Study Sheet (14.1)
Functions of Several Variables
FUNCTIONS OF SEVERAL VARIABLES:
In this section, we study functions of two or more variables from four points of view:
§ Verbally (a description in words)
§ Numerically (a table of values)
§ Algebraically (an explicit formula)
§ Visually (a graph or level curves)
FUNCTIONS OF TWO VARIABLES:
Examples:
The temperature T at a point on the surface of the earth at any given time depends on the longitude x
and latitude y of the point.
§ We can think of T as being a function of the two variables x and y, or as a function of
the pair (x, y).
§ We indicate this functional dependence by writing:
T = f(x, y)
Isothermals are lines on a map that connect points of equal temperature. Since latitude plays a large
part of in controlling temperature variations, isotherms often run east to west. There are exceptions to
this rule, of course, since other factors that control temperatures include land and water, ocean
currents, and elevation.
Later we’ll call this line a contour line or level curve of the function.
The volume V of a circular cylinder depends on its radius r and its height h.
§ In fact, we know that V = πr2h.
§ We say that V is a function of r and h.
§ We write V(r, h) = πr2h.
Concentration of a Drug in the Blood:
When a drug is injected into muscle tissue, it diffuses into bloodstream. The concentration of the drug
in the blood increases until it reaches a maximum, and then decreases. The concentration, C (in mg
per liter), of the drug in the blood is a function of two variables: x, the amount ( in mg) of the drug
given in the injection, and t, the time ( in hours) since the injection was administered. We are told that
C = f (x, t) = te−t (5−x )
for 0 ≤ x ≤ 4
and t ≥ 0
Definition: A function f of two variables is a rule that assigns to each ordered pair of real
numbers (x, y) in a set D a unique real number denoted by (x, y).
§ The set D is the domain of f.
§ Its range is the set of values that f takes on, that is, { f (x, y) (x, y) ∈ D}
We often write z = f (x, y) to make explicit the value taken on by f at the general point (x, y).
§
§
§
The variables x and y are independent variables.
z is the dependent variable.
Compare this with the notation y = f (x) for functions of a single variable.
A function of two variables is just a function whose:
§ Domain is a subset of R2
§ Range is a subset of R
One way of visualizing such a function is by means of an arrow diagram, where
the domain D is represented as a subset of the xy-plane.
If a function f is given by a formula and no domain is specified, then the domain of f is understood to
be:
§ The set of all pairs (x, y) for which the given expression is a well-defined real number.
Example 1: For each of the following functions, evaluate f(3, 2) and find the domain.
(a) f (x, y) =
x + y +1
x −1
(b) f (x, y) = xln(y 2 − x)
Not all functions are given by explicit formulas.
§ The function in the next example is described verbally and by numerical estimates of its
values.
Example 2: Consider the table shown below.
y
x
5
6
7
8
1
2
3
10
11
12
13
12
13
14
15
(a) Write the z-value associated with each ordered pair given below.
f (1,10) =
f (1,12) =
f (2, 5) =
f (3,10) =
(b) Does this table represent z as a function of x and y?
(c) Find a formula for this function.
Example 3: The height (in feet) of and an object thrown upwards from an initial height of 8 ft
depends on both the time, t (in second), and the initial velocity, v0 (in meters/sec), so
h = f (t, v0 ) . The table below gives the height at various times and for various initial
velocities.
Time(sec)
0
0
0.5
1.0
1.5
2.0
8
4
-8
-28
-56
Initial Velocity (ft/sec)
10
20
8
9
2
-13
-36
8
14
12
2
-16
30
8
19
22
17
4
(a) What is the height after 1.5 sec if the initial velocity is 20 ft/sec.
(b) At what times and for which initial velocities was the object 4 ft off the ground?
(c) Use the grid below to sketch the height versus time data when the initial velocity is 30
ft/sec.
Example 4: Consider the function g(x, y) = 9 − x 2 − y 2 .
(a) Find the domain and range of g(x, y) = 9 − x 2 − y 2 .
