Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 16 - Vector Calculus - 16.8 Stokes' Theorem - 16.8 Exercises - Page 1179: 16

Answer

$$\dfrac{2 A}{\sqrt 3}$$

Work Step by Step

We have: $curl F=3i+j-2k$ Need to re-write the equation as: $\int_{C} F \cdot dr=\int_{C} (z i-2 x j+3y k) \cdot (dx i+dy j+dz k)$ Let us suppose that $S$ be the part of the plane $x+y+z=1$ and this is the region enclosed by the loop $C$. Stokes' Theorem states that $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr $ Now, $$\int_{C} F \cdot dr=\iint_{S} F \cdot dS= \iint curl F \cdot n dS \\=\dfrac{1}{\sqrt 3} \times \iint_{S}(3 i+j -2k) \cdot (i+j+k) dS \\=\dfrac{1}{\sqrt 3} \times \iint_{S}(3+1 -2) dS \\=\dfrac{2}{\sqrt 3} \iint_{S} dS \\=\dfrac{2 A}{\sqrt 3}$$
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