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Unformatted text preview: Math 317, Section 202, Homework no. 3 (due Friday January 29, 2010) [5 problems, 32 points] Problem 1 (2 points) . Suppose C is a smooth curve with a smooth parameterization r ( t ), a ≤ t ≤ b . Show that the parameter t is arc length if and only if | r ( t ) | = 1 for all t . [Hint: The question says ’if and only if’ and so you have to show: (1) if t is arc-length, then | r ( t ) | = 1 for all t ; and (2) if | r ( t ) | = 1 for all t , then t is arc-length.] Problem 2 (6 points) . (a) (4 points) Suppose the curve C in the plane is given implicitly as the solution of the equation f ( x,y ) = 0. You may assume that f has continuous second order partial derivatives and that for all ( x,y ) ∈ C , f satisfies f x ( x,y ) 6 = 0 or f y ( x,y ) 6 = 0. Show that the curvature of C is given by κ ( x,y ) = | 2 f x f y f xy- f 2 y f xx- f 2 x f yy | ( f 2 x + f 2 y ) 3 / 2 (1) for all ( x,y ) ∈ C . [Hint: Review partial derivatives and the chain rule for functions of several variables. The partial derivatives of f ( x,y ) are denoted by f x = ∂f ∂x , f y = ∂f ∂y , f yx = ∂ 2 f ∂x∂y , etc. You may use the following result: Whenever ( x,y ) ∈ C is such that f y ( x,y ) 6 = 0, there is a parameterization r ( x ) = h x,y ( x ) i of C near ( x,y ) with some function y ( x ) that is differentiable twice. The equation that defines C then reads f ( x,y ( x )) = 0. Compute the first and second derivatives of this equation and use)) = 0....
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This note was uploaded on 03/18/2010 for the course MAATH 317 taught by Professor Phfiier during the Spring '10 term at UBC.
- Spring '10