# Level curves and functions of three variables

I would like to ask just a quick question. Say for example I give you a function of two variables $z = f(x,y)$ = $x^2 + y^2$ which represents a paraboloid. If I want the level curves $f(x,y) = c$, then these now represent concentric circles in the $x-y$ plane centered at the origin of radius $\sqrt{c}$.

Now here's my question. Say I have $w = f(x,y,z)$ now a function of three variables, i.e. it is a hypersurface in $\mathbb{R}^4$. If I have a level "curve" say $w = f(x,y,z) = 0$, does this then represent now a level "surface" in $\mathbb{R}^3$?

Thanks, Ben

Absolutely - and this method can be extended to any number of dimensions. Here is the Wikipedia article on the general concept, called a "level set".

For example, if $w=f(x,y,z)=x^2+y^2+z^2$, then the level surfaces $f(x,y,z)=c$ represent the concentric spheres of radius $\sqrt{c}$ centered at the origin.

Here are the level surfaces of $f(x,y,z)=x^2-y^2-z^2$, for

$f(x,y,z)=1$

here http://www3.wolframalpha.com/Calculate/MSP/MSP388119f46d7361i3966h000046ac79g6ge037cg3?MSPStoreType=image/gif&s=13&w=273&h=300&cdf=Rotation

$f(x,y,z)=4$

here http://www3.wolframalpha.com/Calculate/MSP/MSP26219f46h62f84da1ag00001ac8h48162gb4i8e?MSPStoreType=image/gif&s=13&w=273&h=300&cdf=Rotation

$f(x,y,z)=16$

here http://www3.wolframalpha.com/Calculate/MSP/MSP209219f46fb1ifh4308200000di10292120dhe16?MSPStoreType=image/gif&s=3&w=273&h=300&cdf=Rotation

• Ok thanks Zev that makes things clear now. So If I give you $w = f(x_1, x_2 \ldots x_n) = (x_1^2 + \ldots x_n^2)$, a hypersurface in $\mathbb{R}^n$, then the level "curves" $w=c$ are hyperspheres in $\mathbb{R}^{n-1}$ of radius $\sqrt{c}$? – user38268 Apr 10 '11 at 23:54
• @David: Yes, they are. – Alex Becker Apr 10 '11 at 23:58
• @David - I've added some level surfaces for a function other than $x_1^2+\cdots+x_n^2$, just to give another example. You can play around with level sets on WolframAlpha: go to http://www.wolframalpha.com/input/?i=ContourPlot3D%5Bx^2-y^2-z^2%3D%3D1%2C{x%2C-8%2C8}%2C{y%2C-8%2C8}%2C{z%2C-8%2C8}%5D, it should be clear how to change it to graph what you want. – Zev Chonoles Apr 11 '11 at 0:04
• Ok thanks I'll type the codes into Mathematica and try a few functions. – user38268 Apr 11 '11 at 1:04