Level Surfaces

A function of 3 variables is a function of the form U(x,y,z) whose inputs are points in R3 and whose outputs are numbers. For example, U( x,y,z) = x2yz is a function of 3 variables. In this section, we generalize many of the concepts from chapter 10 to concepts involving functions of 3 variables.

Given a function of 3 variables U( x,y,z) , we define the level surface of U( x,y,z) of level k to be the set of all points in R3 which are solutions to
U( x,y,z) = k
Indeed, many of the most familiar surfaces are level surfaces of functions of 3 variables.           

 

EXAMPLE 1    Find the equation of a sphere of radius R centered at the origin.      

Solution: Every point ( x,y,z) on the sphere must be a distance R from the origin. Thus, the length of every vector with initial point ( 0,0,0) and final point ( x,y,z) is R, which means that
( x-0) 2+( y-0) 2+( z-0) 2
= R
This in turn simplifies to x2+y2+z2 = R2.   

A quadric surface is a level surface of a second degree polynomial Q( x,y) . Indeed, the sphere of radius R centered at the origin is a level surface of level k = R2 of the second degree polynomial
Q( x,y) = x2+y2+z2
Moreover, a sphere is a special type of ellipsoid, which is a surface of the form
 x2
a2
 +  y2
b2
 +  z2
c2
 = 1     

Other quadric surfaces include the elliptic paraboloids, which are defined by equations of the form
 z
c
 =  x2
a2
 +  y2
b2
 ,       

and the hyperbolic paraboloids, which are defined by equations of the form
 z
c
 =  x2
a2
 -  y2
b2
 ,    

In addition, there are the hyperboloids, where a hyperboloid in one sheet has an equation of the form
 x2
a2
 +  y2
b2
 -  z2
c2
 = 1,  

and a hyperboloid in 2 sheets has an equation of the form
 z2
c2
 -  x2
a2
 -  y2
b2
 = 1      

And finally, one of the most important classes of quadric surfaces are the elliptic cones, which are surfaces defined by equations of the form
 x2
a2
 +  y2
b2
 =  z2
c2
 ,          

For example, the surface defined by x2+y2 = z2 is a right cylindrical cone, of the type used to define the conics (see the end of this section):

       

Check your Reading: What type of quadric surface is given by the equation
x2-y2+z2 = 4