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 ) = 〈 e z 2 , 2 y + sin ( x 2 z ) , 4 z + x 2 + 9 y 2 〉 , S is the region x 2 + y 2 ≤ z ≤ 8 − x 2 − y 2 . 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 ( e z 2 ) + ∂ ∂ y ( 2 y + sin ( x 2 z ) ) + ∂ ∂ z ( 4 z + x 2 + 9 y 2 ) = 0 + 2 + 4 = 6 ∬ S F ⋅ d S = ∭ V ( 6 ) d x d y d z It is given that x 2 + y 2 ≤ z ≤ 8 − x 2 − y 2 Now, change the variables x , y and z in r , z and θ because and we get the polar coordinates: x = r cos θ , y = r sin θ , z = z and d V = r d z d r d θ x 2 + y 2 = r 2 So, the triple integral will become: ∬ S F ⋅ d S = ∭ V ( 6 ) r d z d r d θ So, the region can be written as: x 2 + y 2 ≤ z ≤ 8 − x 2 − y 2 r 2 ≤ z ≤ 8 − r 2 The intersection of the curve is: x 2 + y 2 = 8 − x 2 − y 2 2 x 2 + 2 y 2 = 8 x 2 + y 2 = 4 x 2 + y 2 = 2 2 So, the intersection of the curves is a circle of radius 2 centered at the origin...

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Chapter 18.3, Problem 16E

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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 15E

Chapter 18.3, Problem 17E

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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, with normal pointing outwards.

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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, with normal pointing outwards.

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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.