# Integration of vectors

## Homework Statement

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

## The Attempt at a Solution

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

## The Attempt at a Solution

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 Helper
Gold Member

## Homework Statement

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

## The Attempt at a Solution

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

## 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:

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

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

Matterwave
Gold Member
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?

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?

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
Homework Helper
Gold Member
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
Homework Helper
Gold Member
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 