1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Divergence Theorem: Multiplied by Scalar Field

  1. Oct 28, 2012 #1
    1. The problem statement, all variables and given/known data


    2. Relevant equations

    Definitely related to the divergence theorem (we're working on it):


    3. The attempt at a solution

    I'm a bit confused about multiplying a scalar field f into those integrals on the RHS, and I'm not sure if they can be taken out or not. If they can be, I evaluated the RHS out to be 0 (zero), which doesn't make sense with my evaluation of the LHS, which is just grad f dotted into F.

    On the other hand, if it CAN'T be taken out of the integral, I'm at a loss as of how this relates to the divergence theorem..
    I'm not sure what I'm missing here :( Help would be very much appreciated!
  2. jcsd
  3. Oct 28, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Start by using the divergence theorem on the first term on the right$$
    \iint_{\partial R}(f\vec F)\cdot \hat n\, dA = \iiint_R\nabla \cdot (f\vec F)\, dV$$Work out that ##\nabla \cdot (f\vec F)## in the integrand and go from there.
  4. Oct 28, 2012 #3


    User Avatar
    Science Advisor
    Homework Helper

    First show,
    [tex]\nabla \cdot (fF)=\nabla f \cdot F+f \nabla \cdot F[/tex]
    If you write it out in components it's just the product rule.
  5. Oct 28, 2012 #4


    User Avatar
    Homework Helper

    This is the product rule
    [tex]\nabla \cdot (\psi\mathbf{A}) = \mathbf{A} \cdot\nabla\psi + \psi\nabla \cdot \mathbf{A}[/tex]
    wrapped up with the divergence theorem.
  6. Oct 28, 2012 #5
    For the integrand, I'm getting:
    [itex]\partial (f A) / \partial x + \partial (f B) / \partial y + \partial (f C) / \partial z [/itex], where [itex]\vec F = (A, B, C)[/itex]

    Am I on the right track?
  7. Oct 28, 2012 #6
    Double Post, but:

    Ahh so if I use the Divergence Thm as LCKurtz suggested on the first term on the RHS, I get the product rule in the form:

    [tex]\nabla f \cdot F = \nabla \cdot (fF) - f \nabla \cdot F[/tex]

    Except with the terms as integrands. I'm not sure if this is sufficient to prove the validity of the equation though? I'm sorry guys, I feel like you guys are putting the answer right in my face but I'm just not getting it :(
  8. Oct 28, 2012 #7


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Assuming ##\vec F = \langle A,B,C\rangle##, Yes. Keep going...
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook