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 _________.

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 directional derivative of $$f\left( {x,y} \right) = {{xy} \over {\sqrt 2 }}\left( {x + y} \right)$$ at $$(1, 1)$$ in the direction of the unit vector at an angle of $${\pi \over 4}$$ with $$y-$$axis, is given by ________.

The series $$\sum\limits_{n = 0}^\infty {{1 \over {n!}}\,} $$ converges to