13.1 day 2 level curves
Level curves
One way to visualize a function of two
variables is to use a scalar field in which
the scalar z = f(x,y) is assigned to the point
(x,y). A scalar field can be characterized
by level curves or contour lines.
(yesterday we learned how to create a
“wire” frame by looking at constant
values of x then constant values of y)
Examples of level curves
These level curves shows
lines of equal pressure in
millibars
These level curves show the
lines of equal
temperature in degrees F
Level Curves
These level curves show
lines of equal elevation
above sea level
Alfred B. Thomas/Earth Scenes
USGS
For an animation of this concept visit:
http://archives.math.utk.edu/ICTCM/VOL10/C009/lc.gif
and
http://www.math.umn.edu/~nykamp/m2374/readings/levelset/index.html
and
http://archives.math.utk.edu/ICTCM/VOL10/C009/paper.html#Level%20curves%20
and%20level%20surfaces
Example 3
the hemisphere given by
Draw level curves for z = 0,1,2, ….8
Example 3
x
Graphing on TI-89
• Change the format
Select wire frame
Select f1 (tools) scroll
down to 9 (format)
select wire frame.
Graph the equation
normally.
You can rotate the
surface with the arrow
keys.
Graphing level curves
To graph level curves on
the TI-89
Option 1
Do the same procedure
as before but select
contour levels
Graph normally.
Recall: Select f1 (tools)
scroll down to 9
(format) select level
curves.
You can rotate this to
see the “height” of the
contours.
Graphing level curves on TI-89
Option 2
Select function mode.
Find the equation of the
individual level curves
and graph them as
functions.
Use copy and paste to
avoid retyping.
Graph normally
Graphing level curves on TI-89
Use zoom square
to obtain a better
picture of the
curves
This method is
slower but lets
you select the
number and
z-value of the level
curves.
Example 4
Draw the level curves for this surface:
Example 4
(note: c is the value for z)
The concept of a
level curve can be
extended by one
dimension to define
a level surface.
f(x,y,z)=c is a level
surface of the
function f.
With computers,
engineers and
scientists have
developed other ways
to view functions with 3
variables. For instance,
this figure shows a
computer simulation
that uses color to
represent the optimal
strain on a car door.
Reprinted with permission. © 1997 Automotive Engineering Magazine. Society of Automotive Engineers, Inc.
Scientists have
expanded the use of
color into higher
dimensions. This
system represents a
function of 3
independent
variables and 1
dependent variable
denoted by color.
Example, this class
room. The three
independent
variables could be
length width and
height with
temperature as the
dependent variable.
Example 6
Describe the level surfaces of the function.
Example 6
Describe the level surfaces
of the function.
Level surfaces also be
depicted as follows:
Examples of level surfaces:
For more information
visit:
http://www.math.umn.edu/~nykam
p/m2374/readings/levelcurve/
Teamwork is important…
A chain is only as strong as its weakest
link.
You will need to support each other as
We go through this class.
(Hopefully better than the people in this
comic.)