Explanation Formula used: The divergence of the vector field F ( f , g , h ) is ∇ ⋅ F = ∂ f ∂ x + ∂ g ∂ y + ∂ h ∂ z . The curl of a vector field F ( f , g , h ) is ∇ × F = ( ∂ h ∂ y − ∂ g ∂ z ) i + ( ∂ f ∂ z − ∂ h ∂ x ) j + ( ∂ g ∂ x − ∂ f ∂ y ) k . Proof: Suppose F ( f , g , h ) and G ( k , m , n ) are the two vector fields. Compute the value of F × G . F × G = | i j k f g h k m n | = i ( g n − h m ) − j ( f n − k h ) + k ( f m − k g ) = 〈 g n − h m , f n − k h , f m − k g 〉 Use the curl formula to compute the value of ∇ × ( F × G ) . ∇ × ( F × G ) = ∇ × 〈 g n − h m , f n − k h , f m − k g 〉 = ( ( ∂ ∂ y ( f m − k g ) − ∂ ∂ z ( f n − k h ) ) i + ( ∂ ∂ z ( g n − h m ) − ∂ ∂ x ( f m − k g ) ) j + ( ∂ ∂ x ( f n − k h ) − ∂ ∂ y ( g n − h m ) ) k ) = 〈 ∂ ∂ y ( f m − k g ) − ∂ ∂ z ( f n − k h ) , ∂ ∂ z ( g n − h m ) − ∂ ∂ x ( f m − k g ) , ∂ ∂ x ( f n − k h ) − ∂ ∂ y ( g n − h m ) 〉 Thus, the value of ∇ × ( F × G ) is 〈 ∂ ∂ y ( f m − k g ) − ∂ ∂ z ( f n − k h ) , ∂ ∂ z ( g n − h m ) − ∂ ∂ x ( f m − k g ) , ∂ ∂ x ( f n − k h ) − ∂ ∂ y ( g n − h m ) 〉 . Compute the value of ( G ⋅ ∇ ) F . ( G ⋅ ∇ ) F = ( k ∂ ∂ x + m ∂ ∂ y + n ∂ ∂ z ) 〈 f , g , h 〉 = 〈 k ∂ f ∂ x + m ∂ f ∂ y + n ∂ f ∂ z , k ∂ g ∂ x + m ∂ g ∂ y + n ∂ g ∂ z , k ∂ h ∂ x + m ∂ h ∂ y + n ∂ h ∂ z 〉 Use divergence formula to compute the value of G ( ∇ ⋅ F ) . G ( ∇ ⋅ F ) = 〈 k , m , n 〉 ( ∂ f ∂ x + ∂ g ∂ y + ∂ h ∂ z ) = 〈 k ∂ f ∂ x + k ∂ g ∂ y + k ∂ h ∂ z , m ∂ f ∂ x + m ∂ g ∂ y + m ∂ h ∂ z , n ∂ f ∂ x + n ∂ g ∂ y + n ∂ h ∂ z 〉 Compute the value of ( F ⋅ ∇ ) G . ( F ⋅ ∇ ) G = ( f ∂ ∂ x + g ∂ ∂ y + h ∂ ∂ z ) 〈 k , m , n 〉 = 〈 f ∂ k ∂ x + g ∂ k ∂ y + h ∂ k ∂ z , f ∂ m ∂ x + g ∂ m ∂ y + h ∂ m ∂ z , f ∂ n ∂ x + g ∂ n ∂ y + h ∂ n ∂ z 〉 Use divergence formula to compute the value of F ( ∇ ⋅ G )

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Question

Chapter 14.5, Problem 70E

To determine

To prove: The identity ∇×(F×G)=(G⋅∇)F−G(∇⋅F)+(F⋅∇)G−F(∇⋅G) .

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Chapter 14.5, Problem 69E

Chapter 14.5, Problem 71E

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Chapter 14 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 14.1 - How is a vector field F = f, g, h used to describe...Ch. 14.1 - Sketch the vector field F = x, y.Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Interpret the gradient field of the temperature...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...

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To prove: The identity ∇ × ( F × G ) = ( G ⋅ ∇ ) F − G ( ∇ ⋅ F ) + ( F ⋅ ∇ ) G − F ( ∇ ⋅ G ) .

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To prove: The identity ∇×(F×G)=(G⋅∇)F−G(∇⋅F)+(F⋅∇)G−F(∇⋅G) .

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Chapter 14 Solutions

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