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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

    Dick

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    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.
     
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