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 2 − z 2 , e z 2 − cos x , y 3 〉 , S is the boundary of the region bounded by x + 2 y + 4 z = 12 and the coordinate planes in the first octant. 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 2 − z 2 ) + ∂ ∂ y ( e z 2 − cos x ) + ∂ ∂ z ( y 3 ) = 2 x + 0 + 0 = 2 x ∬ S F ⋅ d S = ∭ V ( 2 x ) d x d y d z Now, ∬ S F ⋅ d S = ∭ V ( 2 x ) d x d y d z ...

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Chapter 18.3, Problem 14E

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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 13E

Chapter 18.3, Problem 15E

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Chapter 18 Solutions

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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.