Applying the Divergence Theorem to the Volume of a Ball with a Given Radius

[s.perkins]

Homework Statement

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.

Homework Equations

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

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.

 

[Dick]

Homework Statement

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.

Homework Equations

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

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.

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.