7. Given below are three vector fields, F1, F2, and F3. Let C1 and C2 denote circles of radius 1 and 2, respectively, both centered at (0,0) and oriented counterclockwise. Also, let C3 denote the line segment starting at (-3, –3) and ending at (2, 2), and let C4 denote the line segment starting at (2, 2) and ending at (2, –2). F (r, y) F:(x, y) 3 3 3 2 2 1 1 1 -1 -1 -2 -2 -2 3 -3 -3 -2 -1 2 3 -3 -2 -1 о 1 2 3 -3 -2 -1 0 1 2 3 (a) For each of the four curves C1, C2, C3, and C4, and for each of the vector fields Fi, F2, and F3, decide whether f F.dr appears to be positive, negative, or zero. Note that there are 12 different integrals to be examined. (b) Which is larger, fc. F1 · dr or fc. F · dr? Explain.

7. Given below are three vector fields, F1, F2, and F3. Let C1 and C2 denote circles of radius 1 and 2, respectively, both centered at (0,0) and oriented counterclockwise. Also, let C3 denote the line segment starting at (-3, –3) and ending at (2, 2), and let C4 denote the line segment starting at (2, 2) and ending at (2, –2). F (r, y) F:(x, y) 3 3 3 2 2 1 1 1 -1 -1 -2 -2 -2 3 -3 -3 -2 -1 2 3 -3 -2 -1 о 1 2 3 -3 -2 -1 0 1 2 3 (a) For each of the four curves C1, C2, C3, and C4, and for each of the vector fields Fi, F2, and F3, decide whether f F.dr appears to be positive, negative, or zero. Note that there are 12 different integrals to be examined. (b) Which is larger, fc. F1 · dr or fc. F · dr? Explain.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 24EQ

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