3.1. Consider the vector field F(x, y, z) = (2yz, y sin z, 1+ cos z).(a) Find a vector field G whose curl is F.(b) Let S be the half-ellipsoid 4x² + 4y² + z² = 4, z ≥ 0, oriented by the upward normal.Use Stokes's theorem to find ffs F. ds.-(c) Find fF.dS if S is the portion of the surface z = 1 – x² − y² above the xy-plane,oriented by the upward normal. (Hint: Take advantage of what you've already done.)

Answered: Image /qna-images/answer/228e32d8-ed55-415f-905f-2195ca29aeff.jpg

Answered: Consider the vector field F(x, y, z) =…

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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s) Given vector field F (x, y) = 2y i + 3x j .Find the circulation of the vector field F over the circle with radius 2, centered at (2, 3) oriented counterclockwise. Parametric equations of the circle are x(t) 3. = 2 + 2 cos t, y(t) = 3+ 2 sin t, 0
Compute the curl of F = (3y², xy, 22).
Exercise 1. A student is asked to prove If m is even and n is odd, then m² +n² is odd. The student provides the following proof: Proof. If m is even and n is odd, then m² + n² is odd. Since m is even, m = = 2k for any integer k. Since n is odd, n = 2k + 1. Note that m² + n² = (2k)² + (2k + 1)² = 8k² + 4k +1 So m² + n² is odd because it is an even number plus 1. (a) Identify the errors in the proof above. There are multiple errors. Explain each error. Your explanation should be written to the student who made the error and should try to help the student understand why what they wrote is incorrect. (b) In addition to errors, provide suggestions to improve the student's proof writing. That is, explain what parts of the proof are unclear or unjustified. Explain how such parts of the proof prevent some students from fully following the ment made. (c) Finally, provide a correct proof of the theorem above.
r = 2 on the hemispherical surface defined by 0 ≤ Ɵ ≤ π/2 and 0 ≤ɸ ≤ 2π F = 100cosƟar Verify Stokes' theorem for the vector field given by.
Find the directional derivative Duf (2,0) where, f(x, y) = xexy + y and u is the unit vector in the direction of 27/3. C) Ele tiple integral the left in each of the. (a) (b) [¹'. 24
Use Stokes' Theorem to find the circulation of F = 3yi + 2zj + 2xk around the triangle obtained by tracing out the path (4, 0, 0) to (4, 0, 6), to (4, 6, 6) back to (4, 0, 0). Circulation = [ F . dr =
Explain in your own words what does “divergence of vector field” mean. Calculate V. f, where f = sin(x) a + sin(y) ay. (The vectors a, ay are unit vec- tors.). Plot the divergence function. What do the maximum and minimum values represent?
Prove Stokes theorem for the vector field P over the triangle area bounded by this path by the path indicated by the arrows in the xy planet in the figure.
Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction 3₁ F = 2yi +5xj + z³k; C: the counterclockwise path around the perimeter of the triangle in the x-y plane formed from the x-axis, y- axis, and the line y = 5 - 6x
Calculate the work done by the force: F(x, y, z) = (y − z)î + (z + x)ĵ + (x + y)k in moving a particle along the curve parameterised as: r(t) = (2 cost, √2 sin t, √2 sin t) with t = [0, 2π]. Comment on what can be inferred from this result on the properties of this vector field.

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