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**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 (by a description in words)

• numerically (by a table of values)

• algebraically (by an explicit formula)

• visually (by a graph or level curves)

### Functions of Two Variables

### Functions of Two Variables

*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).*

*The volume V of a circular cylinder depends on its radius r*
*and its height h. In fact, we know that V = *π*r*^{2}*h. We say that *

### Functions of Two Variables

*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 and z is ****the dependent variable. [Compare this with the notation **
*y = f(x) for functions of a single variable.]*

### Example 2

*In regions with severe winter weather, the wind-chill index *
is often used to describe the apparent severity of the cold.

*This index W is a subjective temperature that depends on *
*the actual temperature T and the wind speed v. *

*So W is a function of T and v, and we can write W = f(T, v).*

7

### Example 2

*Table 1 records values of W compiled by the US National *
Weather Service and the Meteorological Service of Canada.

Wind-chill index as a function of air temperature and wind speed

cont’d

### Example 2

For instance, the table shows that if the temperature is –5°C and the wind speed is 50 km/h, then subjectively it

would feel as cold as a temperature of about –15°C with no wind.

So

*f(–5, 50) = –15*

cont’d

### Example 3

In 1928 Charles Cobb and Paul Douglas published a study in which they modeled the growth of the American

economy during the period 1899–1922.

They considered a simplified view of the economy in which production output is determined by the amount of labor

involved and the amount of capital invested.

While there are many other factors affecting economic

performance, their model proved to be remarkably accurate.

### Example 3

The function they used to model production was of the form
*P(L, K) = bL*^{α}*K*^{1 –}^{α}

*where P is the total production (the monetary value of all *
*goods produced in a year), L is the amount of labor (the *
*total number of person-hours worked in a year), and K is *
the amount of capital invested (the monetary worth of all
machinery, equipment, and buildings).

cont’d

### Example 3

Cobb and Douglas used economic data published by the government to obtain Table 2.

**Table 2**

cont’d

### Example 3

*They took the year 1899 as a baseline and P, L, and K for *
1899 were each assigned the value 100.

The values for other years were expressed as percentages of the 1899 figures.

Cobb and Douglas used the method of least squares to fit the data of Table 2 to the function

*P(L, K) = 1.01L*^{0.75}*K*^{0.25}

cont’d

### Example 3

If we use the model given by the function in Equation 2 to compute the production in the years 1910 and 1920, we get the values

*P(147, 208) = 1.01(147)*^{0.75}(208)^{0.25} ≈ 161.9
*P(194, 407) = 1.01(194)*^{0.75}(407)^{0.25} ≈ 235.8

which are quite close to the actual values, 159 and 231.

The production function (1) has subsequently been used in many settings, ranging from individual firms to global

**economics. It has become known as the Cobb-Douglas **
**production function.**

cont’d

### Example 3

*Its domain is {(L, K) | L ≥ 0, K ≥ 0} because L and K *
represent labor and capital and are therefore never
negative.

cont’d

### Graphs

### Graphs

Another way of visualizing the behavior of a function of two variables is to consider its graph.

*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). *

### Graphs

*We can visualize the graph S of f as lying directly above or *
*below its domain D in the xy-plane (see Figure 5).*

**Figure 5**

### Graphs

**The function f(x, y) = ax + by + c is called as a linear ****function. **

The graph of such a function has the equation
*z = ax + by + c or ax + by – z + c = 0*

so it is a plane. In much the same way that linear functions of one variable are important in single-variable calculus, we will see that linear functions of two variables play a central role in multivariable calculus.

### Example 6

Sketch the graph of Solution:

The graph has equation We square
*both sides of this equation to obtain z*^{2} *= 9 – x*^{2} *– y*^{2}, or
*x*^{2} *+ y*^{2} *+ z*^{2} = 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).

Graph of

### Level Curves

### 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 on which points of constant
*elevation are joined to form contour curves, or level curves.*

*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. *

*In other words, it shows where the graph of f has height k.*

### Level Curves

You can see from Figure 11 the relation between level curves and horizontal traces.

