Explanation Given: The vector field F ( x , y , z ) = 〈 0 , y , − z 〉 . The surface S is the side of the surface x 2 + y 2 = 4 ; 0 ≤ z ≤ 2 oriented by out pointing normal. Formula used: Stokes’ Theorem, ∫ C F . d r = ∬ S C u r l ( F ) ⋅ d S Calculation: Using the Stokes’ Theorem, we can write the surface integral as a linear integral first and then we can evaluate the linear integral easily: ∬ S F ⋅ d S = ∬ S C u r l ( A ) ⋅ d S = ∫ C A . d r Let us now find the vector A such that F = C u r l ( A ) = 〈 0 , y , − z 〉 Let us consider, − ∂ A 2 ∂ z = 0 ⇒ A 2 = − B 2 ( x , y ) ∂ A 1 ∂ z = y ⇒ A 1 = y z + B 1 ( x , y ) Where, we have, A 2 x − A 1 y = − z ⇒ − B 2 x ( x , y ) − z − B 1 x ( x , y ) = − z B 2 x ( x , y ) = − B 1 x ( x , y ) Thus, we can write, A = 〈 y z , 1 , 0 〉 Thus, we can convert the surface integral to line integral as: ∬ S F ⋅ d S = ∬ S C u r l ( A ) ⋅ d S = ∫ C A . d r The integral ∫ C A . d r represents the line integral equivalent to the flux of the given vector field F ( x , y , z ) = 〈 0 , y , − z 〉 , around the boundary x 2 + y 2 = 4 ...

4th Edition

ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

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Chapter 18, Problem 21CRE

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The given surface integral using Stokes’ Theorem.

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The given surface integral using Stokes’ Theorem.

Solution Summary: The author explains how Stokes' Theorem can be used to evaluate a given surface integral.

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To evaluate: The given surface integral using Stokes’ Theorem.

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Chapter 18 Solutions

To evaluate:

The given surface integral using Stokes’ Theorem.