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n=-2 n=1 1 1 n=-1 n=2 n=0 0 n=3

n=-2 n=1 1 1 n=-1 n=2 n=0 0 n=3

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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Could someone answer last part  (detailed solutions) thanks

-1 -1 n=-2 0 n=1 1 1 -1 -1 n=-1 0 n = 2 -1 n=0 0 n=3 0 1 1
Transcribed Image Text:-1 -1 n=-2 0 n=1 1 1 -1 -1 n=-1 0 n = 2 -1 n=0 0 n=3 0 1 1
6. Consider the family of rotating vector fields V(x, y, z) = (x² + y²)n/2 - (²-OV ₂ VxX= i + as shown over the page. (i) Compute the curl of V; i.e. Vx X. Simplify your answer as much as possible. (ii) Convert the Cartesian forms of V and V x V into cylindrical forms. Simplify your answers as much as possible. (iii) State the domains for V and Vx V- it will change with n. (iv) Using the cylindrical form of V confirm your curl answer in (ii) with the cylindrical form of curl, I (x² + y²)n/2³ + Ok, 8V/₂2 (Ə(pV₂) +7 -V) e ép- es + др (v) For a cylindrical vector field Green's Theorem in the (zy) plane can be written = // vxv.è, ds. nĘ Z ᎯᏙ . Әр ᎯᏙᎥ " ap (This is really Stokes's Theorem with n = è.). Evaluate the line integral and the surface integral above on the unit circle centred at the origin. Explain the discrepancy in the two values and suggest a way to correct it. Hint: consider the domain.
Transcribed Image Text:6. Consider the family of rotating vector fields V(x, y, z) = (x² + y²)n/2 - (²-OV ₂ VxX= i + as shown over the page. (i) Compute the curl of V; i.e. Vx X. Simplify your answer as much as possible. (ii) Convert the Cartesian forms of V and V x V into cylindrical forms. Simplify your answers as much as possible. (iii) State the domains for V and Vx V- it will change with n. (iv) Using the cylindrical form of V confirm your curl answer in (ii) with the cylindrical form of curl, I (x² + y²)n/2³ + Ok, 8V/₂2 (Ə(pV₂) +7 -V) e ép- es + др (v) For a cylindrical vector field Green's Theorem in the (zy) plane can be written = // vxv.è, ds. nĘ Z ᎯᏙ . Әр ᎯᏙᎥ " ap (This is really Stokes's Theorem with n = è.). Evaluate the line integral and the surface integral above on the unit circle centred at the origin. Explain the discrepancy in the two values and suggest a way to correct it. Hint: consider the domain.
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