What are the level surfaces of the function f(x,y,z) = z + sqrt(x^2 + y^2)?

[kasse]

Homework Statement

Describe the level surfaces of f(x,y,z) = z + sqrt(x^2 + y^2)

The Attempt at a Solution

First of all, what is actually a level surface? Just a normal surface in space?

I followed an example I found on the internet, and this is my attempt at a solution:

First replace f(x,y,z) with a constant

k = z + sqrt(x^2 + y^2)

Then square (k is now another constant)

k = z^2 + x^2 + y^2

This is an ellipsoid, so the level surfaces are ellipsoids centered at the origin.

Is this the right solution? If so, is it possible to say more about the ellipsoids?

 

[matt grime]

The definition of a level surface of function of f(x,y,z) is the solutions to f(x,y,z)=k for a constant k.

Now, please don't tell me that you think (a+b)^2=a^2+b^2, as you wrote above...

 

[kasse]

OK, so

k = z^2 + 2sqrt(x^2+y^2) + x^2 +y^2

then.

Is this one easy to recognize as a 3D-figure?

 

[Dick]

[
k = z + sqrt(x^2 + y^2)

Get the radical all by itself on one side of the equation before you square. It will be MUCH easier to recognize.

 

[kasse]

Get the radical all by itself on one side of the equation before you square. It will be MUCH easier to recognize.

Then i get x^2 + y^2 - z^2 = k^2 - 2kz

Still doesn't resemble anything I'm familiar with.

 

[Dick]

Leave it as x^2+y^2=(k-z)^2. x^2+y^2 is often called r^2, right? So write this as r=|z-k|. Can you describe it now?

 

[kasse]

A sphere, isn't is?

 

[Dick]

A sphere, isn't is?

Nope. Look, r is the radius in the xy plane, right? If r=0 then z=k. What is z for r=1? Note k-z must be positive. Why?