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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Topic Video
Question
Can you help with 28 please?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,