Solution Found!
Shear transformations in The transformation T in given by
Chapter 14, Problem 58E(choose chapter or problem)
Shear transformations in \(\mathbf{R}^{2}\) The transformation T in \(\mathbf{R}^{2}\) 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 \(\{(u, v): 0 \leq u \leq 1,0 \leq v \leq 1\}\). Let R = T(S) 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 l ).
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 (\(\left(\frac{1}{2}, \frac{1}{2}\right)\))).
e. Find an analogous transformation that gives a shear in the y-direction,
Questions & Answers
QUESTION:
Shear transformations in \(\mathbf{R}^{2}\) The transformation T in \(\mathbf{R}^{2}\) 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 \(\{(u, v): 0 \leq u \leq 1,0 \leq v \leq 1\}\). Let R = T(S) 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 l ).
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 (\(\left(\frac{1}{2}, \frac{1}{2}\right)\))).
e. Find an analogous transformation that gives a shear in the y-direction,
ANSWER:Solution 58E
Step 1
