Question

Using divergence theorem, evaluate ∫∫_s vector F.ndS where vector F = x³i + y³j + z³k and S is the surface of the sphere x² + y² + z² = a².

Answer 1

∴ by divergence theorem, we get

Since, V is the volume of the sphere we transform the above triple integral into spherical polar coordinates (r, θ, φ).

For the spherical polar coordinates (r, θ, φ), we have

x² + y² + z² = r² and dx dy dz = dV

∴ dV = r²sin θ dr dθ d φ

Also, 0 ≤ r ≤ a, 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π

Therefore (1) reduces to,