Solved: Meaning of the Jacobian The Jacobian is a

Chapter 13, Problem 61

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Meaning of the Jacobian The Jacobian is a magnification (or reduction) factor that relates the area of a small region near the point 1u, v2 to the area of the image of that region near the point 1x, y2. a. Suppose S is a rectangle in the uv-plane with vertices O10, 02, P1_u, 02, 1_u, _v2, and Q10, _v2 (see figure). The image of S under the transformation x = g1u, v2, y = h1u, v2 is a region R in the xy-plane. Let O_, P_, and Q_ be the images of O, P, and Q, respectively, in the xy-plane, where O_, P_, and Q_ do not all lie on the same line. Explain why the coordinates of O_, P_, and Q_ are 1g10, 02, h10, 022, 1g1_u, 02, h1_u, 022, and 1g10, _v2, h10, _v22, respectively. b. Use a Taylor series in both variables to show that g1_u, 02 _ g10, 02 + gu10, 02_u g10, _v2 _ g10, 02 + gv10, 02_v h1_u, 02 _ h10, 02 + hu10, 02_u h10, _v2 _ h10, 02 + hv10, 02_v where gu10, 02 is 0x 0u evaluated at 10, 02, with similar meanings for gv, hu, and hv. c. Consider the vectors Or_P_ and Or_Q_ and the parallelogram, two of whose sides are Or_P_ and Or_Q_. Use the cross product to show that the area of the parallelogram is approximately _ J1u, v2 _ _u _v. d. Explain why the ratio of the area of R to the area of S is approximately _ J1u, v2 _.