Explanation Given: 1) Vector field F = 〈 − z 2 , 2 z x , 4 y − x 2 〉 . 2) C is a simple closed curve in the plane x + y + z = 4 that encloses region of area 16 as shown in given figure. 3) C is oriented in counter clockwise direction when viewed from above the plane. 4) Given figure is as follows Formulae used: 1) Stoke’s theorem: If F is any continuously differentiable vector function and S is a surface enclosed by a curve C , then ∫ C F . d r = ∬ S ( ∇ × F ) . n d s Where n is the unit normal vector at any point of S 2) n = ∇ ϕ | ∇ ϕ | Where ϕ represents the palne ϕ having equation a x + b y + c x − d = 0 And | ∇ ϕ | = a 2 + b 2 + c 2 Calculation: ∫ C 1 F . d r = ∬ S ( ∇ × F ) . n d s ... ( i ) [Stokes theorem] F = 〈 − z 2 , 2 z x , 4 y − x 2 〉 [Given] Calculating ∇ × F ∇ × F = | i ^ j ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z − z 2 2 z x 4 y − x 2 | = i [ ∂ ∂ y ( 4 y − x 2 ) − ∂ ∂ z ( 2 z x ) ] − j [ ∂ ∂ x ( 4 y − x 2 ) − ∂ ∂ z ( − z 2 ) ] + k [ ∂ ∂ x ( 2 z x ) − ∂ ∂ y ( − z 2 ) ] = i [ 4 − 2 x >&#...

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 18.2, Problem 26E

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Calculation of $\oint_{C} F . d r$ for a given vector $F$ and closed curve $C$ .

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Chapter 18.2, Problem 25E

Chapter 18.2, Problem 27E

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Chapter 18 Solutions

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Calculation of ∮ C F . d r for a given vector F and closed curve C .

Solution Summary: The author explains the Stoke's theorem: if F is any continuously differentiable vector function and S is a surface enclosed by curve C, then C F.

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Calculation of $\oint_{C} F . d r$ for a given vector $F$ and closed curve $C$ .

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Chapter 18 Solutions

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Solution Summary: The author explains the Stoke's theorem: if F is any continuously differentiable vector function and S is a surface enclosed by curve C, then C F.

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