The magnitude of the gradient for the function $$f\left( {x,y,z} \right) = {x^2} + 3{y^2} + {z^3}\,\,$$ at the point $$(1,1,1)$$ is _________.

If $$\,\overrightarrow r = x\widehat a{}_x + y\widehat a{}_y + z\widehat a{}_z\,\,\,\,$$ and $$\,\left| {\overrightarrow r } \right| = r,$$ then div $$\left( {{r^2}\nabla \left( {\ln \,r} \right)} \right) $$ = ________.

Consider a vector field $$\overrightarrow A \left( {\overrightarrow r } \right).$$ The closed loop line integral $$\oint {\overrightarrow A \bullet \overrightarrow {dl} } $$ can be expressed as

Consider a closed surface $$'S'$$ surrounding a volume $$V.$$ If $$\overrightarrow r $$ is the position vector of a point inside $$S$$ with $$\overrightarrow n $$ the unit normal on $$'S',$$ the value of the integral is