MATH 213 Final, Fall 2003

} Lib akRY | OF BEIRUT AMERICAN UNIVERSITY OF BEIRUT The Department of Mathematics Math 213 Final Examination January 30, 2004 1. (15 pts.) If a, 8,7, 6 are the four angles of a convex quadrilateral, prove that sinasin Bsinysiné < “apy. ur Formulate, without proof, a generalization of this inequality. 2. (25 pts.) Leta : (-b,b) + R? be a regular curve in R? parametrized by arc length. Suppose that ajla(s) — ui? + azla(s) — u2|? + a3lafs) — a3!? = ¢ where uj, U2, U3 are fixed vectors, and @1,@2,a3, and c,are positive constants. (a) Prove that the trace of a lies on a sphere. (b) Show that the curvature x and the torsion 7 of a satisfy Jo 4 pp K?(s) t#(s) K(s) (c) If a(0) = 0, show that x(s) > jee 3. (20 pts.) Let C and D be two points on the x-axis, and () the circle of center 8 and diameter CD. Let 4 and B be two points in the upper half-plane, inverse of each other with respect to (8). et (a) be the semicircle in the upper half-plane which passes through 4 and B, RPI ON aN ml COl(P) aad (a). Tahe a pout )'}2 = constant. bye ata ee STs UE Ie Line oe et OH puna aud WIUde Lote UW Lies OLY E on (B) and draw EH perpendicular to the x-axis.Denote by G the intersection of CF and EH. (a) Taking C as center of inversion, and CE as radius of inversion, find, with justification, the inverse of (3), and the inverse of (a). (b) Find, with justification, the inverse point 4, of A, and the inverse point B; of B, under the same inversion. (c) Find the Euclidean distances GA;,G8,,HA,, HB, and prove that A;B, is parallel to the X-axis. . (d) Prove that any point Mf on (8) has equal hyperbolic distances from A and B. 4. (15 pts.) Let H? = {(x,y) : y > 0}, the upper half-plane, with metric ds = ey ~ (a) Specify, without proof, four kinds of transformations under which ds is mvariant. (b) If ABC is the hyperbolic triangle whose sides are given by the three equations: y= yl—x?,x =-l,andy = /4-(x+1)*, sketch the triangle, and find the lengths of the sides, the measures of the angles, and the area. 5. (25 pts.) Leta : 7 + R? be a regular curve in R* parametrized by arc length, and B : J + R} the curve given by B(s) = als) + (e -s)t(s) where, c is a real constant, s is the arc length on a and t,n, b are the tangent, normal, and binormal to a. Denote by x,7 the curvature and torsion of a, and by x,,7, the curvature and torsion of B. ; (a) Show that x? = =>) (c-3)?K? (b) Show that the unit binormal of B is b; = xb+rt Kesher” (c) Find the torsion of B 6. ( Extra credit ) Let S be the helicoid surface given by uc toss ~ * ASF & i LIBRARY | OF BEIRUT x(u,v) = (ucosv,usiny, bv), b = 0. (a) Show that Sis a minimal surface and find its Gaussian curvature. (b) If Z, is the intersection of S with the plane z = c, find the curvature and torsion of L..