dghw3 - Math 454, Spring '08, homework assignment #3, due...

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Math 454, Spring ’08, homework assignment #3, due Feb 28 Problem 1 Let C be a regular parametrized simple space curve of class C 3 . Prove that for any point P on C , there exists a plane π passing through P and such that all nearby points on C lie on the same side of π . ( Hint: it is one of the “coordinate planes” determined by the orthonormal frame { t , n , b } . Use the local canonical form.) Problem 2 Use vector analysis and the Frenet–Serret formulas for a regular space curve γ ( s ) to compute the following quantities in terms of the vectors t ( s ) , n ( s ) , b ( s ) and the curvature and torsion, κ ( s ) , τ ( s ): 1. b ( s ) × t ( s ); 2. n 0 ( s ); 3. b 00 ( s ); 4. γ 000 ( s ). Prime denotes differentiation with respect to s and you may assume that all the nec- essary derivatives exist. Problem 3 Let γ ( t ) be a circular helix as in Pressley, p. 26. Find the equations of the osculating plane, rectifying plane, and the normal plane at the point P = ( x 0 , y 0 , z
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This homework help was uploaded on 02/24/2008 for the course MATH 4540 taught by Professor Protsak during the Spring '08 term at Cornell University (Engineering School).

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dghw3 - Math 454, Spring '08, homework assignment #3, due...

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