18.02 Pset 7

18.02 Pset 7 - Let a b c be the radian measures of the...

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18.02 HOMEWORK #7, DUE NOVEMBER 18, 2010 BJORN POONEN 1. Part A (After Nov. 4) p. 986: 40* (10 pts.; “the solid” means only the part in the ﬁrst octant; assume constant density 1) (After Nov. 4) 5A-2b* (10 pts.) (After Nov. 5) p. 843: 5, 11, 21, 49, 58* (10 pts.) (After Nov. 5) p. 993: 7, 9, 39 (After Nov. 5) p. 1000: 3, 15 (After Nov. 5) 5B-1 (After Nov. 5) 5C-1* (8+5 pts.; in case it’s not clear, V is the part of a radius a sphere lying inside a solid cone with vertex at the sphere’s center; assume that we want the gravitational force on a unit mass at the vertex ), 5C-5* (12 pts.) (After Nov. 9) 6A-3, 6A-4 (After Nov. 9) 6B-7* (8 pts.), 6B-9* (10 pts.) 2. Part B B.1) (After Nov. 5, 10 pts.) The latitude of Boston in approximately 42 . What is the average latitude of the points in the Northern Hemisphere? B.2) (After Nov. 5) A spherical triangle is formed by three arcs of great circles on a sphere.
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Unformatted text preview: Let a, b, c be the radian measures of the sides of a spherical triangle. Let C be the angle of the triangle opposite side c , i.e., between sides α and β , which means that C is the angle between the tangent vectors to the two arcs of great circles meeting at that vertex. (a) (10 pts.) Prove the spherical law of cosines , which says that cos c = cos a cos b + sin a sin b cos C. (b) (7 pts.) What does this relation say approximately when a, b, c are very small? (Expand the functions of a, b, c in power series, and approximate by discarding any term involving a product of three or more factors a, b, c , such as b 2 c .) Reminder: Please write “Sources consulted: none” at the top of your homework, or list your (animate and inanimate) sources. See the course information sheet for details. 1...
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