A function of two variables is defined similar to a function of one variable. It has a domain (in the plane) and a range. The graph of such a function is a surface in space and we try to sketch some.

1. Section 9.6
Functions and Surfaces
Math 21a
February 13, 2008
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2. Outline
Functions of more than one variable
Domain and Range
Graphs
Traces
Quadric Surfaces
Review of Conic Sections
Examples of Quadric Surfaces

3. What is a function?
A function is a box which changes numbers to numbers, or vectors
to vectors, or dogs to cats, or whatever. There are lots of functions
which naturally have multiple inputs and a single output.

4. What is a function?
A function is a box which changes numbers to numbers, or vectors
to vectors, or dogs to cats, or whatever. There are lots of functions
which naturally have multiple inputs and a single output.
The temperature in this room is a function of position and
time.
The production of an economy is a function of capital (money
and goods invested) and labor
I derive utility (happiness) from eating bacon and eggs for
breakfast.

5. 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 f (x, y ). The set D is the domain of f and its range is
the set of values that f takes on. That is { f (x, y ) | (x, y ) ∈ D }.

6. Example
Example
√
Find the domain and range of f (x, y ) = xy .

7. Example
Example
√
Find the domain and range of f (x, y ) = xy .
Solution
Working from the outside in, we see that xy must be
nonnegative, which means x ≥ 0 and y ≥ 0 or x ≤ 0 and
y ≤ 0. Thus the domain is the union of the coordinate axes,
and the first and third quadrants.
The range of f is the set of all “outputs” of f . Clearly the
range of f is restricted to the set of nonnegative numbers. To
make sure that we can get all nonnegative numbers x, notice
x = f (x 2 , 1).

8. Worksheet #1

9. Outline
Functions of more than one variable
Domain and Range
Graphs
Traces
Quadric Surfaces
Review of Conic Sections
Examples of Quadric Surfaces

10. 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 with z = f (x, y ) and
(x, y ) ∈ D.

11. 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 with z = f (x, y ) and
(x, y ) ∈ D.
Functions of one variable are easy to graph on the Cartesian plane.
Functions of two variables need a three-dimensional space. Our
goal is to understand functions of two variables and how to graph
them.

12. Example (Worksheet 2(i))
Sketch the graph of the function f (x, y ) = 3.

13. Example (Worksheet 2(i))
Sketch the graph of the function f (x, y ) = 3.
Example (Worksheet 2(ii))
Sketch the graph of the function f (x, y ) = 6 − 3x − 2y .
Example (Worksheet 2(iii))
Sketch the graph of the function f (x, y ) = x 2 + y 2 .

14. Traces
A trace of a surface is the intersection of it with a plane. The
result is a curve.
Multiple traces give multiple curves which help sketch the
function
choices for traces:
coordinate planes x = 0, y = 0, z = 0
parallel planes, e.g., z = k for many k

15. Worksheet #3–4

16. Outline
Functions of more than one variable
Domain and Range
Graphs
Traces
Quadric Surfaces
Review of Conic Sections
Examples of Quadric Surfaces

17. Conic Sections
Circle
Ellipse
Parabola
Hyperbola

18. Quadric Surfaces
Sphere
Ellipsoid
Elliptic paraboloid
Hyperbolic paraboloid
Hyperboloid of one sheet
Hyperboloid of two sheets