1. Introduction to Functions of Several Variables
13.1

2. Functions of Several Variables
The work done by a force (W = FD) and the volume of a right circular cylinder (V = ) are both functions of two variables. The volume of a rectangular solid (V = lwh) is a function of three variables.
The notation for a function of two or more variables is similar to that for a function of a single variable. Here are two examples.
and
Function of two variables
Function of three variables

3. Functions of Several Variables
For the function given by z = f(x, y), x and y are called the independent variables and z is called the dependent variable.

4. Functions of Several Variables
A function that can be written as a sum of functions of the form cxmyn (where c is a real number m and n are nonnegative integers) is called a polynomial function of two variables.
For instance, the functions given by
are polynomial functions of two variables.
A rational function is the quotient of two polynomial functions. Similar terminology is used for functions of more than two variables.

5. Example – Domains of Functions of Several Variables
Find the domain of each function.
a. f(x, y) = x2y – 2xy – xy2 b.
c d.
Solution:
a. Polynomials are defined for all x and y. So, the domain is R2.

6. Example – Solution
cont’d
b. f(x, y) is a rational function in terms of x and y. The domain must exclude points at which the denominator is zero. So, the domain is {(x, y): x ≠ y}.
c. 4 – x2 – y2 ≥ 0 or x2 + y2 ≤ 4. So, the domain is {(x, y): x2 + y2 ≤ 4}, which is the set of points on or within a circle with radius 2 (disk of radius 2).

7. Example - Solution

8. Example - Solution
d. x2 + y2 + z2 – 1 ≥ 0 or x2 + y2 + z2 ≥ 1. The points are on and outside of a unit sphere. The domain is {(x, y, z): x2 + y2 + z2 ≥ 1}.

9. Functions of Several Variables
Functions of several variables can be combined in the same ways as functions of single variables.
For instance, you can form the sum, difference, product,
and quotient of two functions of two variables as follows.

10. Functions of Several Variables
You cannot form the composite of two functions of several variables.
However, if h is a function of several variables and g is a
function of a single variable, you can form the composite
function as follows.

11. The Graph of a Function of Two Variables

12. The Graph of a Function of Two Variables
The graph of a function f of two variables is the set of all points (x, y, z) for which z = f(x, y) and (x, y) is in the domain of f.
This graph can be interpreted
geometrically as a surface
in space. In Figure 13.2,
note that the graph of z = f (x, y)
is a surface whose projection
onto the xy-plane is the D,
the domain of f.
Figure 13.2

13. The Graph of a Function of Two Variables
To each point (x, y) in D there corresponds a point (x, y, z)
on the surface, and, conversely, to each point (x, y, z) on
the surface there corresponds a point (x, y) in D.

14. Graphs of Functions
Like functions of one variable, functions of two variables must pass the vertical line test. A relation of the form F(x, y, z) = 0 is a function provided every line parallel to the z-axis intersects the graph of F at most once.

15. Example
The Ellipoid is NOT the graph of a function since vertical lines intersect the surface twice.

16. Example
Elliptic Paraboloid of the form z = ax2 + by2 does represent a function.

17. Example
Find the domain and range of the functions and graph them. a. f(x, y) = 2x + 3y - 12
Let z = f(x, y), then z = 2x + 3y – 12 or 2x + 3y – z = 12, which describes a plane with normal vector n = <2, 3, -1>. The domain consists of all points in R2 and the range is R.
x-intercept: (6, 0, 0), let y = z = 0
y-intercept: (0, 4, 0), let x = z = 0
z-intercept: (0, 0, -12), let x = y = 0

18. Example

19. Example
f(x, y) = x2 + y2
Let z = f(x, y), then z = x2 + y2 describes an elliptic paraboloid that opens upward with vertex (0, 0, 0). The domain is R2 and the range consists of all nonnegative real numbers.

20. Example
The domain is R2 since the quantity under the square root is always positive.
Note, 1 + x2 + y2 ≥ 1, so the range is {z: z ≥ 1}.
Squaring both sides, we obtain z2 = 1 + x2 + y2 or –x2 – y2 + z2 = 1. This is an equation of a hyperboloid of two sheets that opens along the z-axis.

21. Example
Since the range is {z: z ≥ 1}, the given function represents only the upper sheet of the hyperboloid.

22. Example Sketch the graph of Solution:
The graph has equation We square both sides of this equation to obtain z2 = 9 – x2 – y2, or x2 + y2 + z2 = 9, which we recognize as an equation of the sphere with center the origin and radius 3.
But, since z  0, the graph of g is just the top half of this sphere (see Figure 7).
Figure 7
Graph of

23. Level Curves

24. Level Curves
A second way to visualize a function of two variables is to use a scalar field in which the scalar z = f(x, y) is assigned to the point (x, y).
A scalar field can be characterized by level curves (or contour lines) along which the value of f(x, y) is constant.
For instance, the weather map
in Figure 13.5 shows level curves
of equal pressure called isobars.
Level curves show the lines of equal pressure (isobars) measured in millibars.
Figure 13.5

25. Level Curves
In weather maps for which the level curves represent points of equal temperature, the level curves are called isotherms, as shown in Figure 13.6.
Another common use of level curves is in representing electric potential fields.
In this type of map, the level curves are called equipotential lines.
Level curves show the lines of equal (isobars) measured in millibars. temperature (isotherms) measured in degrees Fahrenheit.
Figure 13.6

26. Level Curves
Contour maps are commonly used to show regions on Earth’s surface, with the level curves representing the height above sea level. This type of map is called a topographic map.

