Find a function of two variables with these propertes. .

[Vampire]

Homework Statement

Find a function of two variables whose level curves are parabolas with vertex (0,0) with a hole in the parabolas at the origin.

Homework Equations

No special equations come to mind.
4ay=x² may be a little useful.

The Attempt at a Solution

The first thing I thought of was y=x³/x, but that doesn't include the third variable. I know the parabolas can have any value for a or face any direction, but I do not know how to include z so that the function will remain having level curves as stated.

Are there any other functions with these properties? Any input will be appreciated.

 

[lanedance]

so i assume by z you mean the function z = F(x,y) we are trying to find?

so we have the level curves of z = F(x,y), given by F(x,y) = c, for some constant a, are given by y = ax^2, for some constant a.

What if we set the constants the same?

F(x,y) = a --> y = a^x2
hopefully this will get you started...

a few other things to consider.
- The solution for F(x,y) is not uniquie - can you find the full family of solutions?
- The gradient of F(x,y) will be perpindicular to a level curve at any point, useful check
- The levels curevs have a "hole" at (0,0), why & what does this imply for F(x,y)?