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Starting from the definition of curl and divergence in Cartesian coordinates, show that the fol- lowing properties hold for a differentiable scalar function f(x, y, z) and a differentiable vector field v(x, y, z) = v₁(x, y, z)i + v₂(x, y, z)j + v3(x, y, z)k. a) V. (fv) = Vf.v + f(V.v) b) ▼x (fv) =Vfxv + f(V x V).

Starting from the definition of curl and divergence in Cartesian coordinates, show that the fol- lowing properties hold for a differentiable scalar function f(x, y, z) and a differentiable vector field v(x, y, z) = v₁(x, y, z)i + v₂(x, y, z)j + v3(x, y, z)k. a) V. (fv) = Vf.v + f(V.v) b) ▼x (fv) =Vfxv + f(V x V).

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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Starting from the definition of curl and divergence in Cartesian coordinates, show that the fol- lowing properties hold for a differentiable scalar function f(x, y, z) and a differentiable vector field v(x, y, z) = v₁(x, y, z)i + v₂(x, y, z)j + v3(x, y, z)k. a) V. (fv) = Vf.v + f(V.v) b) ▼x (fv) =Vfxv + f(V x V).
Transcribed Image Text:Starting from the definition of curl and divergence in Cartesian coordinates, show that the fol- lowing properties hold for a differentiable scalar function f(x, y, z) and a differentiable vector field v(x, y, z) = v₁(x, y, z)i + v₂(x, y, z)j + v3(x, y, z)k. a) V. (fv) = Vf.v + f(V.v) b) ▼x (fv) =Vfxv + f(V x V).
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