### Level Curves

*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. *

So if you draw the level curves of a function and visualize them being lifted up to the surface at the indicated height, then you can mentally piece together a picture of the graph.

The surface is steep where the level curves are close

together. It is somewhat flatter where they are farther apart.

### Level Curves

One common example of level curves occurs in

topographic maps of mountainous regions, such as the map in Figure 12.

### Level Curves

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 this section.

**Here the level curves are called isothermals and join **
locations with the same temperature.

### Level Curves

Figure 13 shows a weather map of the world indicating the average July temperatures. The isothermals are the curves that separate the colored bands.

### Level Curves

In weather maps of atmospheric pressure at a given time
as a function of longitude and latitude, the level curves are
**called isobars and join locations with the same pressure.**

Surface winds tend to flow from areas of high pressure across the isobars toward areas of low pressure, and are strongest where the isobars are tightly packed.

### Level Curves

A contour map of world-wide precipitation is shown in Figure 14.

**Figure 14**

Precipitation

### Level Curves

For some purposes, a contour map is more useful than a graph. It is true in estimating function values. Figure 20 shows some computer-generated level curves together with the corresponding computer-generated graphs.

**Figure 20**

### Level Curves

Notice that the level curves in part (c) crowd together near

**Figure 20**

cont’d

### Functions of Three or More

### Variables

### Functions of Three or More Variables

**A function of three variables, f, is a rule that assigns to ***each ordered triple (x, y, z) in a domain 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 14

*Find the domain of f if*

*f(x, y, z) = ln(z – y) + xy sin z*

Solution:

*The expression for f(x, y, z) is defined as long as z – y > 0, *
*so the domain of f is*

*D = {(x, y, z) ∈* *| z > y}*

**This is a half-space consisting of all points that lie above **
*the plane z = y.*

### Functions of Three or More Variables

*It’s very difficult to visualize a function f of three variables *
by its graph, since that would lie in a four-dimensional

space.

*However, we do gain some insight into f by examining its *
**level surfaces, which are 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.

Functions of any number of variables can be considered.

**A function of n variables is a rule that assigns a number **

### Functions of Three or More Variables

*For example, if a company uses n different ingredients in *
*making a food product, c*_{i}*is the cost per unit of the i th *

*ingredient, and x*_{i}*units of the i th ingredient are used, then *
*the total cost C of the ingredients is a function of the n*

*variables x*_{1}*, x*_{2}*, . . . , x** _{n}*:

*C = f(x*_{1}*, x*_{2}*, . . . , x*_{n}*) = c*_{1}*x*_{1} *+ c*_{2}*x*_{2} + ^{. . .}*+ c*_{n}*x*_{n}

*The function f is a real-valued function whose domain is a *
subset of .

### Functions of Three or More Variables

Sometimes we will use vector notation to write such

**functions more compactly: If x = **

### 〈

^{x}_{1}

^{, x}_{2}

*, . . . , x*

_{n}### 〉

, we often

**write f(x) in place of f(x**_{1}

*, x*

_{2}

*, . . . , x*

*).*

_{n}With this notation we can rewrite the function defined in Equation 3 as

* f(x) = c*

**x**

**where c = **

### 〈

^{c}_{1}

^{, c}_{2}

*, . . . , c*

_{n}### 〉

^{and c}^{}

**x denotes the dot product**

**of the vectors c and x in V***.*

_{n}### Functions of Three or More Variables

In view of the one-to-one correspondence between points
*(x*_{1}*, x*_{2}*, . . . , x** _{n}*) in and their position vectors

**x = **

### 〈

^{x}_{1}

^{, x}_{2}

*, . . . , x*

_{n}### 〉

^{in V}*, we have three ways of looking at*

_{n}*a function f defined on a subset of :*

**1. As a function of n real variables x**_{1}*, x*_{2}*, . . . , x*_{n}

**2. As a function of a single point variable (x**_{1}*, x*_{2}*, . . . , x** _{n}*)

**3. As a function of a single vector variable**

**x = **

### 〈

^{x}_{1}

^{, x}_{2}

*, . . . , x*

_{n}