Explanation Given: We have been given the velocity vector of a fluid is F ( x , y , z ) = 〈 x 2 + y 2 , 0 , z 2 〉 . W is the region between hemisphere S : x 2 + y 2 + z 2 = 1 ; z ≥ 0 and disk D : x 2 + y 2 ≤ 1 in xy plane. Calculation: Let us consider g = x 2 + y 2 − 1 . Thus, we have, ∇ g = 〈 0 , 0 , − 1 〉 We can now set up ⇒ ∬ S F ⋅ d S = ∬ D F ⋅ ∇ To determine (b) To compute: The flow rate across S .

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 18, Problem 33CRE

To determine

(a)

To show:

That flow rate across D is zero.

To determine

(b)

To compute:

The flow rate across S.

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Chapter 18, Problem 32CRE

Chapter 18, Problem 34CRE

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Chapter 18 Solutions

CALCULUS (CLOTH)

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(a) To show: That flow rate across D is zero.

Solution Summary: The author explains that the flow rate across D is zero. The value of flux has been evaluated using the Divergence Theorem.