Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 15 - Multiple Integrals - 15.10 Exercises - Page 1071: 12

Answer

$x=\dfrac{u+2v}{5}$ and $y=\dfrac{3v-u}{10}$

Work Step by Step

The parallelogram with vertices $(0,0), (4,3), (2,4), (-2,1)$ is: $-10 \lt 3x-4y \lt 0$; $0 \lt x+2y \lt 10$ Plug $u=3x-4y$ and $v=x+2y$ and $-10 \lt 3x-4y \lt 0$; $0 \lt x+2y \lt 10 \implies -10 \lt u \lt 0$ and $0 \lt v \lt 10$ This shows that the rectangle is in the uv plane. Further, we have $3x-4y+2x+4y=u+2v$ $\implies x=\dfrac{u+2v}{5}$ and t $3x-4y-3x-6y=u-3v \implies y=\dfrac{3v-u}{10}$ Hence, $x=\dfrac{u+2v}{5}$ and $y=\dfrac{3v-u}{10}$
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