Let r = (x? + y?)!/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 (rA(-yi + xj)), and show that it can be written in the form curl F = pak, 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 = -4: direction = A = 0; direction = (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 = -4, the circulation is ? If A = 0, the circulation is ?

Let r = (x? + y?)!/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 (rA(-yi + xj)), and show that it can be written in the form curl F = pak, 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 = -4: direction = A = 0; direction = (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 = -4, the circulation is ? If A = 0, the circulation is ?

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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A: solution

Question

Let r = (x? + y?)!/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 (rA(-yi + xj)), and show that it can be written in the form
curl F =
pak, 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 = -4: direction =
A = 0; direction =
(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 = -4, the circulation is ?
If A = 0, the circulation is ?

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