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Showing a set is compact

  • Thread starter MuIotaTau
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  • #1
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Homework Statement



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)##.

Homework Equations



None

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?
 

Answers and Replies

  • #2
pasmith
Homework Helper
1,735
410

Homework Statement



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)##.

Homework Equations



None

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 $$K = \{(x,y,z)|g(x,y,z) = 0\}$$
What is g?

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?
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.
 
  • #3
33,072
4,773

Homework Statement



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)##.

Homework Equations



None

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?
What's g(x, y, z)?
 
  • #4
82
3
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.
 
  • #5
Dick
Science Advisor
Homework Helper
26,258
618
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.
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.
 
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