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Unformatted text preview: Mathematics 20E - Spring 2009 - Homework 3 May 15, 2009 Please complete the following problems and turn them in to the homework drop box in the sixth floor of AP&M labelled “Math 20E / Ryan Szypowski” by Friday, May 8 at 5:30pm. Only a subset of the problems will be graded; leave some blank at your own risk. 1. In this problem we will determine an affine transformation which maps the unit square [0 , 1] × [0 , 1] onto an arbitrary parallelogram. First, let P be the parallelogram with one vertex at ( a, b ) connected to the the other two vertices at ( a, b )+ q and ( a, b )+ r where q and r are not scalar multiples of each other. The fourth vertex is then at ( a, b )+ q + r . See the diagram below for clarity. ( a, b ) r x y P q (a) Determine the entries of a 2 × 2 matrix A so that T 1 ( u, v ) = A parenleftbigg u v parenrightbigg 1 maps [0 , 1] × [0 , 1] onto the parallelogram described above with ( a, b ) = (0 , 0). (b) Show that T 2 ( u, v ) = T 1 ( u, v ) + parenleftbigg a b parenrightbigg maps (0 , 0) to ( a, b ). Thus, T 2 maps [0 , 1] × [0 , 1] onto P . (c) Suppose P has vertices at (1 , 2), (3 , 3), (2 , 5), and (4 , 6). Give the transformation which maps [0 , 1] × [0 , 1] onto P . (d) Use the above transformation to compute integraldisplayintegraldisplay P x + y dA x,y where P is the parallelogram in (c)....
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This note was uploaded on 10/22/2009 for the course CHEM 140A taught by Professor Whiteshell during the Spring '04 term at UCSD.
- Spring '04