Many thanks to Professor T.J. Murphy of University of Oklohoma
and to Professor John F. Putz of Alma College, Michigan for
preparing the animations mentioned here, and for making them
available to the whole world, including us.
<br>

<a 
href="http://www.math.ou.edu/%7Etjmurphy/Teaching/2443/LevelCurves/levelCurves.html">Professor 
Murphy's animation</a> 
shows the level curves and the graph of the 
function f(x,y)=x^2+y^2 and the intersection of the graph with 
planes z=constant and x=constant. Please remember that the level 
curves are defined in the xy plane and NOT in three dimensional 
space. Yes, they look 
like the intersection curves of the graph with planes 
z=constant, because they are the PROJECTIONS of these curves 
onto the xy plane.
<br>
<br>

<a 
href="http://archives.math.utk.edu/ICTCM/VOL10/C009/paper.html"?>
Professor Putz' animation </a> shows us level surfaces of 
various functions of three variables (these particular surfaces 
are in 
fact quadric surfaces), and their intersections with various 
planes.
You will also see graphs and level curves of some other 
functions of two variables including one which has a "saddle 
point" (to be discussed in connection with the topic of second 
order 
derivatives), and some animations which should 
help us understand the gradient and the directional derivative 
of a function of TWO variables, when we come to that topic. 
<br>
Warning: Many students get confused between the meanings of 
gradient and directional derivative for functions of TWO 
variables and for functions of THREE variables. What you 
see here is for TWO variables.