Explanation G is a coordinate transformation from x y -plane to u v -plane then G ( u , v ) = ( x , y ) ---------------- (i) Given: G ( u , v ) = ( 3 u + 4 v , u − 2 v ) ---------- (ii) Comparing (i) and (ii), x = 3 u + 4 v ; y = u − 2 v Jacobian of the transformation is J ( u , v ) = ∂ ( x , y ) ∂ ( u , v ) = | ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v | --------- (iii) Where ∂ x ∂ u , ∂ x ∂ v , ∂ y ∂ u , ∂ y ∂ v are first order partial derivatives. ∂ x ∂ u = ∂ ∂ u ( 3 u + 4 v ) = ∂ ∂ u ( 3 u ) + ∂ ∂ u ( 4 v ) = 3 ∂ ∂ u ( u ) + 0 [While differentiating partially with respect to u treat v as constant] ∂ x ∂ u = 3 ( 1 ) = 3 ∂ x ∂ v = >&#x...

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Chapter 16.6, Problem 13E

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The Jacobian

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Chapter 16.6, Problem 12E

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The Jacobian

Solution Summary: The author explains that G is a coordinate transformation from x y to u v -plane.

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Chapter 16 Solutions