# Vector analysis identity

1. May 16, 2007

### neelakash

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

we are to show a=(1/2) closed loop integral over [r x dl]

2. Relevant equations

3. The attempt at a solution

I suppose this can be done formally from the alternative form of Stokes' theorem that can be obtained by replacing the vector field in curl theorem by VxC where C is a constant vector

The identity is :

surface int [(da x grad) x V]=closed loop integral over [dl x V]

The RHS matches.But how to show that LHS leads to the required value?

2. May 16, 2007

### Meir Achuz

There is an identity:
$$\oint{\bf dr\times V}={\bf \int(\nabla V )\cdot dS - \int dS(\nabla\cdot V)}$$.
This can be derived by dotting the left hand side by a constatn vector, and then applying Stokes' theorem.
Applying this with V=r works.

Last edited: May 16, 2007
3. May 16, 2007

### neelakash

First I was sceptical about the grad V in your RHS...However,I started from the very beginning by dotting c with the required integral and it worked well.

4. May 17, 2007

### Meir Achuz

There is an easier way I overlooked. Just take
$${\vec k}\cdot\oint{\vec r}\times{\vec dr}$$
where k is a constant vector, and apply Stokes' theorem.

Last edited: May 17, 2007
5. May 17, 2007

### neelakash

I did just that...your dr reolaced by dl...