For the following scalar fields, answer the following:

a$.$ For the scalar field $f=2x^{2}y-6x{y}^2+3z$

i$.$ Calculate the gradient gradf $(5$ points$)$

ii$.$ Prove that the gradient of $f$ in the $z-$direction does not depend on $z(5$ points$)$ easy way

b$.$ For the temperature field $T=3x^2-y^3+2z^2-4xz+6xy$

i$.$ Calculate the gradient gradT $(5$ points$)$

ii$.$ For the point $P(3,5,2),$ which directions have the maximum temperature gain and loss? Also provide the magnitude of maximum temperature gain and loss. Note that the temperature is in $C$ and distance is in m $(5$ points$)$