Let c be the unit circle in the xy-plane, oriented counterclockwise as seen from above. the divergence of the vector field f~ = (z, x, y) is zero, and as a result the flux through every surface with boundary c should be the same. confirm that this is the case with the upper half of the unit sphere, the lower half of the unit sphere, and the unit disk in the xy-plane

Upper half of the unit sphere (call it ): parameterize by

with and . Take the normal vector to be

Then the flux of over this surface is

Lower half of the sphere (call it ): all the details remain the same as above, but with . The flux is again .

Unit disk (call it ): parameterize the disk by

with and . Take the normal vector to be

Then the flux across is

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is

at least two

step-by-step explanation: