worksheet2.20100123.4b5b48d5e94e09.72437805

# worksheet2.20100123.4b5b48d5e94e09.72437805 - Worksheet on...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## 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.

### Page1 / 9

worksheet2.20100123.4b5b48d5e94e09.72437805 - Worksheet on...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online