Consider the temperature distribution in space given by the formula $T(x, y, z) = x^3 + 2y + 3z$,
where x, y, and z are Cartesian coordinates.
a) Find \nabla T
b) Let k is the heat conductivity of the medium. Find the vector of heat flux at the point (1,1,1)
c) Find the rate of conduction heat transfer in the positive z-direction through a surface of unit
area located around the point (1,1,1) and parallel to x- and y-axes of the coordinate system
d) Find the total heat flux out of the cube $0 \le x \le 2$, $0 \le y \le 2$, $0 \le z \le 2$ through its
surface.