Given $$\,\,\overrightarrow F = z\widehat a{}_x + x\widehat a{}_y + y\widehat a{}_z.\,\,$$ If $$S$$ represents the portion of the sphere $${x^2} + {y^2} + {z^2} = 1$$ for $$\,z \ge 0,$$ then $$\int\limits_s {\left( {\nabla \times \overrightarrow F .} \right)\overrightarrow {ds} \,\,} $$ is ________.

The divergence of the vector field $$\,\overrightarrow A = x\widehat a{}_x + y\widehat a{}_y + z\widehat a{}_z\,\,$$ is

The direction of vector $$A$$ is radially outward
from the origin, with $$\left| A \right| = K\,{r^n}$$
where $${r^2} = {x^2} + {y^2} + {z^2}$$ and $$K$$ is constant.
The value of $$n$$ for which $$\nabla .A = 0\,\,$$ is

If $$\overrightarrow A = xy\,\widehat a{}_x + {x^2}\widehat a{}_y\,\,$$ then $$\,\,\oint {\overrightarrow A .d\overrightarrow r \,\,} $$ over the path shown in the figure is