Explanation Given information: There is a region R bounded by parallelogram with vertices ( 0 , 0 ) , ( 2 , 0 ) , ( 1 , 1 ) , ( − 1 , 1 ) . Formula used: The Jacobian for change if variable is, J ( u , v ) = ∂ ( x , y ) ∂ ( u , v ) = ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v Calculation: Consider the provided statement thatR is the region bounded by parallelogram with vertices ( 0 , 0 ) , ( 2 , 0 ) , ( 1 , 1 ) , ( − 1 , 1 ) . The parallelogram is bounded by the equations. Recall that two point form of equation of line is Line of equation between the points y − y 1 = y 2 − y 1 x 2 − x 1 ( x − x 1 ) where ( x 1 , y 1 ) and ( x 2 , y 2 ) are point through which line passes. Line of equation between the points ( 0 , 0 ) and ( 2 , 0 ) is y = 0 . Line of equation between the points ( − 1 , 1 ) and ( 1 , 1 ) is y = 1 . Line of equation between the points ( 0 , 0 ) and ( − 1 , 1 ) is x + y = 0 . Line of equation between the points ( 2 , 0 ) and ( 1 , 1 ) is x + y = 2 . Take the transformation as x + y = u and y = v where 0 ≤ u ≤ 2 and 0 ≤ v ≤ 1 . Solve the above equations for x and y. The equation obtained are, u − v = x and y = v

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ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 12.8, Problem 57PS

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To show:When the change of variable of the form x=au+bv,y=cu+dv the double integral is ∬Rf(x+y)dydx=∫02f(t)dt where R is the region bounded by parallelogram with vertices (0,0),(2,0),(1,1),(−1,1) .

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Chapter 12.8, Problem 56PS

Chapter 12.8, Problem 58PS

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When the change of variable of the form x = a u + b v , y = c u + d v the double integral is ∬ R f ( x + y ) d y d x = ∫ 0 2 f ( t ) d t where R is the region bounded by parallelogram with vertices ( 0 , 0 ) , ( 2 , 0 ) , ( 1 , 1 ) , ( − 1 , 1 ) .

Solution Summary: The author explains the Jacobian for change if variable is cJ(u,v). The parallelogram is bounded by the equation

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To show:When the change of variable of the form x=au+bv,y=cu+dv the double integral is ∬Rf(x+y)dydx=∫02f(t)dt where R is the region bounded by parallelogram with vertices (0,0),(2,0),(1,1),(−1,1) .

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Chapter 12 Solutions