1.6. Exercises 23 (c) Use Gauss’s flux theorem to determine the force F(r) acting on mass m at point P due to the gravitational attraction of a uniform solid sphere of mass M, density ρ, and radius R. Be sure to consider the case where point P is inside the shell (r R) as well as outside the shell (r R). (d) Integrate the above force (starting at a reference point at infinity) to rederive the potential energy in Exercise 1.6.2. 1.6.6. First fundamental forms. Determine the first fundamental form for three of the following seven surfaces. The surfaces you may choose from are: (a) the helicoid x = u cos v , y = u sin v , z = a v (1.55) (b) the torus x = (b + a cos u) cos v , x = (b + a cos u) cos v , z = a sin u (1.56) (c) the catenoid x = a cosh u a cos v , y = a cosh u a sin v , z = u (1.57) (d) the general surface of revolution x = f(u) cos v , y = f(u) sin v , z = g(u) (1.58) (e) the sphere (with alternate parameterization) x = 4a2u 4a2 + u2 + v2 , y = 4a2v 4a2 + u2 + v2 , (1.59) z = a 4a2 u2 v2 4a2 + u2 + v2 (f) the ellipsoid x = a cos u cos v , y = b cos u sin v , z = c sin u (1.60) (g) the hyperbolic paraboloid x = a(u + v) , y = b(u v) , z = uv . (1.61)
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