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(a) Prove that F(x, y, z) = (y² cos r+z³)i+(2y sin r-4)j+(3r2+2)k is a conservative vector field. (b) Avsume that Fir ₂) Vi ✓ CULLC

(a) Prove that F(x, y, z) = (y² cos r+z³)i+(2y sin r-4)j+(3r2+2)k is a conservative vector field. (b) Avsume that Fir ₂) Vi ✓ CULLC

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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V2 Question Three (a) Prove that F(x, y, z) = (y² cos x+2³)i+(2y sinx-4)j +(3xz+2)k is a conservative vector field. (b) Assume that F(x, y, z) = ei +rej + (z + 1)eºk is a vector field defined on a smooth curve C given by r(t) = ti + t²j+t³k for 0 < t < 1. (c) Let S be a surface described by a² + y² + 32² = 1, z ≤0, and let F(x, y, z) = yi - rj+za³y²k. Prove that (VXF) [3,3,3,3] (i) Using the fundamental theorem of calculus, evaluate the line integral fF.dr. (ii) Show whether foF dr is independent of path. x F). dS = 2T Question Four Page 1 of 2 [3,4,4] 27
Transcribed Image Text:V2 Question Three (a) Prove that F(x, y, z) = (y² cos x+2³)i+(2y sinx-4)j +(3xz+2)k is a conservative vector field. (b) Assume that F(x, y, z) = ei +rej + (z + 1)eºk is a vector field defined on a smooth curve C given by r(t) = ti + t²j+t³k for 0 < t < 1. (c) Let S be a surface described by a² + y² + 32² = 1, z ≤0, and let F(x, y, z) = yi - rj+za³y²k. Prove that (VXF) [3,3,3,3] (i) Using the fundamental theorem of calculus, evaluate the line integral fF.dr. (ii) Show whether foF dr is independent of path. x F). dS = 2T Question Four Page 1 of 2 [3,4,4] 27
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