Level curves and functions of three variables - Mathematics Stack Exchange most recent 30 from math.stackexchange.com 2020-01-27T06:22:04Z https://math.stackexchange.com/feeds/question/32176 https://creativecommons.org/licenses/by-sa/4.0/rdf https://math.stackexchange.com/q/32176 3 Level curves and functions of three variables user38268 https://math.stackexchange.com/users/0 2011-04-10T23:44:02Z 2011-04-10T23:58:57Z <p>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 $\sqrt{c}$. </p> <p>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 $\mathbb{R}^3$?</p> <p>Thanks, Ben</p> https://math.stackexchange.com/questions/32176/-/32179#32179 5 Answer by Zev Chonoles for Level curves and functions of three variables Zev Chonoles https://math.stackexchange.com/users/264 2011-04-10T23:48:00Z 2011-04-10T23:58:57Z <p>Absolutely - and this method can be extended to any number of dimensions. <a href="http://en.wikipedia.org/wiki/Level_set" rel="nofollow noreferrer">Here</a> is the Wikipedia article on the general concept, called a "level set". </p> <p>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. </p> <p>Here are the level surfaces of $f(x,y,z)=x^2-y^2-z^2$, for</p> <p>$f(x,y,z)=1$</p> <p><a href="http://www3.wolframalpha.com/Calculate/MSP/MSP388119f46d7361i3966h000046ac79g6ge037cg3?MSPStoreType=image/gif&amp;s=13&amp;w=273&amp;h=300&amp;cdf=Rotation" rel="nofollow noreferrer">here http://www3.wolframalpha.com/Calculate/MSP/MSP388119f46d7361i3966h000046ac79g6ge037cg3?MSPStoreType=image/gif&amp;s=13&amp;w=273&amp;h=300&amp;cdf=Rotation</a></p> <p>$f(x,y,z)=4$</p> <p><a href="http://www3.wolframalpha.com/Calculate/MSP/MSP26219f46h62f84da1ag00001ac8h48162gb4i8e?MSPStoreType=image/gif&amp;s=13&amp;w=273&amp;h=300&amp;cdf=Rotation" rel="nofollow noreferrer">here http://www3.wolframalpha.com/Calculate/MSP/MSP26219f46h62f84da1ag00001ac8h48162gb4i8e?MSPStoreType=image/gif&amp;s=13&amp;w=273&amp;h=300&amp;cdf=Rotation</a></p> <p>$f(x,y,z)=16$</p> <p><a href="http://www3.wolframalpha.com/Calculate/MSP/MSP209219f46fb1ifh4308200000di10292120dhe16?MSPStoreType=image/gif&amp;s=3&amp;w=273&amp;h=300&amp;cdf=Rotation" rel="nofollow noreferrer">here http://www3.wolframalpha.com/Calculate/MSP/MSP209219f46fb1ifh4308200000di10292120dhe16?MSPStoreType=image/gif&amp;s=3&amp;w=273&amp;h=300&amp;cdf=Rotation</a></p>