13.1 day 2 level curves

Level curvesOne 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

andhttp://archives.math.utk.edu/ICTCM/VOL10/C009/paper.html#Level%20curves

%20and%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 frameSelect 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/~nykamp/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.)