Explanation Let the vector field be F → = x i ^ + y j ^ and S is the hemisphere z = 1 − x 2 − y 2 . We will use the divergence theorem, so div F ^ = ∂ ∂ x ( x ) + ∂ ∂ y ( y ) + ∂ ∂ z ( 0 ) = 2 Thus, ∬ S F → ⋅ N → d S = ∭ R div F ^ d V = ∫ 0 1 ∫ 0 1 − x 2 ∫ 0 1 − x 2 − y 2 ( 2 ) d z d y d x = ∫ 0 1 ∫ 0 1 − x 2 2 [ z ] 0 1 − x 2 − y 2 d y d x =

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 13.5, Problem 35PS

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To calculate: The value of the integral ∬SF→⋅N→dS for the vector field F→=xi^+yj^ and S is the hemisphere z=1−x2−y2 .

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Chapter 13.5, Problem 34PS

Chapter 13.5, Problem 36PS

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Chapter 13 Solutions

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The value of the integral ∬ S F → ⋅ N → d S for the vector field F → = x i ^ + y j ^ and S is the hemisphere z = 1 − x 2 − y 2 .

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To calculate: The value of the integral ∬SF→⋅N→dS for the vector field F→=xi^+yj^ and S is the hemisphere z=1−x2−y2 .

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Chapter 13 Solutions