Vector Calculus: Solving a Solenoidal Field Problem

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To show that the integral of A . grad(phi) over volume V equals zero, where A is a solenoidal field and its normal component at the boundary vanishes, one must find a function whose divergence is A . grad(phi). The integrand does not necessarily vanish, but by taking the divergence of (phi A), the integral can be transformed into an expression involving the surface integral of phi A over the boundary. Since the normal component of A at the boundary is zero, this surface integral vanishes, confirming the original statement. The approach taken is correct and aligns with the conditions provided.
Kolahal Bhattacharya
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Homework Statement



I am to show
int{A . grad(phi)}dV=0 where A is a solinoidal field and normal component of A at the boundary vanishes.
I expressed A as curl F and suspect that the (curl F).(grad phi)=0
So,I am done.Is it correct?

Homework Equations





The Attempt at a Solution

 
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You aren't supposed to show the integrand vanishes. In general, it doesn't. You need to find something whose divergence is A.grad(phi) and then argue that integrating that something over the boundary vanishes.
 
I hyope i got it.I took the divergence of (phi A),and what comes out is that the given integral transforms to Int{phi A}.dS where I can apply the additional condition
 
Yes. That's it.
 

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