1. Heat flows from areas of higher temperature to areas of lower temperature in the direction of greatest change. As a result, measuring heat flux involves the gradient of the temperature. The flux depends on the area of the surface. It is the normal direction to the surface that is important, because heat that flows in directions tangential to the surface will give no heat loss. So, assume that the heat flux across a portion of the surface of area ΔS is given by ΔH ≈ -k∇T · ň ΔS , where T is the temperature, ň is the unit normal to the surface in the direction of the heat flow, and k is the thermal diffusivity of the material (a constant). The heat flux across the surface S is given by H = ∬ -k∇T · ň dS.
Consider a single heat source located at the origin with temperature T(x, y, z) = 30 / √(x² + y² + z²)
Calculate the heat flux across the surface with the parameterization ŕ(u, v) = vī + cos u ĵ + sin u ḱ, π/6 ≤ u ≤ π/4, 0 ≤ v ≤ 1.