Question Three(a) Prove that F(x, y, z) = (y2cos x+z3)i+(2y sin x−4)j+(3xz+2)k is a conservativevector field.(b) Assume that F(x, y, z) = eyi + xeyj + (z + 1)ezk is a vector field defined on asmooth curve C given by r(t) = ti + t2j + t3k for 0 ≤ t ≤ 1.P(i) Using the fundamental theorem of calculus, evaluate the line integralRC F · dr.(ii) Show whether RC F · dr is independent of path.(c) Let S be a surface described by x2 + y2 + 3z2 = 1, z ≤ 0, and letF(x, y, z) = yi − xj + zx3y2k. Prove thatZS(∇ × F) · dS = 2π[ Question Three(a) Prove that F(x, y, z) = (y² cos x+z³)i+(2y sin x-4)j+(3xz+2)k is a conservativevector field.(b) Assume that F(x, y, z) = ei + xe³j + (z + 1)e²k is a vector field defined on asmooth curve C given by r(t) = ti+t²j+t³k for 0 ≤ t ≤ 1.(i) Using the fundamental theorem of calculus, evaluate the line integralJF. dr.(ii) Show whether fF. dr is independent of path.(c) Let S be a surface described by x² + y² + 3z² = 1, z ≤ 0, and letF(x, y, z) = yi-xj+zx³y²k. Prove that(((VXF)Page 1 of 2(V x F). dS = 2″

First question is wrong. F is not conservative.

Answered: Question Three (a) Prove that F(x, y,…

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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Question Three (a) Prove that F(x, y, z) = (y 2 cos x+z 3 )i+(2y sin x−4)j+(3xz+2)k is a conservative vector field. (b) Assume that F(x, y, z) = e y i + xey j + (z + 1)e zk is a vector field defined on a smooth curve C given by r(t) = ti + t 2 j + t 3k for 0 ≤ t ≤ 1. P(i) Using the fundamental theorem of calculus, evaluate the line integral R C F · dr. (ii) Show whether R C F · dr is independent of path. (c) Let S be a surface described by x 2 + y 2 + 3z 2 = 1, z ≤ 0, and let F(x, y, z) = yi − xj + zx3 y 2k. Prove that Z S (∇ × F) · dS = 2π [