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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).)

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Vectors In Two And Three Dimensions
Section9.FOM: Focus On Modeling: Vectors Fields
Problem 11P
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leave answers in vector form, thank you.

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).)
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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