Explanation Given: The given equation is z = x 2 + 2 y 2 . Calculation: As the given equation z = x 2 + 2 y 2 . As ∂ 2 z ∂ r ∂ y = ( ∂ z ∂ r ) ( ∂ z ∂ y ) ( ∂ z ∂ y ) r = ∂ ( r 2 + y 2 ) ∂ r | r ( ∂ z ∂ y ) r = 2 y Now, from exercise-11, ∂ 2 z ∂ r ∂ y = ( ∂ ∂ r ) ( ∂ z ∂ y ) = ( ∂ ∂ r ( 2 y ) ) = 0 Now, repeat z = r 2 tan 2 θ , where x = r cos θ and y = r sin θ . Therefore, x 2 + y 2 = r 2 ( cos 2 θ + sin 2 θ ) = r 2 ............. ( 1 ) y x = r sin θ r cos θ = tan θ ........................ ( 2 ) . Since, z = r 2 tan 2 θ ......................... ( 3 )

Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

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Chapter 4.1, Problem 19P

To determine

Tofind:the value of ∂2z∂r∂y at the given points in the given equation.

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Chapter 4.1, Problem 18P

Chapter 4.1, Problem 20P

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Chapter 4 Solutions

Mathematical Methods in the Physical Sciences

Ch. 4.1 - If u=x2/x2+y2, find u/x,u/y.Ch. 4.1 - Prob. 2P Ch. 4.1 - If z=lnu2+v2+w2, find z/u,z/v,z/w.Ch. 4.1 - For w=x3y32xy+6, find 2w/ax2 and 2w/ay2 at the...Ch. 4.1 - For w=8x4+y42xy2, find 2w/x2 and 2w/y2 at the...Ch. 4.1 - For u=excosy, (a) verify that 2u/xy=2u/yx; (b)...Ch. 4.1 - Prob. 7P Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...Ch. 4.1 - If z=x2+2y2,x=rcos,y=rsin, find the following...

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Tofind: the value of ∂ 2 z ∂ r ∂ y at the given points in the given equation.

Solution Summary: The author explains how to find the value of partial2zprtial y at the given points in a given equation.

Tofind:the value of ∂2z∂r∂y at the given points in the given equation.

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Chapter 4 Solutions