Explanation Given: The given Laplacian is, Δ u = ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 Calculation: Consider theLaplacian, Δ u = ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 Consider the equation. u = u ( x , y ) … ( 1 ) Differentiate equation ( 1 ) partially with respect to x . u x = ∂ u ∂ x = ∂ u ∂ r ∂ r ∂ x + ∂ u ∂ θ ∂ θ ∂ x … (2) Differentiate equation ( 2 ) partially with respect to x . u x x = ∂ 2 u ∂ x 2 = ( u r ) x r x + u r r x x + ( u θ ) x θ x + u θ θ x x = ( u r r r x + u r θ θ x ) r x + u r r x x + ( u θ r r x + u θ θ θ x ) θ x + u θ θ x x … (3) For polar coordinates, x = r cos θ y = r sin θ r = x 2 + y 2 θ = tan − 1 ( y x ) Differentiate r partially with respect to x . ∂ r ∂ x = ∂ ∂ x x 2 + y 2 = 2 x 2 x 2 + y 2 = x r … (4) Differentiate Equation ( 4 ) partially with respect to x . ∂ 2 r ∂ x 2 = y 2 r 3 Differentiate θ partially with respect to x . ∂ θ ∂ x = ∂ ∂ x tan − 1 ( y x ) = − y r 2 … (5) Differentiate Equation ( 5 ) partially with respect to x . ∂ 2 θ ∂ x 2 = 2 x y r 4 Substitute the values in Equation ( 3 )

Fundamentals of Differential Equations [With CDROM] - 7th Edition

7th Edition

ISBN: 9780321410481

Author: Saff, Edward B., Snider, Arthur David, Nagle, R. Kent

Publisher: Addison Wesley

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Question

Chapter 10.7, Problem 6E

To determine

To derive:

The polar coordinate form of the given Laplacian.

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Chapter 10.7, Problem 5E

Chapter 10.7, Problem 7E

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Chapter 10 Solutions

Fundamentals of Differential Equations [With CDROM] - 7th Edition

Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 6E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...

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To derive: The polar coordinate form of the given Laplacian.

Solution Summary: The author explains how to derive the polar coordinate form of the given Laplacian.

To derive: The polar coordinate form of the given Laplacian.

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Chapter 10 Solutions

To derive:

The polar coordinate form of the given Laplacian.