Explanation G is a coordinate transformation from x y -plane to r t -plane then G ( r , t ) = ( x , y ) ---------------- (i) Given: G ( r , t ) = ( r sin t , r − cos t ) ---------- (ii) Comparing (i) and (ii), x = r sin t ; y = r − cos t Calculation: Jacobian of the transformation is J ( r , t ) = ∂ ( x , y ) ∂ ( r , t ) = | ∂ x ∂ r ∂ x ∂ t ∂ y ∂ r ∂ y ∂ t | --------- (iii) Where ∂ x ∂ r , ∂ x ∂ t , ∂ y ∂ r , ∂ y ∂ t are first order partial derivatives. ∂ x ∂ r = ∂ ∂ r ( r sin t ) = ( sin t ) ∂ ∂ r ( r ) = ( sin t ) ( 1 ) = sin t [While differentiating partially with respect to r treat t as constant] ∂ x ∂ r = sin t ∂ x ∂ t = ∂ ∂ t ( r sin t ) = r ∂ ∂ t ( sin t ) = r ( cos t ) = r cos t , [using ∂ ∂ t ( sin t ) = cos t ] While differentiating partially with respect to t treat r as constant ∂ x ∂...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 16.6, Problem 15E

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The Jacobian

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Chapter 16.6, Problem 14E

Chapter 16.6, Problem 16E

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Chapter 16 Solutions

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The Jacobian

Solution Summary: The author explains that the Jacobian solution is J ( r, t ) = 1 at 1.

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Chapter 16 Solutions