Explanation Divergence Theorem says about the relation between surface and volume integral. Let S be a closed smooth surface, which is boundary of V, with normal pointing outwards. Let F = 〈 x , y , z 〉 on a region containing V, then the flux is equal to triple integral of div F over the solid region V. ∬ S F ⋅ d S = ∭ V d i v ( F ) d V Given: F ( x , y , z ) = 〈 x + y , z , z − x 〉 , S is the boundary of the region between the parabolic z = 9 − x 2 − y 2 and the xy-plane. Formula Used: ∬ S F ⋅ d S = ∭ V d i v ( F ) d V Calculation: First, we have to calculate the div F and we get: d i v ( F ) = ∂ ∂ x ( x + y ) + ∂ ∂ y ( z ) + ∂ ∂ z ( z − x ) = 1 + 0 + 1 = 2 ∬ S F ⋅ d S = ∭ V ( 2 ) d x d y d z Theregion between the parabolic z = 9 − x 2 − y 2 and the xy-plane is described by: 0 ≤ z ≤ 9 − x 2 − y 2 , − 9 − x 2 ≤ y ≤ 9 − x 2 , − 3 ≤ x ≤ 3 ∬ S F ⋅ d S = ∭ V ( 2 x )

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ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 18.3, Problem 15E

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The flux $\iint_{S} F \cdot d S$ by using the Divergence Theorem.

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Chapter 18.3, Problem 14E

Chapter 18.3, Problem 16E

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The flux ∬ S F ⋅ d S by using the Divergence Theorem.

Solution Summary: The author calculates the flux by using the Divergence Theorem. Let S be a closed smooth surface, which is boundary of V.

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To calculate:

The flux $\iint_{S} F \cdot d S$ by using the Divergence Theorem.

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Chapter 18 Solutions

The flux ∬ S F ⋅ d S by using the Divergence Theorem.

Solution Summary: The author calculates the flux by using the Divergence Theorem. Let S be a closed smooth surface, which is boundary of V.

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To calculate: The flux ∬SF⋅dS by using the Divergence Theorem.

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Chapter 18 Solutions

To calculate:

The flux $\iint_{S} F \cdot d S$ by using the Divergence Theorem.