Background: The heat passing through a given surface can be formulated as a flux integral,in which the integrand is the temperature gradient multiplied by the thermal conductivity.Question: Imagine that within the space (x,y,z), there is a steady temperature field ofT(x,y,z)=(x2+y2)z.An engineer wants to divide the space by a set of closed surfacesand maximize the outgoing heat flux from the inside to the outside of the closed surfaces.The engineer considers 4 options for this design. The first option is a closed cylindrical shellwith the following equation:S2:x2+y2+z2=1(0,0,0)S3:x2+y2+z2=1x2+y2+z2=4S4:{x2+y2=z2;0z<4x2+y216;z=4