Consider the family of rotating vector fields X(², ²) = (²+²)/2 + (x²+²)/²³ +0k, než as shown over the page. (i) Compute the curl of V; i.e. Vx V. Simplify your answer as much as possible. (ii) Convert the Cartesian forms of Y and Vx V into cylindrical forms. Simplify your answers as much as possible. (iii) State the domains for Y and Vx X-it will change with n. (iv) Using the cylindrical form of V confirm your curl answer in (ii) with the cylindrical form of curl, vxX 7xX= - (²016-02) ₂ - (-) és + - (012 - 2V) e. + = (³x(OVD) OVA) è (v) For a cylindrical vector field Green's Theorem in the (zy) plane can be written fx-de-VXX-a, ds. dr (This is really Stokes's Theorem with = 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.
Consider the family of rotating vector fields X(², ²) = (²+²)/2 + (x²+²)/²³ +0k, než as shown over the page. (i) Compute the curl of V; i.e. Vx V. Simplify your answer as much as possible. (ii) Convert the Cartesian forms of Y and Vx V into cylindrical forms. Simplify your answers as much as possible. (iii) State the domains for Y and Vx X-it will change with n. (iv) Using the cylindrical form of V confirm your curl answer in (ii) with the cylindrical form of curl, vxX 7xX= - (²016-02) ₂ - (-) és + - (012 - 2V) e. + = (³x(OVD) OVA) è (v) For a cylindrical vector field Green's Theorem in the (zy) plane can be written fx-de-VXX-a, ds. dr (This is really Stokes's Theorem with = 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.
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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i need help with part 4 please
Transcribed Image Text:Consider the family of rotating vector fields
-V
V(z. y, z)=
(2²+3²)n/2
*
+
(z²+p²js/z3+0k, n€Z
as shown over the page.
(i) Compute the curl of V; i.e. Vx V. Simplify your answer as much as possible.
(ii) Convert the Cartesian forms of V and Vx V into cylindrical forms. Simplify your answers as
much as possible.
(iii) State the domains for Y and Vx X-it will change with n.
(iv) Using the cylindrical form of V confirm your curl answer in (ii) with the cylindrical form of curl,
xxx = (2016-01²) a
ap
- 01/2) es + = (2x(012) 31) 2
x)。
(V₂) V)
, ᎧᏙ ;
1
ap
(v) For a cylindrical vector field Green's Theorem in the (zy) plane can be written
f. X. dr = // Vx X-eds.
-OV/³) E₁ - (OV
「 ᎧᏙ
(This is really Stokes's Theorem with = 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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