(b) Sketch the graph of g(x, y) = 9 − x 2 − y 2 .
GRAPHS:
Another way of visualizing the behavior of a function of two variables is to consider its graph.
Definition: If f is a function of two variables with domain D, then the graph of f is the set of all points
(x, y, z) in R3 such that z = f(x, y) and (x,y) is in D.
Just as the graph of a function f of one variable is a curve C with equation y = f(x), so the graph of a
function f of two variables is a surface S with equation z = f (x, y) .
We can visualize the graph S of f as lying directly above or below its domain D
in the xy-plane.
Example 5: Sketch the graph of the function f (x, y) = 6 − 3x − 2y .
Example 6: Find the domain and range and sketch the graph of h(x, y) = 4x 2 + y 2
The figure shows computer-generated graphs of several functions.
Notice that we get an especially good picture of a function when rotation is used to give views from
different vantage points.
In (a) and (b), the graph of f is very flat and close to the xy-plane except near the origin.
§ This is because e –x2–y2 is small when x or y is large.
LEVEL CURVES:
So far, we have two methods for visualizing functions, arrow diagrams and graphs.
§ A third method, borrowed from mapmakers, is a contour map.
§ When the traces parallel to the xy-plane or in other words, the traces found when z or
f(x,y) is set equal to a constant, are drawn in the xy-plane, the traces are called level
curves. When several level curves, also called contour lines, are drawn together in the
xy-plane the image is called a contour map.
Definition: The level curves of a function f of two variables are the curves with equations
f(x, y) = k
where k is a constant (in the range of f).
A level curve f(x, y) = k is the set of all points in the domain of f at which f takes on a given value k.
§ That is, it shows where the graph of f has height k.
You can see from the figure the relation between level curves and horizontal traces.
The level curves f(x, y) = k are just the traces of the graph of f in the horizontal plane z = k projected
down to the xy-plane.
The surface is:
§ Steep where the level curves are close together.
§ Somewhat flatter where the level curves are farther apart.
One common example of level curves occurs in topographic maps
of mountainous regions, such as shown.
The level curves are curves of constant elevation above sea level.
§ If you walk along one of these contour lines, you neither ascend nor descend.
Another common example is the temperature function introduced in the opening paragraph of the
section.
§ Here, the level curves are called isothermals.
§ They join locations with the same temperature.
Example 9: A contour map for a function f is shown. Use it to estimate the values of f(1, 3) and f(4,5).
Example 10: Sketch the level curves of the function f (x, y) = 6 − 3x − 2y for the values k = –6, 0, 6, 12.
Example 11: Sketch the level curves of the function g(x, y) = 9 − x 2 − y 2 for k = 0, 1, 2, 3
Example 12: Sketch some level curves of the function h(x, y) = 4x 2 + y 2 +1 .
The following figure shows some computer-generated level curves together with the corresponding
computer-generated graphs.
Notice that the level curves in (c) crowd together near the origin.
That corresponds to the fact that the graph in (d) is very steep near the origin.
FUNCTION OF THREE VARIABLES:
A function of three variables, f, is a rule that assigns to each ordered triple (x, y, z) in a domain D ⊂ R3
a unique real number denoted by f(x, y, z).
For instance, the temperature T at a point on the surface of the earth depends on the longitude x and
latitude y of the point and on the time t.
So, we could write:
T = f(x, y, t)
Example 13: Find the domain of f if f (x, y, z) = ln(z − y) + xysin z .
MULTIPLE VARIABLE FUNCTIONS:
It’s very difficult to visualize a function f of three variables by its graph. That would lie in a fourdimensional space.
However, we do gain some insight into f by examining its level surfaces—the surfaces with equations
f(x, y, z) = k, where k is a constant. If the point (x, y, z) moves along a level surface, the value of f(x, y,
z) remains fixed.
Example 15
Find the level surfaces of the function f (x, y, z) = x 2 + y 2 + z 2 .