The following surface integral is to be evaluated over a

The following surface integral is to be evaluated over a sphere for the given steady velocity vector field F = xî + yĵ + zk̂ defined with respect to a Cartesian coordinate system having i, j and k as unit base vectors.
∬_s ( F.n )dA
where S is the sphere, x² + y² + z³ = 1 and n is the outward unit normal vector to the sphere. The value of the surface integral is

Use Gauss-Divergence theorem,

I = ∬_s	1	F.n̂dA
	4

= ∭_v ∇.FdV
∇.F = 1 + 1 + 1 = 3

∴ I =	1	× 3V =	3	×	4	π(1)³ = π
	4		4		3