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the other, of charge 1 C, is placeu at (A, I is needed in order to move the 1-C charge along the straight line IPUIII (1, U charge remains at the origin? 19. Consider the force field F(x, y) = (y, 0). Compute the work done on a particle by the force Eie the particle moves from (0, 0) to (1, 1) in each of the following ways: (a) Along the r-axis to (1, 0), then vertically up to (1, 1) (b) Along the parabolic path y = x² (c) Along the path y = x %3D (d) Along the straight line (e) Along the path y = sin (7 x/2) 4 (f) Along the y-axis to (0, 1), then horizontally to (1, 1) Interpret your results. 20. Compute fe 3(x + y)dx along the path c(t) = (e' + 1, e' – 2), 0

the other, of charge 1 C, is placeu at (A, I is needed in order to move the 1-C charge along the straight line IPUIII (1, U charge remains at the origin? 19. Consider the force field F(x, y) = (y, 0). Compute the work done on a particle by the force Eie the particle moves from (0, 0) to (1, 1) in each of the following ways: (a) Along the r-axis to (1, 0), then vertically up to (1, 1) (b) Along the parabolic path y = x² (c) Along the path y = x %3D (d) Along the straight line (e) Along the path y = sin (7 x/2) 4 (f) Along the y-axis to (0, 1), then horizontally to (1, 1) Interpret your results. 20. Compute fe 3(x + y)dx along the path c(t) = (e' + 1, e' – 2), 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section: Chapter Questions
Problem 18T
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the other, of charge 1 C, is placeu at (A, I is needed in order to move the 1-C charge along the straight line IPUIII (1, U charge remains at the origin? 19. Consider the force field F(x, y) = (y, 0). Compute the work done on a particle by the force Eie the particle moves from (0, 0) to (1, 1) in each of the following ways: (a) Along the r-axis to (1, 0), then vertically up to (1, 1) (b) Along the parabolic path y = x² (c) Along the path y = x %3D (d) Along the straight line (e) Along the path y = sin (7 x/2) 4 (f) Along the y-axis to (0, 1), then horizontally to (1, 1) Interpret your results. 20. Compute fe 3(x + y)dx along the path c(t) = (e' + 1, e' – 2), 0 <t < 1. 21. Compute Jydx +xdy)/(x² + y²), where c is the circle centered at the origin of radius 2. oriented counterclockwise. 22. Compute f xydx + ye*dy, where c is the rectangle with vertices (0, 0), (1, 0), (1, 1), and (0, 1), oriented counterclockwise. 23. Consider e xydx + 2ydy , where c; is the straight-line segment joining the points (0, 0) and (1, 1) parametrized in the following ways: (a) c1(1) = (1, t), t e [0, 1] (b) c2(t) = (sin 1, sin t), 1 e [0, 1 /2] %3D (c) C3(1) = (cos t, cos t), t E [0, A/2] Are all the results the same? Explain why or why not. 24. Compute , M(x, y, z)dx , where M is a continuous function and c is any curve contained in a plane parallel to the yz-plane. 25. Show that the assumption "c is a C' curve" in Theorem 5.3 can be replaced by "e is a piecewise C curve." Exercises 26 to 29: In R2, the flux (flow) of a vector field F across a smooth closed curve c is defined as F.n ds, where n denotes the outward unit normal vector field along c. The circulation of F is given by F. ds. Compute the flux and the circulation for the vector field F and the curve c. 26. F(x, y) = 4xi- 2yj, c is a circle of radius r, oriented clock %3D 27. F(x, y) = xi + yj, c is a circle of radius ori junc ounto 28. F(x, y)= x²i+ y²j, ¢ is the semicirele of ra line segment back to (r, 0), oriented counterclo followed straight-
Transcribed Image Text:the other, of charge 1 C, is placeu at (A, I is needed in order to move the 1-C charge along the straight line IPUIII (1, U charge remains at the origin? 19. Consider the force field F(x, y) = (y, 0). Compute the work done on a particle by the force Eie the particle moves from (0, 0) to (1, 1) in each of the following ways: (a) Along the r-axis to (1, 0), then vertically up to (1, 1) (b) Along the parabolic path y = x² (c) Along the path y = x %3D (d) Along the straight line (e) Along the path y = sin (7 x/2) 4 (f) Along the y-axis to (0, 1), then horizontally to (1, 1) Interpret your results. 20. Compute fe 3(x + y)dx along the path c(t) = (e' + 1, e' – 2), 0 <t < 1. 21. Compute Jydx +xdy)/(x² + y²), where c is the circle centered at the origin of radius 2. oriented counterclockwise. 22. Compute f xydx + ye*dy, where c is the rectangle with vertices (0, 0), (1, 0), (1, 1), and (0, 1), oriented counterclockwise. 23. Consider e xydx + 2ydy , where c; is the straight-line segment joining the points (0, 0) and (1, 1) parametrized in the following ways: (a) c1(1) = (1, t), t e [0, 1] (b) c2(t) = (sin 1, sin t), 1 e [0, 1 /2] %3D (c) C3(1) = (cos t, cos t), t E [0, A/2] Are all the results the same? Explain why or why not. 24. Compute , M(x, y, z)dx , where M is a continuous function and c is any curve contained in a plane parallel to the yz-plane. 25. Show that the assumption "c is a C' curve" in Theorem 5.3 can be replaced by "e is a piecewise C curve." Exercises 26 to 29: In R2, the flux (flow) of a vector field F across a smooth closed curve c is defined as F.n ds, where n denotes the outward unit normal vector field along c. The circulation of F is given by F. ds. Compute the flux and the circulation for the vector field F and the curve c. 26. F(x, y) = 4xi- 2yj, c is a circle of radius r, oriented clock %3D 27. F(x, y) = xi + yj, c is a circle of radius ori junc ounto 28. F(x, y)= x²i+ y²j, ¢ is the semicirele of ra line segment back to (r, 0), oriented counterclo followed straight-
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