Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 17 - Line and Surface Integrals - 17.2 Line Integrals - Exercises - Page 933: 41

Answer

Figure A: the line integral of the vector field is zero Figure B: the line integral of the vector field is negative Figure C: the line integral of the vector field is zero

Work Step by Step

Figure A: the line integral of the vector field is zero Explanation: the dot products ${\bf{F}}\cdot{\bf{T}}$ are negative on the upper half of the circle because the angle between the vectors are obtuse. However, the dot products ${\bf{F}}\cdot{\bf{T}}$ are positive on the lower half of the circle because the angle between the vectors are acute. Since ${\bf{F}}$ is the same on both halves, the dot products cancel out. Hence, total line integral is zero. Figure B: the line integral of the vector field is negative Explanation: the dot products ${\bf{F}}\cdot{\bf{T}}$ are negative on the upper half of the circle because the angle between the vectors are obtuse. However, the dot products ${\bf{F}}\cdot{\bf{T}}$ are positive on the lower half of the circle because the angle between the vectors are acute. Since ${\bf{F}}$ is stronger at the upper half, total line integral is negative. Figure C: the line integral of the vector field is zero Explanation: the dot products ${\bf{F}}\cdot{\bf{T}}$ are zero since the angle between the vectors are $90^\circ $ along the circle.
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