Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 16: Integrals and Vector Fields - Section 16.4 - Green's Theorem in the Plane - Exercises 16.4 - Page 979: 37



Work Step by Step

The tangential form for Green's Theorem can be written as: Counterclockwise Circulation: $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy$ Now $\dfrac{\partial^2 f}{\partial y^2} =\dfrac{\partial M}{\partial y}$ and $- \dfrac{\partial^2 f}{\partial x^2} =\dfrac{\partial N}{\partial x}$ Now $\oint_C \dfrac{\partial f}{\partial y} \ dx - \dfrac{\partial f}{\partial x} \ dy=\iint_{R} (- \dfrac{\partial^2 f}{\partial x^2} - \dfrac{\partial^2 f}{\partial y^2}) \ dx \ dy $ or, $\oint_C \dfrac{\partial f}{\partial y} \ dx - \dfrac{\partial f}{\partial x} \ dy=0$
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