7. a) 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 AS is given by AH ~ -CVT . n AS, where I is the temperature, n is the unit normal to the surface in the direction of the heat flow, and c is the thermal diffusivity of the material (a constant). The heat flux across the surface S is given by H = SS -CVT . n ds. 400 Consider a single heat source located at the origin with temperature T(x, y, z) = x2+y2+z2 Calculate the heat flux (in terms of c) across the surface with the parameterization T(u, v) = vi+ cosuj + sinuk, - Sus-,Osus1 b) A cross section of earth's magnetic field can be represented as a vector field in which the center of earth is located at the origin and the positive y-axis points in the direction of the magnetic north pole. The equation for this field is m F(x, V) = (242)5/2 (12xyi + (8yz - 4x2)j) where m is the magnetic moment of earth. Show that this vector field is conservative. c) Using what you know from b, find the work in moving a particle from (100, 300) to (200, 400) in terms of m