If $$\,\phi = 2{x^3}{y^2}{z^4}$$ then $${\nabla ^2}\phi $$ is

The line integral $$\int\limits_{{P_1}}^{{P_2}} {\left( {ydx + xdy} \right)} $$ from $${P_1}\left( {{x_1},{y_1}} \right)$$ to $${P_2}\left( {{x_2},{y_2}} \right)$$ along the semi-circle $${P_1}$$ $${P_2}$$ shown in the figure is

If $$T(x, y, z)$$ $$ = {x^2} + {y^2} + 2{z^2}$$ defines the temperature at any location $$(x, y, z)$$ then the magnitude of the temperature gradient at point $$P(1,1,1)$$ is _________.

The angle (in degrees) between two planar vectors $$\vec a = {{\sqrt 3 } \over 2}\widehat i + {1 \over 2}\widehat j$$ and $$\vec b = {{ - \sqrt 3 } \over 2}\widehat i + {1 \over 2}\widehat j$$ is