How to Use Vector Analysis Identity to Solve a Closed Loop Integral?

[neelakash]

Homework Statement

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

Homework Equations

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?

 

[Meir Achuz]

There is an identity:
$$\oint{\bf dr\times V}={\bf \int(\nabla V )\cdot dS<br /> - \int dS(\nabla\cdot V)}$$.
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.

 

[neelakash]

OK,thank you.Your method worked nicely...
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.

 

[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.

 

[neelakash]

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