Explanation Given Data: Write the given integral as follows. ∬ S ( 2 x + 2 y + z 2 ) d S (1) Here, S is the sphere. Write the given equation of the sphere as follows. x 2 + y 2 + z 2 = 1 Formula used: Write the expression to find flux of the vector field F ( x , y , z ) across the surface S . ∬ S F ⋅ d S = ∬ S F ⋅ n d S (2) Write the expression to find ∬ S F ⋅ n d S . ∬ S F ⋅ n d S = ∭ B d i v F d V (3) Here, B is the boundary of the unit ball. Write the expression to find divergence of vector field F ( x , y , z ) = P i + Q j + R k . d i v F = ∂ ∂ x P + ∂ ∂ y Q + ∂ ∂ z R (4) Write the expression to find volume of the sphere with the radius r . V = 4 3 π r 3 (5) From equation (1) and (2), it is clear that the expression ( F ⋅ n ) is equals to the expression ( 2 x + 2 y + z 2 ) . F ⋅ n = 2 x + 2 y + z 2 (6) For the surface S , the normal n is considered as follows. n = x i + y j + z k x 2 + y 2 + z 2 As the surface is a sphere with the equation x 2 + y 2 + z 2 = 1 , substitute 1 for x 2 + y 2 + z 2 . n = x i + y j + z k 1 = x i + y j + z k If the normal n is x i + y j + z k , the field vector F must be 2 i + 2 j + z k in order to satisfy the equation (6)

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ISBN: 9781337772198

Author: Stewart

Publisher: VST

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Chapter 13.9, Problem 24E

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To evaluate: The integral ∬S(2x+2y+z2)dS .

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Chapter 13.9, Problem 23E

Chapter 13.9, Problem 25E

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The integral ∬ S ( 2 x + 2 y + z 2 ) d S .

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To evaluate: The integral ∬S(2x+2y+z2)dS .

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Chapter 13 Solutions