Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 15 - Section 15.9 - Change of Variables in Multiple Integrals - 15.9 Exercise - Page 1060: 6



Work Step by Step

Here, we have $\dfrac{\partial x}{\partial u}=1; \dfrac{\partial x}{\partial v}=w$ and $ \dfrac{\partial x}{\partial w}=v$ Also, $\dfrac{\partial y}{\partial u}=w; \dfrac{\partial y}{\partial v}=1$ and $ \dfrac{\partial y}{\partial w}=u$ Now, $Jacobian =\begin{vmatrix} \dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v}&\dfrac{\partial x}{\partial w}\\\dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}&\dfrac{\partial y}{\partial w}\\\dfrac{\partial z}{\partial u}&\dfrac{\partial z}{\partial v}&\dfrac{\partial z}{\partial w}\end{vmatrix}=\begin{vmatrix} 1&w&v\\w&1&u\\v&u&1\end{vmatrix}=1(1-u^2)-w(w-uv)+v(uw-v)=1+2uvw-u^2-v^2-w^2$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.