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Choosing a surface that makes the flux of F maximal,

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2

$begingroup$

For a closed surface S in $R^3$, consider the flux of F, given by the usual flux integral. For what choice of S will the flux be maximal?

So, I want to apply the divergence theorem and instead look at the triple integral of the divergence of the given vector field F (over a solid that is enclosed by S.)

Since the flux integral = the divergence integral, I can aim to maximize the divergence integral.

My computation of the divergence gives me -3($x^2 + y^2 + z^2 - frac53$), so this would be the integrand in the triple integral.

It sounds reasonable that in order to maximize this triple integral, I should maximize the integrand. How do I do that?

My attempt was to make $x^2 + y^2 + z^2$ - $frac53$ < 0, since there's a factor of -3 to consider. negative * negative will give me a positive integrand - which would be good for maximizing the flux of F.

Then this tells me that I should choose a sphere of radius $sqrt(frac53)$.

I carried out my work and it looks like I got the correct answer.

But I feel like I chose my sphere ...by luck.

How do I know for sure that I've maximized the integrand, simply by making $x^2 + y^2 + z^2$ - $frac53$ < 0? Could I have done even better, achieving a better maximum? I simply knew that this factor had to be < 0, since it was being multiplied by -3.

Thanks,

calculus real-analysis multivariable-calculus optimization

share|cite|improve this question

asked Jun 17 '15 at 0:30

user248707user248707

133

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Can you provide the actual vector field equation?
  $endgroup$
  – user237392
  Jun 17 '15 at 1:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sure, @Bey -- it's F(x,y,z) = $(2x - y^2 - x^3, 3y-y^3, -x-z^3)$.
  $endgroup$
  – user248707
  Jun 17 '15 at 1:17
  

  
  
  
  

add a comment | 

2

$begingroup$

For a closed surface S in $R^3$, consider the flux of F, given by the usual flux integral. For what choice of S will the flux be maximal?

So, I want to apply the divergence theorem and instead look at the triple integral of the divergence of the given vector field F (over a solid that is enclosed by S.)

Since the flux integral = the divergence integral, I can aim to maximize the divergence integral.

My computation of the divergence gives me -3($x^2 + y^2 + z^2 - frac53$), so this would be the integrand in the triple integral.

It sounds reasonable that in order to maximize this triple integral, I should maximize the integrand. How do I do that?

My attempt was to make $x^2 + y^2 + z^2$ - $frac53$ < 0, since there's a factor of -3 to consider. negative * negative will give me a positive integrand - which would be good for maximizing the flux of F.

Then this tells me that I should choose a sphere of radius $sqrt(frac53)$.

I carried out my work and it looks like I got the correct answer.

But I feel like I chose my sphere ...by luck.

How do I know for sure that I've maximized the integrand, simply by making $x^2 + y^2 + z^2$ - $frac53$ < 0? Could I have done even better, achieving a better maximum? I simply knew that this factor had to be < 0, since it was being multiplied by -3.

Thanks,

calculus real-analysis multivariable-calculus optimization

share|cite|improve this question

asked Jun 17 '15 at 0:30

user248707user248707

133

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Can you provide the actual vector field equation?
  $endgroup$
  – user237392
  Jun 17 '15 at 1:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sure, @Bey -- it's F(x,y,z) = $(2x - y^2 - x^3, 3y-y^3, -x-z^3)$.
  $endgroup$
  – user248707
  Jun 17 '15 at 1:17
  

  
  
  
  

add a comment | 

$begingroup$

For a closed surface S in $R^3$, consider the flux of F, given by the usual flux integral. For what choice of S will the flux be maximal?

So, I want to apply the divergence theorem and instead look at the triple integral of the divergence of the given vector field F (over a solid that is enclosed by S.)

Since the flux integral = the divergence integral, I can aim to maximize the divergence integral.

My computation of the divergence gives me -3($x^2 + y^2 + z^2 - frac53$), so this would be the integrand in the triple integral.

It sounds reasonable that in order to maximize this triple integral, I should maximize the integrand. How do I do that?

My attempt was to make $x^2 + y^2 + z^2$ - $frac53$ < 0, since there's a factor of -3 to consider. negative * negative will give me a positive integrand - which would be good for maximizing the flux of F.

Then this tells me that I should choose a sphere of radius $sqrt(frac53)$.

I carried out my work and it looks like I got the correct answer.

But I feel like I chose my sphere ...by luck.

How do I know for sure that I've maximized the integrand, simply by making $x^2 + y^2 + z^2$ - $frac53$ < 0? Could I have done even better, achieving a better maximum? I simply knew that this factor had to be < 0, since it was being multiplied by -3.

Thanks,

calculus real-analysis multivariable-calculus optimization

share|cite|improve this question

asked Jun 17 '15 at 0:30

user248707user248707

133

$endgroup$

share|cite|improve this question

asked Jun 17 '15 at 0:30

user248707user248707

133

share|cite|improve this question

asked Jun 17 '15 at 0:30

user248707user248707

133

asked Jun 17 '15 at 0:30

user248707user248707

133

asked Jun 17 '15 at 0:30

user248707user248707

133