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 - Practice Exercises - Page 1029: 30



Work Step by Step

A vector field is said to be conservative when $ curl \space F=\nabla \times F=0$ Now, $$ curl F= \nabla \times F\\=(\dfrac{\partial z}{\partial y}-\dfrac{\partial y}{\partial z})i+(\dfrac{\partial x}{\partial z}-\dfrac{\partial z}{\partial x})j+(\dfrac{\partial y}{\partial x}-\dfrac{\partial x}{\partial y}) \space k $$ From the given equation, we have $$ curl \space F=-3yz \times (x^2+y^2+z^2)^{-5/2}+3yz \times (x^2+y^2+z^2)^{-5/2}--3xz \times (x^2+y^2+z^2)^{-5/2}+3xz \times (x^2+y^2+z^2)^{-5/2}-3xy \times (x^2+y^2+z^2)^{-5/2}+3xy \times (x^2+y^2+z^2)^{-5/2}\\=0$$ Thus, the given $ F $ field is conservative.
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