27. Level Curves
For example, the mountain shown in Figure 13.7 is represented by the topographic map in Figure 13.8.
Figure 13.7
Figure 13.8

28. Level Curves
A contour map depicts the variation of z with respect to x and y by the spacing between level curves.
Much space between level curves indicates that z is changing slowly, whereas little space indicates a rapid change in z.
Furthermore, to produce a good three-dimensional illusion in a contour map, it is important to choose c-values that are evenly spaced.

29. Level Curves
Consider a surface defined by the function z = f(x, y).

30. Level Curves
Imagine stepping onto the surface and walking along a path on which your elevation has the constant value z = z0. The path you walk on the surface is part of a contour curve; the complete contour curve is the intersection of the surface and the horizontal plane z = z0.
When the contour curve is projected onto the xy-plane, the result is the curve f(x, y) = z0. This curve in the xy-plane is called a level curve.
Note 1: A contour curve is a trace in the plane z = z0.
Note 2: A level curve may not always be a single curve. It might consist of a point (x2 + y2 = 0) or it might consist of several lines or curves (xy = 0).

31. Level Curves
Imagine repeating this process with a different constant value of z, say, z = z1. The path you walk this time when projected onto the xy-plane is part of another level curve f(x, y) = z1. A collection of such level curves corresponding to different values of z, provides a useful two-dimensional representation of the surface.

32. Level Curves
Note: Concentric close level curves indicate either a peak or a depression on the surface.

33. Level Curves

34. Example Find and sketch the level curves of f(x, y) = y – x2 – 1.
The level curves are described by y – x2 – 1 = z0, where z0 is a constant in the range of f.
For all values of z0, these curves are parabolas in the xy-plane, y = x2 + z0 + 1.
Let z0 = 0; y = x2 + 1; along this curve, the surface has an elevation (z-coordinate) of zero.

35. Example
z0 = -1; y = x2; along this curve, the surface has an elevation of -1.
z0 = 1; y = x2 + 2; along which the surface has an elevation of 1.

36. Example

37. Example f(x, y) = e-x^2 – y^2
The level curves satisfy the equation e-x^2 – y^2 = z0, where z0 is a positive constant.
Taking “ln” of both sides, we obtain: x2 + y2 = -ln z0, which describes circular level curves.
These curves can be sketched for all values of z0 with 0 < z0 ≤ 1 since the right side of x2 + y2 = -ln z0 must be nonnegative.
z0 = 1; x2 + y2 = 0, whose solution is (0, 0).
z0 = e-1; x2 + y2 = -ln e-1 = 1, which is a circle centered at (0, 0) with radius of one and e-1 ≈ 0.37.

38. Example
In general, the level curves are circles centered at (0, 0). As the radii of the circles increase, the corresponding z-values decrease.

39. Example f(x, y) = 2 + sin(x – y)
The level curves are f(x, y) = 2 + sin(x – y) = z0 or sin(x – y) = z0 – 2. Since -1 ≤ sin(x – y) ≤ 1, the admissible values of z0 satisfy -1 ≤ z0 – 2 ≤ 1 or 1 ≤ z0 ≤ 3.
z0 = 2; sin(x – y) = 0, which the solutions are x – y = kπ or y = x – kπ, where k is an integer. So, the surface has an elevation of 2 on this set of lines.
z0 = 1 (the minimum value of z), sin(x – y) = -1 and the solutions are x – y = -π/2 + 2kπ, where k is an integer. Along these lines, the surface has an elevation of 1.

40. Example

41. Example
Application: The electric field at points in the xy-plane due to two point charges located at (0, 0) and (1, 0) is related to the electric potential function
Discuss the electric potential function.

42. Example
The domain of φ(x, y) contains all points of R2 except (0, 0) and (1, 0) where the charges are located. As these points are approached, the potential function becomes arbitrarily large.
The potential approaches zero as x or y increase in magnitude. These observations imply the range of φ(x, y) is all positive real numbers. The level curves of φ(x, y) are closed curves, encircling either a single charge (at small distances) or both charges (at larger distances).

43. Example

44. Level Surfaces

45. Level Surfaces
The concept of a level curve can be extended by one dimension to define a level surface.
If f is a function of three variables
and c is a constant, the graph of the
equation f( x, y, z) = c is a
level surface of the function f,
as shown in Figure
Level surfaces of f
Figure 13.14

46. Level Surfaces
With the help of computers, engineers and scientists have developed other ways to view functions of three variables.
For instance, Figure 13.15
shows a computer simulation
that uses color to represent
the temperature distribution
of fluid inside a pipe fitting.
One-way coupling of ANSYS CFX™ and ANSYS Mechanical™ for thermal stress analysis
Figure 13.15

47. Example – Level Surfaces
Describe the level surfaces of the function
Solution:
Each level surface has an equation of the form
So, the level surfaces are ellipsoids (whose cross sections parallel to the yz-plane are circles).
As c increases, the radii of the circular cross sections increase according to the square root of c.
Equation of level surface

48. Example – Solution
cont’d
For example, the level surfaces corresponding to the
values c = 0, c = 4, c = 16 and are as follows.
Level surface for c = 0 (single point)
Level surface for c = 4 (ellipsoid)
Level surface for c = 16 (ellipsoid)

49. Example – Solution These level surfaces are shown in Figure 13.16.
cont’d
These level surfaces are shown in Figure
If the function represents the temperature at the point
(x, y, z), the level surfaces shown in Figure would be
called isothermal surfaces.
Figure 13.16