Let r = (x2 + y²)/2 and consider the vector field F = r4(-yi + j), where r + 0 and A is a constant. F has no z-component and is inde Find curl (r4(-yi + xj)), and show that it can be written in the form k[2+A] E rak, where a = , for any constant A. %3D Using your answer to part (a), find the direction of the curl of the vector fields with each of the following values of A (enter your answer as a unit vector ection of the curl): = -7: direction = = 5: direction = 7(x^2+y^2)^5/2k For each values of A in part (b), what (if anything) does your answer to part (b) tell you about the sign of the circulation around a small circle oriented interclockwise when viewed from above, and centered at (1, 1, 1)? 1= -7, the circulation is negative 1= 5, the circulation is positive sure you can say how your answer in part (c) would change if the question were about a small circle centered at (0, 0,0).)
Let r = (x2 + y²)/2 and consider the vector field F = r4(-yi + j), where r + 0 and A is a constant. F has no z-component and is inde Find curl (r4(-yi + xj)), and show that it can be written in the form k[2+A] E rak, where a = , for any constant A. %3D Using your answer to part (a), find the direction of the curl of the vector fields with each of the following values of A (enter your answer as a unit vector ection of the curl): = -7: direction = = 5: direction = 7(x^2+y^2)^5/2k For each values of A in part (b), what (if anything) does your answer to part (b) tell you about the sign of the circulation around a small circle oriented interclockwise when viewed from above, and centered at (1, 1, 1)? 1= -7, the circulation is negative 1= 5, the circulation is positive sure you can say how your answer in part (c) would change if the question were about a small circle centered at (0, 0,0).)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let r = (x2 + y²)!/2 and consider the vector field F = rA(-yi + xj), where r + 0 and A is a constant. F has no z-component and is independent
of z.
(a) Find curl (pA(-yi + xj)), and show that it can be written in the form
curl F
k[2+A]
E rak, where a =
, for any constant A.
(b) Using your answer to part (a), find the direction of the curl of the vector fields with each of the following values of A (enter your answer as a unit vector in the
direction of the curl):
A = -7: direction =
A = 5: direction = 7(x^2+y^2)^5/2k
(c) For each values of A in part (b), what (if anything) does your answer to part (b) tell you about the sign of the circulation around a small circle oriented
counterclockwise when viewed from above, and centered at (1, 1, 1)?
If A = -7, the circulation is negative
If A = 5, the circulation is positive
(Be sure you can say how your answer in part (c) would change if the question were about a small circle centered at (0, 0,0).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80006671-cca5-4622-a451-7a3e68f04456%2F5810d6f3-801d-4e87-b761-033a216a9652%2F85lmf1n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let r = (x2 + y²)!/2 and consider the vector field F = rA(-yi + xj), where r + 0 and A is a constant. F has no z-component and is independent
of z.
(a) Find curl (pA(-yi + xj)), and show that it can be written in the form
curl F
k[2+A]
E rak, where a =
, for any constant A.
(b) Using your answer to part (a), find the direction of the curl of the vector fields with each of the following values of A (enter your answer as a unit vector in the
direction of the curl):
A = -7: direction =
A = 5: direction = 7(x^2+y^2)^5/2k
(c) For each values of A in part (b), what (if anything) does your answer to part (b) tell you about the sign of the circulation around a small circle oriented
counterclockwise when viewed from above, and centered at (1, 1, 1)?
If A = -7, the circulation is negative
If A = 5, the circulation is positive
(Be sure you can say how your answer in part (c) would change if the question were about a small circle centered at (0, 0,0).)
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