lecture notes 13

.Example. Suppose that a uniform solid ball B of radius R is centeredat the origin, and at its point (x, y, z) the temperatureu(x, y, z) = c(R 2 − x 2 − y 2 ). Find the rate of the flow of heat across. the sphere S of radius a < R centered at O(0, 0, 0).Solution. The heat flow across the sphereS ∫∫= { (x, y, z) | x∫∫2 + y 2 + z 2 = a 2 } is ∫∫ given byq · n dS = − k∇u · n dS = −k ∇(R 2 − x 2 − y 2 − z 2 ) · n dS =SSS∫∫∫∫(x, y, z)2(x−k (−2x, −2y, −2z) · dS = 2 + y 2 + z 2 )kdS =∫∫SaS a2ak dS = 2ak(4πa 2 ) = 8kπa 3 .S. . . . . .