Explanation Given information: The vector field of fluid flow is. ∮ C V . d R The value of vector V V = y 2 i + tan − 1 z j + ( x 2 + 1 ) k Here, C is the intersection of the cylinder x 2 + y 2 = 2 x and the plane z = y . Calculation: The stokes’ theorem states that: Let S is an oriented surface with unit normal vector field N , and let S be bounded by a smooth curve C compatible with S . If F is a vector field, ∮ C F → . d R → = ∬ S ( c u r l F → . N ^ ) d S Calculate the circulation of ∮ c V → . d R → around the boundary C . Calculate the curl vector field. c u r l V → = | i ^ j ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z y 2 tan − 1 x x 2 + 1 | = 〈 0 , − 2 x , 1 x 2 + 1 − 2 y 〉 Calculate the normal vector. N ^ = 1 2 ( j ^ − k ^ ) = 1 2 〈 0 , 1 , − 1 〉 Multiply c u r l V → . N ^ c u r l V → . N ^ = 1 2 〈 0 , − 2 x , 1 x 2 + 1 − 2 y 〉 . 〈 0 , 1 , − 1 〉 = 1 2 ( − 2 x + 2 y − 1 x 2 + 1 ) The circulation is. ∮ c V → . d R → = ∬ s ( c u r l V → . N ^ ) d S ∮ c V → . d R → = 1 2 ∬ s ( − 2 x + 2 y − 1 x 2 + 1 ) d S Let S is the slanted elliptical surface. This projects a disk D on x y plane ( x − 1 ) 2 + y 2 ≤ 1 . Using equation z = y

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ISBN: 9781465208880

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

Publisher: Kendall Hunt Publishing

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Chapter 13.6, Problem 46PS

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To find: The value of circulation of boundary C .

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Chapter 13.6, Problem 45PS

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The value of circulation of boundary C .

Solution Summary: The author explains the stokes' theorem: Let S be an oriented surface with unit normal vector field N.

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To find: The value of circulation of boundary C .

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Chapter 13 Solutions