Explanation Given information: Le F and G be vector fields. Concept used: Let F = ( F x , F y , F z ) and G = ( G x , G y , G z ) then F × G = | i j k F x F y F z G x G y G z | = i ( F y G z − G y F z ) + j ( F x G z − G x F z ) + k ( F x G y − G x F y ) Curl of two vectors is given by curl F = ∇ × F = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z F x F y F z | Divergence of two vectors is given by Div ( F × G ) = ∇ ⋅ ( F × G ) Proof: Take left side of the above proving statement: Div ( F × G ) = ∇ ⋅ ( F × G ) = ∂ ∂ x ( F y G z − G y F z ) + ∂ ∂ y ( F x G z − G x F z ) + ∂ ∂ z ( F x G y − G x F y ) = F y ∂ G z ∂ x + G z ∂ F y ∂ x − F z ∂ G y ∂ x − G y ∂ F z ∂ x + F z ∂ G x ∂ y + G x ∂ F z ∂ y − F x ∂ G z ∂ y − G z ∂ F x ∂ y + F x ∂ G y ∂ z + G y ∂ F x

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 18.3, Problem 32E

To determine

To prove: div(F×G)=curl(F)⋅G−F⋅curl(G)

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Chapter 18.3, Problem 31E

Chapter 18.3, Problem 33E

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Chapter 18 Solutions

Calculus - Standalone book

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To prove: div ( F × G ) = curl ( F ) ⋅ G − F ⋅ curl ( G )

Solution Summary: The author explains how the curl of two vectors is given by ccurlF=nabla.

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To prove: div(F×G)=curl(F)⋅G−F⋅curl(G)

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Chapter 18 Solutions