# What is the difference between a level set and a level curve?

I have the following queston:

Consider the function $f(x,y)=(4-x^2-y^2)^2$.

a) Sketch all the level curves for $f(x,y)=c$ for $c=0$, $c=4$ and $c=25$

b) Also plot the level set for $f(x,y)=16$.

So, I know to find the level curves I have to solve the equation equal to my value of c. However, I have tried looking online but I can't understand what the difference between a level curve and a level set is. I would naturally just solve it in the same way as a level curve with $c=16$. But the distinction between a level curve and a level set is made so I believe there must be some difference in approach, but I personally cannot see what it is.

Any insight would be appreciated.

• A level set is the generalization of a level curve. Level curves are for functions in two variables, while level sets are for any number of variables. Feb 15, 2017 at 14:01
• According to Wikipedia, in two variables level set and level curve are the same. Font: en.wikipedia.org/wiki/Level_set Feb 15, 2017 at 14:02
• @MichaelBurr Why would the same not be true for c=25? As only one solution would exist since a radius of $\sqrt{-1}$ is not feasible.Also, I guess my understanding may be off but even with $c=16$, you would get a solution of $x^2+y^2=8$, why is this not a curve?
– Evan
Feb 15, 2017 at 14:17

A level curve is a type of level set. For $c=16$, the only point in the solution set is the origin, $x=y=0$. A single point is not a curve.