Lecture ¹1

# Lecture ¹1. The functions of many variables and their properties. The limits and continuity

# Exercises for Lecture #1

The purpose of work:

The acquaintance with the functions of many variables and there properties, the limits and continuity.

Theoretical information and examples

Content

1.1 Basic Concepts

1.1.1 The concept of a function of several variables

1.1.2 Domain of definition of a function:

1.1.3 Level lines and level surfaces of a function

1.2 Continuity:

1.2.1 The limit of a function:

1.2.2 Continuity and points of discontinuity:

1.1 Basic Concepts

1.1.1 The concept of a function of several variables

Functional notation. A variable quantity z is called a single-valued function of the two variables x, y, if there corresponds to each set of their values (x, y) in a given range a unique value of z. The variables x and y are called arguments or independent variables. The functional relation is denoted by

We define functions of three or more arguments in a similar manner.

Example 1. Express the volume of a cone V as a function of its generator x and its base radius y.

Solution. We know from geometry that the volume of a cone is given by

where h is the height of the cone. Since , the required functional relation is

The value of the function z = f(x, y) at a point P(a, b), i.e., when x=a and y=b, is denoted by f(a, b) or f(P). Generally speaking, the geometric representation of a function like z = f(x, y) in a rectangular co-ordinate system X, Y, Z is a surface (Fig. 1).

Example 2. Find f(2, -3) and f(l, y/x) if

Solution. Substituting x = 2 and y = - 3, we find

Setting x = 1 and replacing y by y/x, we find

i.e., f(1, y/x) = f(x, y).

1.1.2 Domain of definition of a function. We understand by the domain of definition of a function z = f(x, y) a set of points (x, y) in an xy-plane in which the given function is defined (i.e., in which it takes on definite real values). In the simplest cases, the domain of definition of a function is a finite or infinite part of the xy-plane bounded by one or several curves (the boundary of the domain).

Similarly, for a function of three variables , the domain of its definition is a volume in xyz-space.

Example 3. Find the domain of definition of the function

Solution. The function has real values if or . The last inequality is satisfied by the co-ordinates of points lying inside a circle of radius 2 with centre at the origin. The domain of definition of .the function is the interior of the circle (Fig. 2).

Example 4. Find the domain of definition of the function

.

Solution. The first term of the function is defined for or . The second term has real values if xy ≥ 0, i.e., in the two cases when

.

The domain of definition of the entire function is shown in Fig. 3 and includes the boundaries of the domain.

1.1.3 Level lines and level surfaces of a function: The level line of a function z = f(x,y) is a line f(x,y)=C (in an x,y-plane) at the points of which the function takes on one and the same value z = C (usually labelled in drawings).

The level surface of a function of three arguments u = f(x. y, z) is the surface f(x,y,z)=C, at the points of which the function takes on a constant value u=C.

Example 5. Construct the level lines of the function

Solution. The equation of the level lines has the form

Setting C = 0, ±1, ± 2, ..., we get a family of level lines (Fig. 4).

1.2 Continuity

1.2.1 The limit of a function. A number A is called the limit of a function z=f(x, y) as the point P'(x, y) approaches the point P(a, b), if there exists for any e > 0 a d > 0 such that, when 0 < r < d, where r = [(x -a)² +(y - b)²]½ is the distance between P and P', we have the inequality

In this case, we write

1.2.2 Continuity and points of discontinuity: A function z = f(x, y) is called continuous at a point P(a, b), if

.

A function which is continuous at all points of a given range is said to be continuous over this range.

A function f(x, y) may cease to be continuous either at separate points (isolated points of discontinuity) or at points which form one or several lines (lines of discontinuity) or at times more complex geometric objects.

Example 1. Find the discontinuities of the function

Solution. The function will have no meaning if the denominator becomes zero. But x² - y = 0 or y = x² is the equation of a parabola, whence the given function has for its discontinuity the parabola y = x².

Control questions:

1. What is called a single-valued function of the two variables x, y?

2. What we understand by the domain of definition of a function z = f(x, y)?

3. What is the level line of a function z = f(x,y)?

4. What is the level surface of a function of three arguments u = f(x. y, z)?

5. What is called the limit of a function z=f(x, y) as the point P'(x, y)