Chapter 5.6, Problem 20ES

1. Newton's Law of Gravitation (an example of an inverse square law) states that the magnitude of the gravitational force between two objects with masses m and M is |F| mMG |r|2 where r is the distance between the objects, and G is the gravitational constant. Assume that the object with mass M is located at the origin in R³. Then, the gravitational force field acting on the object at the point r = (x, y, z) is given by F(x, y, z) = mMG r3 r. mMG mMG Show that the scalar vector field f(x, y, z) = = is a potential function for r √√x² + y² . Fi.e. show that F = Vf. Remark: f is the negative of the physical potential energy, because F = -V(-ƒ).

show your answer in pen and paper Don't use any Al tool show ur answer in pe n and paper then take -2-i Evaluate f² (3xy + iy²)dz a) along the straight line joining from z = i to z = 2 - i Inspiring Excellence b) along the parabola from x = 2t - 2 and y = 1+t-t²

Prove let Aand B submodul of M A is large sub podule A large of B and B large of M. SM B Smale sub module B/A smal of M/A and As Mallof M. Give example and expleain caim. Amonorphism and split d) Determine the following group: Hom, (Q,Z) and Ho M₂ (Q, Q) and Hom (2/12, Q) =