Chapter 14.2, Problem 64E

Heat flux in a plate A square plate R = {(x, y): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} has a temperature distribution T(x, y) = 100 – 50x – 25y.

a.    Sketch two level curves of the temperature in the plate.

b.    Find the gradient of the temperature ▿ T(x, y).

c.    Assume that the flow of heat is given by the vector field F = –▿ T(x, y). Compute F.

d.    Find the outward heat flux across the boundary {(x, y): x= 1, 0 ≤ y ≤ 1}.

e.    Find the outward heat flux across the boundary {(x, y): 0 ≤ x ≤ 1, y = 1}.