00:01
The question is to evaluate the flux of the vector field f of x, y, z, z, is equal to y, i, minus x, j plus 4 z square k through a surface s, where s is the hemisphere x square plus y square plus z square is equal to 1, with z greater than or equal to 0, oriented upwards.
00:28
Now the flux of a vector field f through a spherical surface s can be evaluated as integral over s f dot da is equal to integral over t f of r theta phi dot product with sine phi cos theta i plus sine phi, sine theta j plus cos phi k times r square sine phi d -teta where t is the corresponding surface or region in the theta -phi -phing -fi plane corresponding to the sphere.
01:32
Or the spherical surface s oriented upward.
01:38
Now, first use the spherical coordinates to get x is equal to r cos theta sine phi, y is equal to r sine theta sine phi and z is equal to cos r cos phi.
02:02
Now here the radius of this head hemisphere is 1 therefore r is equal to 1 here so we have x is equal to cos theta sine 5 y is equal to sine theta sine 5 and z is equal to cos phi also for this hemisphere we have 0 less than or equal to theta less than or equal to 2 pi and 0 less than or equal to 5 less than or equal to pi by 2 now f of r theta phi is obtained by substituting the value of x, y and z in terms of theta and phi in the equation of the vector field...