(950, #39) The temperature at the point (x, y, z) in a substance with conductivity K = 6.5 is u(x,y) = 2 y2 + 2 z2 . Find the rate of heat flow inward across the cylindrical surface y2 + z2 = 6, 0 <= x <= 4

Solution: First, the graphic on the handout had the vectors going in the wrong direction.

Next, according to the text,

where K = 6.5.

To continue, if we parmetrize the surface by

then

The negative sign on the y and z components here say the normals are pointinginto the surface.

Note also that for this parametrization

This says that the vector field F and the normal r_u x r_t are parallel.

As for the integral, it is equal to