- #1

s.perkins

- 5

- 0

## 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.