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
Definition
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
Finding vertical traces
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 traces parallel to the
For the function
Vertical Trace for
In a similar fashion, we can substitute the
in the equation
to obtain the traces in the
as listed in the following table.
Vertical traces parallel to the
For the function
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.
Vertical traces of the function
are cosine curves in the
(a) and sine curves in the
(b).
Functions of two variables can produce some striking-looking surfaces. The following figure shows two examples.
Examples of surfaces representing functions of two variables: (a) a combination of a power function and a sine function and (b) a combination of trigonometric, exponential, and logarithmic functions.
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.
Domains for functions of three 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