Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 16 - Multiple Integration - 16.6 Change of Variables - Exercises - Page 905: 15



Work Step by Step

To find the Jacobian, we calculate the determinant of the 2x2 matrix $\frac{\partial(x, y)}{\partial(r, t)}$ as follows $$ \operatorname{Jac}(G)=\frac{\partial(x, y)}{\partial(r, t)}=\left|\begin{array}{ll} {\frac{\partial x}{\partial r}} & {\frac{\partial x}{\partial t}} \\ {\frac{\partial y}{\partial r}} & {\frac{\partial y}{\partial t}} \end{array}\right|=\left|\begin{array}{ll} { \sin t} & {r\cos t} \\ {1} & {\sin t} \end{array}\right| =\sin^2 t-r\cos t. $$ Now, at the point $(r,t)= (1,\pi)$, we have $ \operatorname{Jac}(G)=\sin^2 \pi-\cos \pi=1.$
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