15.1 Graphing functions of several variables

One interesting thing about functions of two variables f=f(x,y) is that we can graph them, similarly with the way we graph ordinary one variable functions. This is done by interpreating the value f(x,y) that f takes at the point (x,y) as a third coordinate z=f(x,y). As we have already observed when we discussed curves, the computers come handy in better understanding some of the properties of functions. As an example, try to match the three graphs below with the functions:

f(x,y) = -x^2 + y^2 (saddle for humans)

f(x,y) = xy^2 - x^3 (saddle for monkeys)

f(x,y) = xy^3 - yx^3 (saddle for dogs)

> with(plots):

>

>

>

> F:=plot3d([x,y,-x^2+y^2],x=-1..1,y=-1..1,style=wireframe,color=blue,numpoints=2000): display(F,axes=frame,labels=[x,y,z]);

> G:=plot3d([x,y,-x^3+x*y^2],x=-1..1,y=-1..1,style=wireframe,color=blue,numpoints=2000): display(G,axes=frame,labels=[x,y,z]);

> H:=plot3d([x,y,x*y^3 - y*x^3],x=-1..1,y=-1..1,style=wireframe,color=blue,numpoints=2000): display(H,axes=frame,labels=[x,y,z]);