Functions of 3 Variables
Just as level curves of a function of 2 variables are curves in the xy -plane, level curves of functions of 3 variables are surfaces
U ( x,y,z ) is a function of 3 variables and k is a constant, then U ( x,y,z ) = k is a level surface of level k of the function U. To plot a level surface in Maple , we use the command "implicitplot3d".
For example, if
U
(
x,y,z
) =
, then level curves of
U
are spheres of radius
centered at the origin. If
k=
1, then the level surface is the unit sphere
To plot
, we use the command "implicitplot3d."
| > | implicitplot3d(x^2+y^2+z^2=1,x=-1..1,y=-1..1,z=-1..1); |
| > |
As another example, consider that
z = f
(
x,y
) is not only the graph of the function
f
, but it is also a level surface of level 0 of the function
U
(
x,y,z
) =
z - f
(
x,y
). For instance, the graph of
z
=
is not only the graph of
f
(
x,y
) =
, but it is also a level surface of
U
(
x,y,z
) =
. Let's graph the surface both as the graph of a function and as a level surface.
| > | pp[1]:=plot3d(x^2+y^2,x=-1..1,y=-1..1,title="As a function"): pp[2]:=implicitplot3d(z-x^2-y^2,x=-1..1,y=-1..1,z=0..2,title="As a level surface"): display(pp[1],pp[2],insequence=true); #Click and choose play to animate |
| > |
Notice that the surfaces are the same, but that the "lines" drawn on the surface are different. We will explore this idea in more detail later in chapter 3. However, suffice it to say for now that a level surface is not necessarily a function. Computers in general and Maple in particular are better suited to working with explicitly defined functions and mappings than with set of points and implicitly defined relationships.