotation by the circle, and we mark c=4. There is the next circle at c=9, and so on.
Thus, we set level lines using cartography techniques and can represent the relief of these surfaces