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Question Two (a) Prove that the vector field F(x, y, z) = (x² + y²)i – 2y(z+=)j + (zy +2²)k is incompressible, and find its vector potential function.

Question Two (a) Prove that the vector field F(x, y, z) = (x² + y²)i – 2y(z+=)j + (zy +2²)k is incompressible, and find its vector potential function.

Advanced Engineering Mathematics
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 One (a) Let U = 3i-j-4k and V = -2i+4j- 3k. Show that U-V is commutative but U x V is not. (b) For any vectors A and B, prove that V. (A x B)=B-(V x A) - A - (V x B) (c) Let F(x, y, z)=z³i+y²j+z³k. Compute the following: (i) V.F (ii) V x F (d) Give one example to illustrate that vectors satisfy the triangle inequality. Justify your answer. Question Two (a) Prove that the vector field F(x, y, z) = (2² + y=)i − 2y(x + =)j + (zy +2²)k is incompressible, and find its vector potential function. (b) Let r = zi+j+zk, and let r= ||r||. If f is a differentiable function of one variable, show that ▼(f(r)r) = rf'(r)+3f(r) (c) If and are smooth scalar fields, show that ▼x (V) = Vox V (d) Let f(x, y) = 2r²+zy-y². Prove that the directional derivative of f(z,y) at point x = (3,-2) in the direction v=i-jis
Transcribed Image Text:Question One (a) Let U = 3i-j-4k and V = -2i+4j- 3k. Show that U-V is commutative but U x V is not. (b) For any vectors A and B, prove that V. (A x B)=B-(V x A) - A - (V x B) (c) Let F(x, y, z)=z³i+y²j+z³k. Compute the following: (i) V.F (ii) V x F (d) Give one example to illustrate that vectors satisfy the triangle inequality. Justify your answer. Question Two (a) Prove that the vector field F(x, y, z) = (2² + y=)i − 2y(x + =)j + (zy +2²)k is incompressible, and find its vector potential function. (b) Let r = zi+j+zk, and let r= ||r||. If f is a differentiable function of one variable, show that ▼(f(r)r) = rf'(r)+3f(r) (c) If and are smooth scalar fields, show that ▼x (V) = Vox V (d) Let f(x, y) = 2r²+zy-y². Prove that the directional derivative of f(z,y) at point x = (3,-2) in the direction v=i-jis
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