# Homework Help: Integration of vectors

1. Mar 13, 2010

### yusukered07

1. The problem statement, all variables and given/known data

$$\int_{S}$$$$\int$$ n dS = 0 for any closed surface S.

2. Relevant equations

3. The attempt at a solution
I can't solve this because I don't have any idea in Vector intregrals.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Mar 13, 2010

### 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.

3. Mar 13, 2010

### LCKurtz

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:

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

4. Mar 13, 2010

### yusukered07

What is i there???

n is normal line, I think

5. Mar 13, 2010

### LCKurtz

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

6. Mar 13, 2010

### 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?

7. Mar 13, 2010

### yusukered07

So, will you help me to make a proof?

8. Mar 13, 2010

### yusukered07

is there any proof you can show??

that's the question..

9. Mar 14, 2010

### gabbagabbahey

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.

10. Mar 14, 2010

### gabbagabbahey

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