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26. Let F(x, y, z) = (x, y, z)/(x² + y² + z²)3/2. Show that curl F = 0. Show that V(x+ y² + z2)-1/2 = -F (see Example 2.40 in Section 2.4). %3D %3D 27. Consider the vector field F(x, y) = -(1+x)ye*i – xe j. %3D (a) Show that F(x, y) is conservative. (b) Using the Fundamental Theorem of Calculus, find the potential energy V(x,0) along the x-axis if V (0, 0) = 0. (c) Using the Fundamental Theorem of Calculus, find the potential energy V (0, y) along the y-axis if V (0, 0) = 0. %3D 28. Let F(x, y, z) = x³ yi + z²k. Does there exist a function ƒ such that F = V f? 29. Find F.ds, where F(x, y) = 2xye'i +x²e°(1+ y)j and c is the straight-line segment from (0,0) to (3, -2), %3D (a) Using a parametrization for c. (b) Using the fact that curl F = 0. 30. Check that F(x, y, z) = 2xy²i+ (2x²y + e²)j+ ye°k is a conservative force field in R³. Find the work done by F on an object that moves from (0, 2, – 1) to (3, 2, 0). %3D 31. Show that the vector field F(r) = ||r||²r is a gradient vector field (r = xi + yj+ zk), and find a function f such that Vf = F. %3D 32. Show that the vector field F(r) = ||r||r is a gradient vector field (r = xi + yj+ zk), and find a function f such that Vf = F. %3D 33. Compute [, F ds, where F = (In (x + y²)+x/(x + y²))i + (2xy/(x + y²))j, and e is the part of the curvey = x' from (1, 1) to (2, 8). Use the fact (check it!) that F is a gradient vector field. %3D

26. Let F(x, y, z) = (x, y, z)/(x² + y² + z²)3/2. Show that curl F = 0. Show that V(x+ y² + z2)-1/2 = -F (see Example 2.40 in Section 2.4). %3D %3D 27. Consider the vector field F(x, y) = -(1+x)ye*i – xe j. %3D (a) Show that F(x, y) is conservative. (b) Using the Fundamental Theorem of Calculus, find the potential energy V(x,0) along the x-axis if V (0, 0) = 0. (c) Using the Fundamental Theorem of Calculus, find the potential energy V (0, y) along the y-axis if V (0, 0) = 0. %3D 28. Let F(x, y, z) = x³ yi + z²k. Does there exist a function ƒ such that F = V f? 29. Find F.ds, where F(x, y) = 2xye'i +x²e°(1+ y)j and c is the straight-line segment from (0,0) to (3, -2), %3D (a) Using a parametrization for c. (b) Using the fact that curl F = 0. 30. Check that F(x, y, z) = 2xy²i+ (2x²y + e²)j+ ye°k is a conservative force field in R³. Find the work done by F on an object that moves from (0, 2, – 1) to (3, 2, 0). %3D 31. Show that the vector field F(r) = ||r||²r is a gradient vector field (r = xi + yj+ zk), and find a function f such that Vf = F. %3D 32. Show that the vector field F(r) = ||r||r is a gradient vector field (r = xi + yj+ zk), and find a function f such that Vf = F. %3D 33. Compute [, F ds, where F = (In (x + y²)+x/(x + y²))i + (2xy/(x + y²))j, and e is the part of the curvey = x' from (1, 1) to (2, 8). Use the fact (check it!) that F is a gradient vector field. %3D

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CM: Cumulative Review
Problem 5CM: Take this test to review the material in Chapters 4 and 5. After you are finished, check your work...
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26. Let F(x, y, z) = (x, y, z)/(x² + y² + z²)3/2. Show that curl F = 0. Show that V(x+ y² + z2)-1/2 = -F (see Example 2.40 in Section 2.4). %3D %3D 27. Consider the vector field F(x, y) = -(1+x)ye*i – xe j. %3D (a) Show that F(x, y) is conservative. (b) Using the Fundamental Theorem of Calculus, find the potential energy V(x,0) along the x-axis if V (0, 0) = 0. (c) Using the Fundamental Theorem of Calculus, find the potential energy V (0, y) along the y-axis if V (0, 0) = 0. %3D 28. Let F(x, y, z) = x³ yi + z²k. Does there exist a function ƒ such that F = V f? 29. Find F.ds, where F(x, y) = 2xye'i +x²e°(1+ y)j and c is the straight-line segment from (0,0) to (3, -2), %3D (a) Using a parametrization for c. (b) Using the fact that curl F = 0. 30. Check that F(x, y, z) = 2xy²i+ (2x²y + e²)j+ ye°k is a conservative force field in R³. Find the work done by F on an object that moves from (0, 2, – 1) to (3, 2, 0). %3D 31. Show that the vector field F(r) = ||r||²r is a gradient vector field (r = xi + yj+ zk), and find a function f such that Vf = F. %3D 32. Show that the vector field F(r) = ||r||r is a gradient vector field (r = xi + yj+ zk), and find a function f such that Vf = F. %3D 33. Compute [, F ds, where F = (In (x + y²)+x/(x + y²))i + (2xy/(x + y²))j, and e is the part of the curvey = x' from (1, 1) to (2, 8). Use the fact (check it!) that F is a gradient vector field. %3D
Transcribed Image Text:26. Let F(x, y, z) = (x, y, z)/(x² + y² + z²)3/2. Show that curl F = 0. Show that V(x+ y² + z2)-1/2 = -F (see Example 2.40 in Section 2.4). %3D %3D 27. Consider the vector field F(x, y) = -(1+x)ye*i – xe j. %3D (a) Show that F(x, y) is conservative. (b) Using the Fundamental Theorem of Calculus, find the potential energy V(x,0) along the x-axis if V (0, 0) = 0. (c) Using the Fundamental Theorem of Calculus, find the potential energy V (0, y) along the y-axis if V (0, 0) = 0. %3D 28. Let F(x, y, z) = x³ yi + z²k. Does there exist a function ƒ such that F = V f? 29. Find F.ds, where F(x, y) = 2xye'i +x²e°(1+ y)j and c is the straight-line segment from (0,0) to (3, -2), %3D (a) Using a parametrization for c. (b) Using the fact that curl F = 0. 30. Check that F(x, y, z) = 2xy²i+ (2x²y + e²)j+ ye°k is a conservative force field in R³. Find the work done by F on an object that moves from (0, 2, – 1) to (3, 2, 0). %3D 31. Show that the vector field F(r) = ||r||²r is a gradient vector field (r = xi + yj+ zk), and find a function f such that Vf = F. %3D 32. Show that the vector field F(r) = ||r||r is a gradient vector field (r = xi + yj+ zk), and find a function f such that Vf = F. %3D 33. Compute [, F ds, where F = (In (x + y²)+x/(x + y²))i + (2xy/(x + y²))j, and e is the part of the curvey = x' from (1, 1) to (2, 8). Use the fact (check it!) that F is a gradient vector field. %3D
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