et σ be the surface 10x+10y+10z=2 in the first octant, oriented upwards. Let C be the oriented boundary of σ. Compute the work done in moving a unit mass particle around the boundary of σ through the vector field F=(x−6y)+(6y−4z)+(4z−x) using line integrals, and using Stokes' Theorem.

Newton's law of cooling states that the rate of change in the temperature t(t) of a body is proportional to the difference between the temperature of the medium m(t) and the temperature of the body. that is, startfraction dt over dt endfraction equals upper k left bracket upper m left parenthesis t right parenthesis minus upper t left parenthesis t right parenthesis right bracket , where k is a constant. let kequals0.04 left parenthesis min right parenthesis superscript negative 1 and the temperature of the medium be constant, m(t) font size decreased by 3 equivalent font size decreased by 3 294 kelvins. if the body is initially at 369 kelvins, use euler's method with hequals0.1 min to approximate the temperature of the body after (a) 30 minutes and (b) 60 minutes.