Answer
The upward flux is equal to the area , that is, projection in the $xy$ plane.
Work Step by Step
Let $Z$ denotes the orientation of a surface of a sphere and is given as: $Z=g(x,y)$
Here, $g(x, y)$ defines the continuous and differentiable function.
Next, the normal vector for the given sphere can be expressed as:
$(-Z_x, -Z_y, 1)$ in the upward direction.
We are given that the vector field is: $F=\lt 0, 0, 1 \gt$
Thus, the Green's Theorem for the upward flux is:
$F=\iint_S F \cdot n \ ds\\=\iint_R \lt 0, 0, 1 \gt \cdot (-Z_x, -Z_y, 1) \ dA \\= \iint_R dA\\=A$
This implies that the upward flux is equal to the area , that is, projection in the $xy$ plane.
