Shear transformations in _3 The transformation T in _3
Chapter 13, Problem 59(choose chapter or problem)
Shear transformations in _3 The transformation T in _3 given by x = au + bv + cw, y = dv + ew, z = w, where a, b, c, d, and e are positive real numbers, is one of many possible shear transformations in _3. Let S be the unit cube 51u, v, w2: 0 u 1, 0 v 1, 0 w 16. Let D = T1S2 be the image of S. a. Explain with pictures and words the effect of T on S. b. Compute the Jacobian of T. c. Find the volume of D and compare it to the volume of S (which is 1). d. Assuming a constant density, find the center of mass of D and compare it to the center of mass of S (which is 112 , 12 , 12 2).
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