Calculus (3rd Edition)

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

Chapter 17 - Line and Surface Integrals - 17.3 Conservative Vector Fields - Preliminary Questions - Page 944: 1

Answer

If ${\bf{F}}$ is a gradient vector field, then the line integral of ${\bf{F}}$ along every closed curve is zero.

Work Step by Step

If ${\bf{F}}$ is a gradient vector field such that ${\bf{F}} = \nabla f$, where $f$ is some potential field for ${\bf{F}}$, then by definition ${\bf{F}}$ is conservative. By Theorem 1, the line integral of ${\bf{F}}$ around a closed curve is zero. So, we must add the word "closed" to the statement so that it becomes true. Thus, the true statement should be: If ${\bf{F}}$ is a gradient vector field, then the line integral of ${\bf{F}}$ along every closed curve is zero.
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