Now we are going to start talking about functions of several variables. First of all, real-valued functions of several independent real variables are defined in much the same way as in the single-variable case. The domains are sets of ordered pairs (triples, n-tuples) of real numbers, and the range is a set of real numbers.

DEFINITION:

Suppose D is a set of n-tuples of real numbers (x ₁, x ₂, . . . x _n ). A real-valued function for D is a rule that assigns a real number w = f (x ₁, x₂, . . . x _n ) to each element of D. The set D is the function's domain. The set of w-values taken on by f is the function's range. The symbol w is the dependent variable of f, and f is said to be a function of the n independent variables x ₁ to x _n.

FACT:

If f is a function of two independent variables x and y, then the domain of f is a region in the xy-plane.

FACT:

If f is a function of three independent variables, x, y, and z, then the domain of f is a region in space.

When determining the domain of a function of more than one variable, we will exclude inputs that lead to complex numbers or division by zero.

 EXAMPLE 1:

Find the domain and range for f (x, y) = 4x ² + 9y ². 

SOLUTION:

Since this function has no denominator or even radical, then the domain for this function will be the following. D: {(x, y) | x Î Â and y Î Â }.

Since this function will always be greater than or equal to zero, the range will be the following. R: {z | z ³ 0}

 EXAMPLE 2:

Find the domain and the range for the following function.

SOLUTION:

Since this function has an even radical in it, we will have to determine what values make the radicand greater than or equal to zero.

9 - x ² - y ² ³ 0 ® 9 ³ x ² + y ²

Therefore, the domain for this function is {(x, y) | x ² + y ² £ 9}.

The surface that we are looking at is the upper half of the sphere, so the range is the following. R: {z | 0 £ z £ 3}

 EXAMPLE 3:

Find the domain and range for f (x, y) = ln (x ² + y ²).

SOLUTION:

Since the value of the argument of a natural log must be greater than zero, this implies that x ² + y ² > 0. Therefore the point (0, 0) is not in the domain of this function. Here is the domain of this function.

D: {(x, y) | (x, y) ¹ (0, 0)}

The range for this function is {z | z Î Â }.

DEFINITION:

A point (x ₀, y ₀) in a region R in the xy-plane is an interior point of R if it is the center of a disk that lies entirely in R (See figure 1)

A point (x ₀, y ₀) is a boundary point of R if every disk centered at (x ₀, y ₀) contains points that lie outside of R as well as points that lie in R. (The boundary point itself need not belong to R.) (See figure 2)

The interior points of a region, as a set, make up the interior of the region, and the region's boundary points make up its boundary. T

The region is open if it consists entirely of interior points, and the region is closed if it contains all of its boundary points.

As with intervals of real numbers, some regions in the plane are neither open nor closed. This means that the region includes some of its boundary points, but not all.

figure 2

DEFINITION:

A region in the plane is bounded if it lies inside a disk of fixed radius. A region is unbounded if it is not bounded.

Here are some examples of bounded sets: line segments, triangles, interiors of triangles, rectangles, and disks.

Here are some examples of unbounded sets: lines, coordinate axes, quadrants, half-planes, and the plane itself.

EXAMPLE 4:

Given f (x, y) = 4x ² + 9y ², (a) find the boundary of the function's domain, (b) determine if the domain is an open region, a closed region, or neither, and (c) decide if the domain is bounded or unbounded.

SOLUTION:

From example 1, the domain is {(x, y) | x Î Â and y Î Â }. Since the domain is defined to be all the points in the xy-plane, there are no boundary points.

Since the xy-plane contains all of its interior points, therefore, the region is opened.

The plane is considered to be unbounded, because we can not draw a disk of fixed radius around it.

EXAMPLE 5:

Given

(a) find the boundary of the function's domain, (b) determine if the domain is an open region, a closed region, or neither, and (c) decide if the domain is bounded or unbounded.

SOLUTION:

From example 2, the domain for this function is {(x, y) | x ² + y ² £ 9}. x ² + y ² = 9 is a circle with radius of 3, therefore the boundary of the domain is the circle x ² + y ² = 9.

Since the boundary points are included in the domain, the region is closed.

The region is bounded because I can bound the region with a disk of fixed radius.

EXAMPLE 6:

Given f (x, y) = ln (x ² + y ²), (a) find the boundary of the function's domain, (b) determine if the domain is an open region, a closed region, or neither, and (c) decide if the domain is bounded or unbounded.

SOLUTION:

From example 3, the domain of this function is {(x, y) | (x, y) ¹ (0, 0)}. Since the point (0, 0) is not included in the domain, then this point is the only boundary point for the domain.

The boundary point is not included in this region, therefore the region is open and is unbounded.

The definitions of interior, boundary, open, closed, bounded, and unbounded for regions in space are similar to those for regions in the plane. To accommodate the extra dimension, we will use balls instead of disks.

FACT:

A closed ball consists of the region of points inside a sphere together with the sphere.

FACT:

An open ball is the region of points inside a sphere without the bounding sphere.

DEFINITION:

A point (x ₀, y₀, z ₀) in a region D in space is an interior point of D if it is the center of a ball that lies entirely in D.

A point is a boundary point on D if every sphere centered at (x ₀, y₀, z ₀) encloses points that lie outside of D as well as points that lie inside D.

The interior of D is the set of interior points of D. The boundary of D is the set of boundary points of D. A region D is open, if it consists entirely of interior points. A region is closed if it contains its entire boundary.

There are two standard ways to picture the values of a function f (x, y). They are (1) draw and label curves in the domain on which f has a constant value, and (2) sketch the surface z = f (x, y) in space. The curves that are drawn in option 1 are called level curves.

DEFINITION:

The set of points in the plane where a function f (x, y) has a constant value, f (x, y) = c, is called a level curve of f.

DEFINITION:

The set of all points (x, y, f (x, y)) in space, for (x, y) in the domain of f, is called the graph of f. The graph of f is also called the surface z = f (x, y).

EXAMPLE 7:

Display the values of the function f (x, y) = x ² + y ² in two ways: (a) by sketching the surface z = f (x, y), and (b) by drawing an assortment of level curves.

SOLUTION:

SURFACE

LEVEL CURVES

EXAMPLE 8:

Display the values of the function f (x, y) = 4 - x ² - y ² in two ways: (a) by sketching the surface z = f (x, y), and (b) by drawing an assortment of level curves.

SOLUTION:

SURFACE

LEVEL CURVES

EXAMPLE 9:

Display the values of the function f (x, y) = 1 - | x | - | y | in two ways: (a) by sketching the surface z = f (x, y), and (b) by drawing an assortment of level curves.

SOLUTION:

SURFACE

LEVEL CURVES

EXAMPLE 10:

Display the values of the function f (x, y) = ln (x ² + y ²) in two ways: (a) by sketching the surface z = f (x, y), and (b) by drawing an assortment of level curves.

LEVEL CURVES

DEFINITION:

The curve in space in which the plane z = c cuts the surface z = f (x, y) is made up of the points that represent the function value f (x, y) = c. It is called the contour line.

Contour lines do not lie in the xy-plane, but level curves do. Remember this important trait.

DEFINITION:

The set of points (x, y, z) in space where a function of three independent variables has a constant value f (x, y, z) = c is called a level surface of f.

EXAMPLE 11:

Sketch a level surface for the function f (x, y, z) = x ² + y ² + z ².

SOLUTION:

EXAMPLE 12:

Sketch a level surface for the function f (x, y, z) = x ² + y ².

SOLUTION:

EXAMPLE 13:

Sketch a level surface for the function f (x, y, z) = z - x ² - y ².

SOLUTION: