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: 38


$$2 \pi $$

Work Step by Step

A vector field is said to be conservative when $\int_C F \cdot dr=0$ We need to consider any closed path. $$\int_C F\cdot dr=\int_{(1,0,0)}^{(1,0,2 \pi)} \nabla (x^2 ze^y) \space dr $$ Now, $$[(x^2 ze^y)]_{1,0,0}^{1,0,2 \pi}-[(x^2 ze^y)]_{1,0,0}^{1,0,2 \pi} =2 \pi -0 \\=2 \pi $$
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