Calculus 8th Edition

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

Chapter 16 - Vector Calculus - 16.5 Curl and Divergence - 16.5 Exercises - Page 1150: 34


$\iint_D (f \nabla^2 g-g \nabla^2 f) dA=\oint_C (f\nabla g-g\nabla f) \cdot n ds$

Work Step by Step

$\iint_Df \nabla^2 g dA=\oint_C f(\nabla g) \cdot n ds-\iint_D \nabla f \cdot \nabla g dA$ ..(Equation 1) The above equation (1) can be re-written as: $\iint_D g \nabla^2 g dA=\oint_C g(\nabla f) \cdot n ds-\iint_D \nabla g \cdot \nabla f dA$ ..(Equation 2) After subtracting equation-1 from the equation 1, we have $\iint_D (f \nabla^2 g-g \nabla^2 f) dA=\oint_C (f\nabla g-g\nabla f) \cdot n ds$ Hence, the result is proved.
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