Vector Integration: Solving \int_{S}\int n dS = 0 on Closed Surfaces

[yusukered07]

Homework Statement

\int_{S}\int 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.

 

[Dustinsfl]

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.

 

[LCKurtz]

Homework Statement

\int_{S}\int 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 V \cdot i. So look at, for example:

i \cdot \int\int_S \hat n\ dS = \int\int_S i\cdot \hat n\ dS

 

[yusukered07]

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

i \cdot \int\int_S \hat n\ dS = \int\int_S i\cdot \hat n\ dS

What is i there?

n is normal line, I think

 

[LCKurtz]

i is the unit vector in the x direction, the usual i,j,k notation. n is the unit outward normal.

 

[Matterwave]

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?

 

[yusukered07]

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?

 

[yusukered07]

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..

 

[gabbagabbahey]

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, \textbf{n} 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.

 

[gabbagabbahey]

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