Integration of vectors

  • #1

Homework Statement



[tex]\int_{S}[/tex][tex]\int[/tex] n dS = 0 for any closed surface S.

Homework Equations





The Attempt at a Solution


I can't solve this because I don't have any idea in Vector intregrals.

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
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It should only equal zero if it is conservative since the partial of n with respect to x equals the partial m with respect to y. I am not sure if this what you are looking for since I am not sure what your question is.
 
  • #3
LCKurtz
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Homework Statement



[tex]\int_{S}[/tex][tex]\int[/tex] n dS = 0 for any closed surface S.

Homework Equations





The Attempt at a Solution


I can't solve this because I don't have any idea in Vector intregrals.

Homework Statement




Homework Equations




The Attempt at a Solution


Hint: The result is a vector, so look at its components. The x component of a vector V is [itex] V \cdot i[/itex]. So look at, for example:

[tex] i \cdot \int\int_S \hat n\ dS = \int\int_S i\cdot \hat n\ dS[/tex]
 
  • #4
Hint: The result is a vector, so look at its components. The x component of a vector V is [itex] V \cdot i[/itex]. So look at, for example:

[tex] i \cdot \int\int_S \hat n\ dS = \int\int_S i\cdot \hat n\ dS[/tex]

What is i there???

n is normal line, I think
 
  • #5
LCKurtz
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i is the unit vector in the x direction, the usual i,j,k notation. n is the unit outward normal.
 
  • #6
Matterwave
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If the vector field is always tangential to the surface (and therefore perpendicular with the normal of the surface), this relation is trivially true. If the vector field is 0 everywhere, then this relation is also trivially true. What is the question exactly? Find all such vector fields n for which this relation holds?
 
  • #7
i is the unit vector in the x direction, the usual i,j,k notation. n is the unit outward normal.

So, will you help me to make a proof?
 
  • #8
If the vector field is always tangential to the surface (and therefore perpendicular with the normal of the surface), this relation is trivially true. If the vector field is 0 everywhere, then this relation is also trivially true. What is the question exactly? Find all such vector fields n for which this relation holds?

is there any proof you can show??

that's the question..
 
  • #9
gabbagabbahey
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If the vector field is always tangential to the surface (and therefore perpendicular with the normal of the surface), this relation is trivially true. If the vector field is 0 everywhere, then this relation is also trivially true. What is the question exactly? Find all such vector fields n for which this relation holds?

As LCKurtz already pointed out, [itex]\textbf{n}[/itex] is the outward unit normal to whichever closed surface is integrated over. The problem is to show that this integral is zero for any closed surface.
 
  • #10
gabbagabbahey
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So, will you help me to make a proof?

We don't make proofs for you here. LCKurtz has given you a very good hint in his first reply, try using it and show us what you get. You should find that the divergence theorem is very useful to you here:wink:
 

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