Shear transformations in _2 The transformation T in _2
Chapter 13, Problem 58(choose chapter or problem)
Shear transformations in _2 The transformation T in _2 given by x = au + bv, y = cv, where a, b, and c are positive real numbers, is a shear transformation. Let S be the unit square 51u, v2: 0 u 1, 0 v 16. Let R = T1S2 be the image of S. a. Explain with pictures the effect of T on S. b. Compute the Jacobian of T. c. Find the area of R and compare it to the area of S (which is 1). d. Assuming a constant density, find the center of mass of R (in terms of a, b, and c) and compare it to the center of mass of S (which is 112 , 12 2). e. Find an analogous transformation that gives a shear in the y-direction.
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