The glass dome of a futuristic greenhouse is shaped like the surface $$ z = 8 - 2 x^{2} - 2 y^{2}. $$ The greenhouse has a flat dirt floor at $ z = 0 $. Suppose that the temperature $ T $, at points $ (x,y,z) $ in and around the greenhouse, varies as $$ T(x,y,z) = x^{2} + y^{2} + 3 (z - 2)^{2}. $$ Then the temperature gives rise to a heat flux density field $ H $ given by $ H = - k \nabla(T) $.
Find the total heat flux outward across the dome and the surface of the ground if $ k = 1 $ on the glass and $ k = 3 $ on the ground.
I know that I need to find the surface integral across the region $ z = 0 $ and across the paraboloid $ z = 8 - 2 x^{2} - 2 y^{2} $ to find the flux, but I’m not sure whether it’s $$ \iint_{S} T ~ \mathrm{d}{\Sigma} $$ or $$ \iint_{S} H ~ \mathrm{d}{\Sigma} $$ that I need to do. Also, how do I parametrize $ T $ (or $ H $) to find the surface integral?
Any help would be really appreciated. Thank you!