Match the following.

List-$${\rm I}$$
$$P.$$ Stoke's Theorem
$$Q.$$ Gauss's Theorem
$$R.$$ Divergence Theorem
$$S.$$ Cauchy's Integral Theorem

List-$${\rm I}{\rm I}$$
$$1.$$
$$2.$$
$$3.$$
$$4.$$

The curl of the gradient of the scalar field defined by $$\,V = 2{x^2}y + 3{y^2}z + 4{z^2}x$$ is

Given a vector field $$\overrightarrow F = {y^2}x\widehat a{}_x - yz\widehat a{}_y - {x^2}\widehat a{}_z,$$ the line integral $$\int {F.dl} $$ evaluated along a segment on the $$x-$$axis from $$x=1$$ to $$x=2$$ 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