Determine whether the following vector field is incompressible or irrotational at the point (0,1,2).

Question

Vector F = [x/x^2+y^2+z^2]i^+[y/x^2+y^2+z^2]j^+[z/x^2+y^2+z^2]k^

Answer 1

incompressible = solenoiodal iff ∇•F = 0

∇•F = ∂F/∂x + ∂f/∂y + ∂f/∂z

= (-x^2+y^2+z^2)/(x^2+y^2+z^2)^2
+ (x^2-y^2+z^2)/(x^2+y^2+z^2)^2
+ (x^2+y^2-z^2)/(x^2+y^2+z^2)^2
= 1/(x^2+y^2+z^2)
∇•F at (0,1,2) = 1/5 ≠ 0

irrotational iff ∇×F = 0
I leave it to you to verify that in fact ∇×F = 0.