Explanation Given: The vector field F ( x , y , z ) = 〈 x y , y z , x 2 z + z 2 〉 . The surface S is the boundary of a unit sphere. Formula used: Divergence Theorem, ∬ S F ⋅ d S = ∭ E d i v F d V Calculation: Let us first find the divergence of the given vector field. ⇒ F ( x , y , z ) = 〈 x y , y z , x 2 z + z 2 〉 ⇒ d i v F = 〈 ∂ ∂ x , ∂ ∂ y , ∂ ∂ z 〉 ⋅ 〈 x y , y z , x 2 z + z 2 〉 ⇒ d i v F = ∂ ∂ x ( x y ) + ∂ ∂ y ( y z ) + ∂ ∂ z ( x 2 z + z 2 ) ⇒ d i v F = y + z + x 2 + 2 z = x 2 + y + 3 z Using divergence theorem, we can convert the surface integral to volume integral as shown below: ⇒ ∬ S F ⋅ d S = ∭ E d i v F d V ⇒ ∬ S F ⋅ d S =

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Chapter 18, Problem 24CRE

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The surface integral $\iint_{S} F \cdot d S$ using Divergence Theorem.

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Chapter 18, Problem 23CRE

Chapter 18, Problem 25CRE

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The surface integral ∬ S F ⋅ d S using Divergence Theorem.

Solution Summary: The author explains how the surface integral is evaluated using the divergence theorem.

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The surface integral $\iint_{S} F \cdot d S$ using Divergence Theorem.

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Solution Summary: The author explains how the surface integral is evaluated using the divergence theorem.

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The surface integral $\iint_{S} F \cdot d S$ using Divergence Theorem.