Explanation Given: 1) Vector field F = 〈 − z 2 , 2 z x , 4 y − x 2 〉 . 2) Need to find equation of plane in following form a x + b y + c z = 0 where a , b , c are not all zero. 3) ∮ C F . d r = 0 for every closed surface C lying in the plane. Formula 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 plane ϕ having equation a x + b y + c x − d = 0 And | ∇ ϕ | = a 2 + b 2 + c 2 Calculation: F = 〈 y 2 , 2 z + x , 2 y 2 〉 [Given] Calculating ∇ × F and n which will be used in further calculation ∇ × F = | i ^ j ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z y 2 2 z + x 2 y 2 | = i [ ∂ ∂ y ( 2 y 2 ) − ∂ ∂ z ( 2 z + x ) ] − j [ ∂ ∂ x ( 2 y 2 ) − ∂ ∂ z ( y 2 ) ] + k [ ∂ ∂ x ( 2 z + x ) − ∂ ∂ y ( y 2 ) ] = i [ 4 y − 2 ] − j [ 0 + 0 ] + k [ 1 − 2 y ] = [ 4 y − 2 ] i −

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Chapter 18.2, Problem 25E

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The plane for a given vector field using stokes’ theorem

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Chapter 18.2, Problem 24E

Chapter 18.2, Problem 26E

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Use Stokes' Theorem to evaluate ∫ C F · dr where F = (x + 9z)i + (7x + y)j + (6y - z)k and C is the curve of intersection of the plane x + 4y + z = 8 with the coordinate planes. (Assume that C is oriented counterclockwise as viewed from above.)

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The plane for a given vector field using stokes’ theorem

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The plane for a given vector field using stokes’ theorem

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