• Support PF! Buy your school textbooks, materials and every day products Here!

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
699
5
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
Science Advisor
Homework Helper
Insights Author
Gold Member
9,517
733

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
Science Advisor
Homework Helper
Insights Author
Gold Member
9,517
733
i is the unit vector in the x direction, the usual i,j,k notation. n is the unit outward normal.
 
  • #6
Matterwave
Science Advisor
Gold Member
3,965
326
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
Homework Helper
Gold Member
5,002
6
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
Homework Helper
Gold Member
5,002
6
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:
 

Related Threads for: Integration of vectors

Replies
8
Views
723
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
9K
Replies
8
Views
3K
Top