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(b) Let r = zi+yj + zk, and let r = ||r. If f is a differentiable function of one variable, show that V(f(r)r) = rf'(r) +3f(r) (c) Ifo and are smooth scalar fields, show that V x (6V) = Vox V₂

(b) Let r = zi+yj + zk, and let r = ||r. If f is a differentiable function of one variable, show that V(f(r)r) = rf'(r) +3f(r) (c) Ifo and are smooth scalar fields, show that V x (6V) = Vox V₂

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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(a) Prove that the vector field F(x, y, z) = (x² + yz)i – 2y(x + z)j + (xy + z²)k is incompressible, and find its vector potential function. ||r. If f is a differentiable function of one (b) Let r = ri + yj + zk, and let r variable, show that ▼ · (ƒ(r)r) = rf'(r) +3ƒ(r) (c) If o and u are smooth scalar fields, show that ▼ × (6V) = ▼¢ × Vv Vox (d) Let f(1, y) = 2r² + xy - y². Prove that the directional derivative of f(x, y) at point x = (3,-2) in the direction v=i-jis.
Transcribed Image Text:(a) Prove that the vector field F(x, y, z) = (x² + yz)i – 2y(x + z)j + (xy + z²)k is incompressible, and find its vector potential function. ||r. If f is a differentiable function of one (b) Let r = ri + yj + zk, and let r variable, show that ▼ · (ƒ(r)r) = rf'(r) +3ƒ(r) (c) If o and u are smooth scalar fields, show that ▼ × (6V) = ▼¢ × Vv Vox (d) Let f(1, y) = 2r² + xy - y². Prove that the directional derivative of f(x, y) at point x = (3,-2) in the direction v=i-jis.
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