Explanation Given information: The u ( x , y ) = a x 3 + b x 2 y + c x y 2 + d y 3 Concept used: The given polynomial u ( x , y ) = a x 3 + b x 2 y + c x y 2 + d y 3 is said to be harmonic iff ∇ 2 u = 0 ⇒ ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0 Calculation: The given polynomial, u ( x , y ) = a x 3 + b x 2 y + c x y 2 + d y 3 On differentiating with respect to x we get, ∂ u ∂ x = a ∂ ∂ x x 3 + b y ∂ ∂ x x 2 + c y 2 ∂ ∂ x x + ∂ ∂ x d y 3 ⇒ ∂ u ∂ x = a ( 3 x 2 ) + b y ( 2 x ) + c y 2 ( 1 ) + ( 0 ) ... a , b , c , d , y a r e c o n s t a n t . ⇒ ∂ u ∂ x = 3 a x 2 + 2 b x y + c y 2 Differentiating again with respect to x we get, ⇒ ∂ 2 u ∂ x 2 = 3 a ∂ ∂ x x 2 + 2 b y ∂ ∂ x x + ∂ ∂ x c y 2 ⇒ ∂ 2 u ∂ x 2 = 3 a ( 2 x ) + 2 b y ( 1 ) + ( 0 ) ⇒ ∂ 2 u ∂ x 2 = 6 a x + 2 b y Now The given polynomial, u ( x , y ) = a x 3 + b x 2 y + c x y 2 + d y 3 On differentiating with respect to y we get, ∂ u ∂ y = ∂ ∂ y a x 3 + b x 2 ∂ ∂ y y + c x ∂ ∂ y y 2 + d ∂ ∂ y y 3 ⇒ ∂ u ∂ y = ( 0 ) + b x 2 ( 1 ) + c x ( 2 y ) + d ( 3 y 2 )

CALCULUS EARLY TRANSCENDENTALS W/ACHIEVE

4th Edition

ISBN: 9781319057664

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 14.3, Problem 76E

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The partial derivatives at P of the function whose contour map is given..

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Chapter 14.3, Problem 75E

Chapter 14.3, Problem 77E

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Chapter 14 Solutions

CALCULUS EARLY TRANSCENDENTALS W/ACHIEVE

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E

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To estimate: The partial derivatives at P of the function whose contour map is given..

Solution Summary: The author calculates the partial derivatives at P of the function whose contour map is given.

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To estimate: The partial derivatives at P of the function whose contour map is given..

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Chapter 14 Solutions

To estimate:

The partial derivatives at P of the function whose contour map is given..