Explanation Given: Two implicit functions are given as x 2 + y 2 = 2 s t − 10 and 2 x y = s 2 − t 2 Formula Used: To find ∂ x ∂ s , ∂ x ∂ t , ∂ y ∂ s , ∂ y ∂ t the following steps should be performed: Find d y and d x by taking the differential of both the equations and solve for d y and d x using cross multiplication method. By chain rule d y = ∂ y ∂ t d t + ∂ y ∂ s d s and d x = ∂ x ∂ t d t + ∂ x ∂ s d s Compare the obtained d y and d x with the chain rule. Calculation: Consider the given equations. x 2 + y 2 = 2 s t − 10 ............ (1) 2 x y = s 2 − t 2 ............ (2) Differentiate (1) and (2) as shown. 2 x d x + 2 y d y = 2 ( s d t + t d s ) x d x + y d y = s d t + t d s ...... ( 3 ) 2 ( x d y + y d x ) = 2 s d s + 2 t d t x d y + y d x = s d s + t d t ...... ( 4 ) Solve equation (3) and (4) for d x by cross multiplication as shown. d x = | s d t + t d s y s d s − t d t x | | x y y x | d x = x ( s d t + t d s ) − y ( s d s − t d t ) x 2 − y 2 d x = s x d t + t x d s − y s d s + y t d t x 2 − y 2 d x = ( t x − y s x 2 − y 2 ) d s + ( s x + y t x 2 − y 2 ) d t ............ (5) Again, by chain rule we have d x = ∂ x ∂ t d t + ∂ x ∂ s d s ............. (6) Comparing equation (5) and (6), we get ∂ x ∂ s = ( t x − y s x 2 − y 2 ) and ∂ x ∂ t = ( s x + y t x 2 − y 2 ) Substitute 4 for x, 2 for y, 5 for s and 3 for t in above values

Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

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Question

Chapter 4.7, Problem 10P

To determine

To find:The value of ∂x∂s,∂x∂t,∂y∂s,∂y∂t at (x,y,s,t)=(4,2,5,3) for the given function.

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Chapter 4.7, Problem 9P

Chapter 4.7, Problem 11P

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Chapter 4 Solutions

Mathematical Methods in the Physical Sciences

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The value of ∂ x ∂ s , ∂ x ∂ t , ∂ y ∂ s , ∂ y ∂ t at ( x , y , s , t ) = ( 4 , 2 , 5 , 3 ) for the given function.

Solution Summary: The author explains how to find the value of lpartial s, dx, and t by taking the differential of both the equations.

To find:The value of ∂x∂s,∂x∂t,∂y∂s,∂y∂t at (x,y,s,t)=(4,2,5,3) for the given function.

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Chapter 4 Solutions