find the flux of the vector field

Question

Let (S) the portion of the sphere " x² +y²+z²=1 " that lies above xy-plane.

Find the flux of the vector field:

F = x i + y j + z k across (S)

Answer 1

The answer is 2π because

1) At each point on the sphere, the magnitude |F| = 1.

2) At each point on the sphere, F is perpendicular to the surface of the sphere.

3) The flux, being the integral of F over the surface, then becomes just

the area of the half-sphere with radius 1, which is 2π.