A particle moves along a curve whose parametric equations are: $$\,x = {t^3} + 2t,\,y = - 3{e^{ - 2t}}\,\,$$ and $$z=2$$ $$sin$$ $$(5t),$$ where $$x, y$$ and $$z$$ show variations of the distance covered by the particle (in cm) with time $$t $$ (in $$s$$). The magnitude of the acceleration of the particle (in cm/s²) at $$t=0$$ is _______.

For a scalar function $$\,f\left( {x,y,z} \right) = {x^2} + 3{y^2} + 2{z^2},\,\,$$ the directional derivative at the point $$P(1,2,-1)$$ in the direction of a vector $$\widehat i - \widehat j + 2\widehat k\,\,$$ is

The velocity vector is given as $${\mkern 1mu} \vec V = 5xy\widehat i + 2{y^2}\widehat j + 3y{z^2}\widehat k.{\mkern 1mu} {\mkern 1mu} $$ The divergence of this velocity vector at $$(1,1,1)$$ is

The directional derivative of $$\,\,f\left( {x,y,z} \right) = 2{x^2} + 3{y^2} + {z^2}\,\,$$ at the point $$P(2,1,3)$$ in the direction of the vector $${\mkern 1mu} \vec a = \widehat i - 2\widehat k{\mkern 1mu} $$ is _________.