Name e-mail
Rec.Teacher Day-hr
Using the Divergence Theorem, evaluate the flux integral
Solution: By the divergence theorem this flux integral is equal to the volume integral
Solution: The surface area is given by the integral We may use the variables and on the surface, the `shadow region' is and the surface measure is
Solution: a) Anything reasonable! It lies over with
b)
Let be the region in three dimensional space consisting of the points with , , and .
Assuming where needed that has unit density, write down integral formulae in terms of spherical coordinates (DO NOT EVALUATE) for
Solution: The region in spherical coordinates is and Thus the three integrals are
Let be `ice-cream cone' consisting of the points in the unit ball (=solid unit sphere) where the `azimuth' (angle with the positive direction of the z-axis) Assuming that it has unit density, compute the gravitational force exerts on a unit mass at the origin.
Solution: Place so that its axis is the z-axis (as indicated). Then in spherical polar coordinates the component (and hence the actual) gravitational force on a unit mass at the origin is