Differentiation of Functions of Several Variables
21 Functions of Several Variables
Learning Objectives
- Recognize a function of two variables and identify its domain and range.
- Sketch a graph of a function of two variables.
- Sketch several traces or level curves of a function of two variables.
- Recognize a function of three or more variables and identify its level surfaces.
Our first step is to explain what a function of more than one variable is, starting with functions of two independent variables. This step includes identifying the domain and range of such functions and learning how to graph them. We also examine ways to relate the graphs of functions in three dimensions to graphs of more familiar planar functions.
Functions of Two Variables
The definition of a function of two variables is very similar to the definition for a function of one variable. The main difference is that, instead of mapping values of one variable to values of another variable, we map ordered pairs of variables to another variable.
A function of two variables maps each ordered pair in a subset of the real plane to a unique real number The set is called the domain of the function. The range of is the set of all real numbers that has at least one ordered pair such that as shown in the following figure.
Determining the domain of a function of two variables involves taking into account any domain restrictions that may exist. Let’s take a look.
Find the domain and range of each of the following functions:
- This is an example of a linear function in two variables. There are no values or combinations of and that cause to be undefined, so the domain of is To determine the range, first pick a value for We need to find a solution to the equation or One such solution can be obtained by first setting which yields the equation The solution to this equation is which gives the ordered pair as a solution to the equation for any value of Therefore, the range of the function is all real numbers, or
- For the function to have a real value, the quantity under the square root must be nonnegative:
This inequality can be written in the form
Therefore, the domain of is The graph of this set of points can be described as a disk of radius centered at the origin. The domain includes the boundary circle as shown in the following graph.
To determine the range of we start with a point on the boundary of the domain, which is defined by the relation It follows that and
If (in other words, then
This is the maximum value of the function. Given any value c between we can find an entire set of points inside the domain of such that
Since this describes a circle of radius centered at the origin. Any point on this circle satisfies the equation Therefore, the range of this function can be written in interval notation as
Find the domain and range of the function
The domain is the shaded circle defined by the inequality which has a circle of radius as its boundary. The range is
Determine the set of ordered pairs that do not make the radicand negative.
Graphing Functions of Two Variables
Suppose we wish to graph the function This function has two independent variables and one dependent variable When graphing a function of one variable, we use the Cartesian plane. We are able to graph any ordered pair in the plane, and every point in the plane has an ordered pair associated with it. With a function of two variables, each ordered pair in the domain of the function is mapped to a real number Therefore, the graph of the function consists of ordered triples The graph of a function of two variables is called a surface.
To understand more completely the concept of plotting a set of ordered triples to obtain a surface in three-dimensional space, imagine the coordinate system laying flat. Then, every point in the domain of the function has a unique associated with it. If is positive, then the graphed point is located above the if is negative, then the graphed point is located below the The set of all the graphed points becomes the two-dimensional surface that is the graph of the function
Create a graph of each of the following functions:
- In (Figure), we determined that the domain of is and the range is When we have Therefore any point on the circle of radius centered at the origin in the maps to in If then so any point on the circle of radius centered at the origin in the maps to in As gets closer to zero, the value of z approaches 3. When then This is the origin in the If is equal to any other value between then equals some other constant between The surface described by this function is a hemisphere centered at the origin with radius as shown in the following graph.
- This function also contains the expression Setting this expression equal to various values starting at zero, we obtain circles of increasing radius. The minimum value of is zero (attained when When the function becomes and when then the function becomes These are cross-sections of the graph, and are parabolas. Recall from Introduction to Vectors in Space that the name of the graph of is a paraboloid. The graph of appears in the following graph.
A profit function for a hardware manufacturer is given by
where is the number of nuts sold per month (measured in thousands) and represents the number of bolts sold per month (measured in thousands). Profit is measured in thousands of dollars. Sketch a graph of this function.
This function is a polynomial function in two variables. The domain of consists of coordinate pairs that yield a nonnegative profit:
This is a disk of radius centered at A further restriction is that both must be nonnegative. When and Note that it is possible for either value to be a noninteger; for example, it is possible to sell thousand nuts in a month. The domain, therefore, contains thousands of points, so we can consider all points within the disk. For any we can solve the equation
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Since we know that so the previous equation describes a circle with radius centered at the point Therefore. the range of is The graph of is also a paraboloid, and this paraboloid points downward as shown.
Level Curves
If hikers walk along rugged trails, they might use a topographical map that shows how steeply the trails change. A topographical map contains curved lines called contour lines. Each contour line corresponds to the points on the map that have equal elevation ((Figure)). A level curve of a function of two variables is completely analogous to a contour line on a topographical map.
Given a function and a number in the range of level curve of a function of two variables for the value is defined to be the set of points satisfying the equation
Returning to the function we can determine the level curves of this function. The range of is the closed interval First, we choose any number in this closed interval—say, The level curve corresponding to is described by the equation
To simplify, square both sides of this equation:
Now, multiply both sides of the equation by and add to each side:
This equation describes a circle centered at the origin with radius Using values of between yields other circles also centered at the origin. If then the circle has radius so it consists solely of the origin. (Figure) is a graph of the level curves of this function corresponding to Note that in the previous derivation it may be possible that we introduced extra solutions by squaring both sides. This is not the case here because the range of the square root function is nonnegative.
A graph of the various level curves of a function is called a contour map.
Given the function find the level curve corresponding to Then create a contour map for this function. What are the domain and range of
To find the level curve for we set and solve. This gives
We then square both sides and multiply both sides of the equation by
Now, we rearrange the terms, putting the terms together and the terms together, and add to each side:
Next, we group the pairs of terms containing the same variable in parentheses, and factor from the first pair:
Then we complete the square in each pair of parentheses and add the correct value to the right-hand side:
Next, we factor the left-hand side and simplify the right-hand side:
Last, we divide both sides by
This equation describes an ellipse centered at The graph of this ellipse appears in the following graph.
We can repeat the same derivation for values of less than Then, (Figure) becomes
for an arbitrary value of (Figure) shows a contour map for using the values When the level curve is the point
Find and graph the level curve of the function corresponding to
The equation of the level curve can be written as which is a circle with radius centered at
First, set and then complete the square.
Another useful tool for understanding the graph of a function of two variables is called a vertical trace. Level curves are always graphed in the but as their name implies, vertical traces are graphed in the – or
Consider a function with domain A vertical trace of the function can be either the set of points that solves the equation for a given constant or for a given constant
Find vertical traces for the function corresponding to and
First set in the equation
This describes a cosine graph in the plane The other values of appear in the following table.
Vertical Trace for | |
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In a similar fashion, we can substitute the in the equation to obtain the traces in the as listed in the following table.
Vertical Trace for | |
---|---|
The three traces in the are cosine functions; the three traces in the are sine functions. These curves appear in the intersections of the surface with the planes and as shown in the following figure.
Determine the equation of the vertical trace of the function corresponding to and describe its graph.
This function describes a parabola opening downward in the plane
Set in the equation and complete the square.
Functions of two variables can produce some striking-looking surfaces. The following figure shows two examples.
Functions of More Than Two Variables
So far, we have examined only functions of two variables. However, it is useful to take a brief look at functions of more than two variables. Two such examples are
and
In the first function, represents a point in space, and the function maps each point in space to a fourth quantity, such as temperature or wind speed. In the second function, can represent a point in the plane, and can represent time. The function might map a point in the plane to a third quantity (for example, pressure) at a given time The method for finding the domain of a function of more than two variables is analogous to the method for functions of one or two variables.
Find the domain of each of the following functions:
- For the function to be defined (and be a real value), two conditions must hold:
- The denominator cannot be zero.
- The radicand cannot be negative.
Combining these conditions leads to the inequality
Moving the variables to the other side and reversing the inequality gives the domain as
which describes a ball of radius centered at the origin. (Note: The surface of the ball is not included in this domain.) - For the function to be defined (and be a real value), two conditions must hold:
- The radicand cannot be negative.
- The denominator cannot be zero.
Since the radicand cannot be negative, this implies and therefore that Since the denominator cannot be zero, or Which can be rewritten as which are the equations of two lines passing through the origin. Therefore, the domain of is
Find the domain of the function
Check for values that make radicands negative or denominators equal to zero.
Functions of two variables have level curves, which are shown as curves in the However, when the function has three variables, the curves become surfaces, so we can define level surfaces for functions of three variables.
Given a function and a number in the range of a level surface of a function of three variables is defined to be the set of points satisfying the equation
Find the level surface for the function corresponding to
The level surface is defined by the equation This equation describes a hyperboloid of one sheet as shown in the following figure.
Find the equation of the level surface of the function
corresponding to and describe the surface, if possible.
describes a sphere of radius centered at the point
Set and complete the square.
Key Concepts
- The graph of a function of two variables is a surface in and can be studied using level curves and vertical traces.
- A set of level curves is called a contour map.
Key Equations
- Vertical trace
for or for - Level surface of a function of three variables
For the following exercises, evaluate each function at the indicated values.
Find
Find
The volume of a right circular cylinder is calculated by a function of two variables, where is the radius of the right circular cylinder and represents the height of the cylinder. Evaluate and explain what this means.
This is the volume when the radius is and the height is
An oxygen tank is constructed of a right cylinder of height and radius with two hemispheres of radius mounted on the top and bottom of the cylinder. Express the volume of the cylinder as a function of two variables, find and explain what this means.
For the following exercises, find the domain of the function.
All points in the
All real ordered pairs in the of the form
Find the range of the functions.
The set
For the following exercises, find the level curves of each function at the indicated value of to visualize the given function.
a hyperbola
a line; line through the origin
three lines
any constant
The level curves are parabolas of the form
For the following exercises, find the vertical traces of the functions at the indicated values of and y, and plot the traces.
a curve in the with rulings parallel to the
Find the domain of the following functions.
All points in
For the following exercises, plot a graph of the function.
Use technology to graph
Sketch the following by finding the level curves. Verify the graph using technology.
Describe the contour lines for several values of for
The contour lines are circles.
Find the level surface for the functions of three variables and describe it.
a sphere of radius
a hyperboloid of one sheet
For the following exercises, find an equation of the level curve of that contains the point
The strength of an electric field at point resulting from an infinitely long charged wire lying along the is given by where is a positive constant. For simplicity, let and find the equations of the level surfaces for
A thin plate made of iron is located in the The temperature in degrees Celsius at a point is inversely proportional to the square of its distance from the origin. Express as a function of
Refer to the preceding problem. Using the temperature function found there, determine the proportionality constant if the temperature at point Use this constant to determine the temperature at point
Refer to the preceding problem. Find the level curves for and describe what the level curves represent.
The level curves represent circles of radii and
Glossary
- contour map
- a plot of the various level curves of a given function
- function of two variables
- a function that maps each ordered pair in a subset of to a unique real number
- graph of a function of two variables
- a set of ordered triples that satisfies the equation plotted in three-dimensional Cartesian space
- level curve of a function of two variables
- the set of points satisfying the equation for some real number in the range of
- level surface of a function of three variables
- the set of points satisfying the equation for some real number in the range of
- surface
- the graph of a function of two variables,
- vertical trace
- the set of ordered triples that solves the equation for a given constant or the set of ordered triples that solves the equation for a given constant