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 problem

  1. Jan 23, 2014 #1
    1. The problem statement, all variables and given/known data

    let Bn be a ball in Rn with radius r. ∂Bn is the boundary. Use divergence theorem to show that:

    V(Bn(r)) = (r/n) * A (∂Bn(r))

    where V(Bn) is volume and A(∂Bn) is surface area.

    2. Relevant equations

    consider the function: u = x1 ^2 + x2 ^2 +....+ xn ^2

    3. The attempt at a solution

    I have defined Bn: {(x1,x2,...,xn) , x1 ^2 + x2 ^2 +....+ xn ^2 < r^2}
    ∂Bn: {(x1,x2,...,xn) , x1 ^2 + x2 ^2 +....+ xn ^2 = r^2}

    i know that ∫(on Bn) of Δu dV = ∫(on ∂Bn) of (∂u/∂n) dA
    where n is the unit normal vector on ∂Bn.

    grad(u) = (2x1,2x2,...,2xn) = 2* (x1,x2,...,xn)
    Δu = div(grad(u)) = 2 (1+1+...+1) = 2n

    That is about all I've got. Thanks for any help.
  2. jcsd
  3. Jan 23, 2014 #2


    User Avatar
    Science Advisor
    Homework Helper

    They want you to apply the divergence theorem to the vector grad(u). What's an expression for the unit normal n? (∂u/∂n) must be the directional derivative. That's the same as the dot product of n with grad(u). What's that? The integrands of both integrals are constants. That should make them easy to integrate over the ball and the boundary.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted