# Generating level surface from 2 variable function

## Homework Statement

given f(x,y)=9/(x+y) find a level surface.

## The Attempt at a Solution

g(x,y,z)=f(x,y)-z=0?
g(x,y,z)=9/(x+y) -z=0?

That answer is wrong. Apparently i must have the following:

g(x,y,z)=z(x+y)=9

How do I solve problems like these?

Another example is to find a function f(x,y,z) whose level surface f=5 is the graph of the paraboloid
g(x,y)=-x^2 -y^2

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LCKurtz
Homework Helper
Gold Member

## Homework Statement

given f(x,y)=9/(x+y) find a level surface.

## The Attempt at a Solution

g(x,y,z)=f(x,y)-z=0?
g(x,y,z)=9/(x+y) -z=0?

That answer is wrong. Apparently i must have the following:

g(x,y,z)=z(x+y)=9

How do I solve problems like these?

Another example is to find a function f(x,y,z) whose level surface f=5 is the graph of the paraboloid
g(x,y)=-x^2 -y^2
You don't need any $z$ variable. To find level surface of a function $f(x,y)$ just plot the graphs of $f(x,y)=C$ for various constants $C$. For a two variable problem like yours, they will be curves in the $xy$ plane, not surfaces.

Wouldn't that be generating level curves? I guess my terminology is bad. What i am asking is how do make f(x,y) into
F(x,y,z)

LCKurtz
You don't need any $z$ variable. To find level surface of a function $f(x,y)$ just plot the graphs of $f(x,y)=C$ for various constants $C$. For a two variable problem like yours, they will be curves in the $xy$ plane, not surfaces.
Yes, if you have a function $f(x,y)$ you would talk about its level curves, not level surfaces as your original question stated. If you want to plot the graph of the function $f(x,y)$ you would to a 3D plot of the equation $z=f(x,y)$, which is the same as $z - f(x,y)=0$ which is one of the level surfaces of $F(x,y,z)=z-f(x,y)$.