Explanation Given: The linear mapping G ( u , v ) maps the unit square to the parallelogram in the x − y plane spanned by the vectors 〈 3 , − 1 〉 and 〈 1 , 4 〉 . If G : D → D 0 and if f is continuous then ∫ ∫ D f ( x , y ) d x d y = ∫ ∫ D 0 f ( x ( u , v ) , y ( u , v ) ) | ∂ ( x , y ) ∂ ( u , v ) | Calculation: It is given that G ( u , v ) is a linear mapping such that G ( 1 , 0 ) = ( 3 , − 1 ) and G ( 0 , 1 ) = ( 1 , 4 ) Let G ( u , v ) = ( A u + B v , C u + D v ) then G ( 1 , 0 ) = ( A , C ) → ( 3 , − 1 ) = ( A , C ) → A = 3 , C = − 1 G ( 0 , 1 ) = ( B , D ) → ( 1 , 4 ) = ( B , D ) → B = 1 , D = 4 Thus, the linear transformation is G ( u , v ) = ( 3 u + v , − u + 4 v ) Consider x = 3 u + v Therefore, ∂ x ∂ u = 3 ∂ x ∂ v = 1 ...

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Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16, Problem 57CRE

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The linear mapping $G (u, v)$ that maps the unit square to the parallelogram in the $x - y$ plane spanned by the vectors $〈 3, - 1 〉$ and $〈 1, 4 〉$ .

The area of the image of the rectangle $R = [0, 4] \times [0, 3]$ under $G$

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Chapter 16, Problem 56CRE

Chapter 16, Problem 58CRE

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Chapter 16 Solutions

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The linear mapping G ( u , v ) that maps the unit square to the parallelogram in the x − y plane spanned by the vectors 〈 3 , − 1 〉 and 〈 1 , 4 〉 . The area of the image of the rectangle R = [ 0 , 4 ] × [ 0 , 3 ] under G

Solution Summary: The linear mapping G ( u, v ) maps the unit square to the parallelogram in the x y plane spanned by the vectors

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To calculate:

The linear mapping $G (u, v)$ that maps the unit square to the parallelogram in the $x - y$ plane spanned by the vectors $〈 3, - 1 〉$ and $〈 1, 4 〉$ .

The area of the image of the rectangle $R = [0, 4] \times [0, 3]$ under $G$

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Chapter 16 Solutions

The linear mapping G ( u , v ) that maps the unit square to the parallelogram in the x − y plane spanned by the vectors 〈 3 , − 1 〉 and 〈 1 , 4 〉 . The area of the image of the rectangle R = [ 0 , 4 ] × [ 0 , 3 ] under G

Solution Summary: The linear mapping G ( u, v ) maps the unit square to the parallelogram in the x y plane spanned by the vectors

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To calculate: The linear mapping G(u,v)that maps the unit square to the parallelogram in the x−y plane spanned by the vectors 〈3,−1〉and 〈1,4〉. The area of the image of the rectangle R=[0,4]×[0,3]under G

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Chapter 16 Solutions

To calculate:

The linear mapping $G (u, v)$ that maps the unit square to the parallelogram in the $x - y$ plane spanned by the vectors $〈 3, - 1 〉$ and $〈 1, 4 〉$ .

The area of the image of the rectangle $R = [0, 4] \times [0, 3]$ under $G$