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Unformatted text preview: Worksheet on coordinate transformations Exploring the inverse function theorem A global chart Given a pair of perpendicular lines labeled X and Y and intersecting at a point O , we saw how a global chart for the configuration of a particle located at P can be constructed using the signed distances from O to the orthogonal projections of P onto X and Y , respectively. We write Ξ for the corresponding one-to-one map and Ξ 1 and Ξ 2 for the corresponding coordinate functions. A linear transformation Define the function f Now consider two new quantities, say Ξ 1 and Ξ 2 and write Ξ = I Ξ 1 , Ξ 2 M T . Define the function f : R 2 fi R 2 such that f H Ξ L = I f 1 I Ξ 1 , Ξ 2 M , f 2 I Ξ 1 , Ξ 2 MM T , where f 1 I Ξ 1 , Ξ 2 M = x + Ξ 1 cos H Θ L + Ξ 2 sin H Θ L and f 2 I Ξ 1 , Ξ 2 M = y- Ξ 1 sin H Θ L + Ξ 2 cos H Θ L , where x , y , and Θ are given constants. For every pair of values of Ξ 1 and Ξ 2 , there is a unique pair of values of the functions f 1 and f 2 . The question is whether the converse holds. Here, f = 8 x0 + Ξ 1t Cos @ Θ D + Ξ 2t Sin @ Θ D , y0- Ξ 1t Sin @ Θ D + Ξ 2t Cos @ Θ D< 8 x0 + Ξ 1t Cos @ Θ D + Ξ 2t Sin @ Θ D , y0 + Ξ 2t Cos @ Θ D- Ξ 1t Sin @ Θ D< Compute the jacobian Let's compute the jacobian matrix ¶ Ξ f H Ξ L of f and its determinant ¶ Ξ f H Ξ L : H jac = D @ f, 88 Ξ 1t , Ξ 2t <<DL MatrixForm K Cos @ Θ D Sin @ Θ D- Sin @ Θ D Cos @ Θ D O Det @ jac D Simplify 1 Find regular and singular points The jacobian matrix is nonsingular for all Ξ 1 and Ξ 2 , since the determinant is everywhere nonzero. It follows that every point Ξ ˛ R 2 is regular. There are no singular points. Apply the inverse function theorem Suppose there exists a configuration of the particle corresponding to a point P * such that Ξ 1 H P * L = Ξ 1 * = x * and Ξ 2 H P * L = Ξ 2 * = y * , i.e., Ξ H P * L = Ξ * , and such that there exists a pair of values of Ξ 1 and Ξ 2 , say Ξ 1 * and Ξ 2 * , such that x * = f 1 I Ξ 1 * , Ξ 2 * M and y * = f 2 I Ξ 1 * , Ξ 2 * M , i.e., Ξ * = f H Ξ * L . Since the point I Ξ 1 * , Ξ 2 * M is a regular point of f , the inverse function theorem implies that there exists a continuously differentiable inverse of f on a small neighborhood of ( x * , y * ), i.e., a function g such that Ξ = g H Ξ L satisfies the equation Ξ = f H Ξ L or, in other words, that Ξ = f H g H Ξ LL provided that Ξ is close to Ξ * . It follows that, for every Ξ close to Ξ * there exists a unique pair of values of Ξ 1 and Ξ 2 . Since there exists a unique Ξ for every configuration of the particle and vice versa, it follows that there exists a unique Ξ for every configuration of the particle near P * and vice versa. Ξ is thus a function of configuration and corresponds to a local chart....
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This note was uploaded on 11/17/2011 for the course TAM 412 taught by Professor Weaver during the Spring '08 term at University of Illinois, Urbana Champaign.
- Spring '08