1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Showing a set is compact

  1. Nov 6, 2013 #1
    1. The problem statement, all variables and given/known data

    Explain why ##f(x,y,z) = x + y - z## must attain both a maximum and a minimum on the sphere ##x^2 + y^2 + z^2 = 81)##.

    2. Relevant equations


    3. The attempt at a solution

    I know that any continuous function attains both a maximum and a minimum on a compact set. I defined the set ##K## as $$K = \{(x,y,z)|g(x,y,z) = 0\}$$ and demonstrated that it is bounded by a closed ball. I then realized, however, that I'm not really sure how to show that the set is closed. I understand the definition of a closed set, but I don't know how to show that it is true in this particular case. Any hints?
  2. jcsd
  3. Nov 6, 2013 #2


    User Avatar
    Homework Helper

    What is g?

    It suffices to show that the sphere is compact. To do that, you can write down a continuous surjection from a compact subset of the plane to the sphere and use the result that the continuous image of a compact space is compact.
  4. Nov 6, 2013 #3


    Staff: Mentor

    What's g(x, y, z)?
  5. Nov 6, 2013 #4
    Oops, I'm sorry, I defined ##g## as ##g(x,y,z) = x^2 + y^z + z^2 - 81##. So the constraint equation.

    We haven't demonstrated that compactness is continuous invariant in class, unfortunately, so I would be required to prove that. If demonstrating the set is closed is more complicated, I'll do it, but I would rather not prove a theorem if it's not strictly necessary.
  6. Nov 6, 2013 #5


    User Avatar
    Science Advisor
    Homework Helper

    Proving that the set of points where g(x,y,z)=0 is closed should be easy. Just use the definition of continuity directly or that the inverse image of a closed set is closed for a continuous function. Now use Bolzano-Weierstrass.
    Last edited: Nov 6, 2013
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted