1. Functions of several variables
Math 200
Week 4 - Monday
Functions of several variables

2. Math 200
Main Goals for Today
Be able to describe and sketch the domain of a function of two or more variables
Domains will be 2D or 3D regions
Know how to evaluate a function of two or more variables
Be able to compute and sketch level curves & surfaces
These are just traces of the form z = k

3. Math 200
Examples and Notation
The following are all functions of several variables
f(x,y) = sin(x) + cos(y)
g(x,y,z) = xyz
z = ln(x2+y2)
A = bh/2
V = lwh

4. Math 200
Definitions
A function of two variables, x and y, is a rule that assigns to each ordered pair, (x,y), exactly one real number.
We assign the value of f(x,y) to z to get a surface
The domain of a function of two variables is the set of ordered pairs (x,y) for which f is defined
A function of three variables, x, y, and z, is a rule that assigns to each ordered triple, (x,y,z), exactly one real number.
The domain of a function of three variables is the set of ordered triples (x,y,z) for which f is defined

5. Domain for a function of two variables
Math 200
Domain for a function of two variables
Consider the function f(x,y) = ln(1-x2-y2)
We know that the input for ln() must be positive
1 - x2 - y2 > 0
x2 + y2 < 1
Let’s sketch the domain along with the graph of f

6. One more 2-variable example
Math 200
One more 2-variable example
Find and sketch the domain for the function
We need the argument of the square root to be greater than or equal to zero
x2 + y2 - 1 ≥ 0
x2 + y2 ≥ 1
All points on and outside the unit circle

7. Math 200
GRAPH
DOMAIN

8. Domain of a function of three variables
Math 200
Domain of a function of three variables
Consider the function f(x,y,z) = arcsin(x2+y2+z2)
Notice that there’s no graph of f - it would be 4D!
But we can still find the domain:
-1 ≤ x2+y2+z2 ≤ 1
x2+y2+z2 ≤ 1
Every point on and inside the unit sphere

9. Level curves and contour plots (or contour maps)
Math 200
Level curves and contour plots (or contour maps)
A level curve for a function f(x,y) is a trace of the form z=constant
It’s often useful to graph and label several level curves together on one set of axes
We call this a contour plot
E.g. consider the function z=x2-y2
z=0: 0 = x2 - y2
x2 = y2
|x| = |y|
z=1: 1 = x2 - y2
x2 = y2 + 1
z=2: x2 = y2 + 2
z=-1: y2 = x2 + 1
z=-2: y2 = x2 + 2

10. Think of this as a topographical map of the surface f(x,y) = z
Math 200
z=0: |x| = |y|
z=1: x2 = y2 + 1
z=2: x2 = y2 + 2
z=-1: y2 = x2 + 1
z=-2: y2 = x2 + 2
z=-1
z=3
z=1
z=1
z=3
z=0
z=2
z=2
z=-1
Think of this as a topographical map of the surface f(x,y) = z

11. Math 200

12. Math 200
z=0
z=1
z=-1
z=2
z=3

13. Level surfaces k = -1 -1 = x2 + y2 - z2 z2 = x2 + y2 +1
Math 200
Level surfaces
While we can’t graph functions of three variables, we can plot their level surfaces
Level surfaces: given f(x,y,z), setting the function equal to a constant yields a level surface
E.g. consider the function f(x,y,z) = x2 + y2 - z2
Set f(x,y,z) = k (const.)
k = -1
-1 = x2 + y2 - z2
z2 = x2 + y2 +1
Hyperboloid of 2 sheets
k = 0
= x2 + y2 - z2
z2 = x2 + y2
Double Cone
k = 1
1 = x2 + y2 - z2
z2 = x2 + y2 -1
Hyperboloid of 1 sheet

14. Math 200
k = -1
-1 = x2 + y2 - z2
z2 = x2 + y2 +1
Hyperboloid of 2 sheets
k = 0
= x2 + y2 - z2
z2 = x2 + y2
Double Cone
k = 1
1 = x2 + y2 - z2
z2 = x2 + y2 -1
Hyperboloid of 1 sheet
We can think of these level surfaces as 3D cross-sections of a 4D object