Explanation Given information: Δ φ = ∂ 2 φ ∂ x 2 + ∂ 2 φ ∂ y 2 + ∂ 2 φ ∂ z 2 , where Δ denotes the Laplace operator. Concept used: Let the vector field F → = A i → + B j → + C k → then the divergence is defined to be, Div F → = ∂ A ∂ x + ∂ B ∂ y + ∂ C ∂ z Proof: Let φ ( x , y , z ) be the vector field So, the divergence is to be: Div φ = ∇ φ = ∂ ∂ x i ^ + ∂ ∂ y j ^ + ∂ ∂ z z ^ If again find the divergence of ∇ φ then: ∇ φ = ∂ φ ∂ x + ∂ φ ∂ y + ∂ φ ∂ z Div ( ∇ φ ) = ( ∂ φ ∂ x + ∂ φ ∂ y + ∂ φ ∂ z ) ( ∂ φ ∂ x + ∂ φ ∂ y + ∂ φ ∂ z ) Div ( ∇ φ ) = ( ∂ φ ∂ x ∂ φ ∂ x + ∂ φ ∂ x

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ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 17.3, Problem 35E

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Prove that div(∇φ)=Δφ if Δφ=∂2φ∂x2+∂2φ∂y2+∂2φ∂z2, where Δ denotes the Laplace operator.

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Chapter 17.3, Problem 34E

Chapter 17.3, Problem 36E

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Chapter 17 Solutions

Calculus: Early Transcendentals

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Prove that div( ∇ φ ) = Δ φ if Δ φ = ∂ 2 φ ∂ x 2 + ∂ 2 φ ∂ y 2 + ∂ 2 φ ∂ z 2 , where Δ denotes the Laplace operator.

Solution Summary: The author explains how the divergence of phi (x,y,z) is defined.

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Prove that div(∇φ)=Δφ if Δφ=∂2φ∂x2+∂2φ∂y2+∂2φ∂z2, where Δ denotes the Laplace operator.

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Chapter 17 Solutions