Explanation Given : v = z k - the velocity field (in meters per second) of a fluid in R 3 , the upper hemisphere ( z ≥ 0 ) of the sphere x 2 + y 2 + z 2 = 1 Calculation : Step 1: We use the spherical coordinates: x = cos θ sin ϕ , y = sin θ sin ϕ , z = cos ϕ With the parameter domain: 0 ≤ θ ≤ 2 π , 0 ≤ θ ≤ π 2 Step 2: The normal vector is: n = T ϕ × T θ = sin ϕ 〈 cos θ sin ϕ , sin θ sin ϕ , cos ϕ 〉 We express the function in terms of the parameters: v = 〈 0 , 0 , z 〉 = 〈 0 , 0 , cos ϕ 〉 Hence, v ⋅ n = 〈 0 , 0 , cos ϕ 〉 ⋅ sin ϕ 〈 cos θ sin ϕ , sin θ sin ϕ , cos ϕ 〉 = sin ϕ cos 2 ϕ Step 3: The flow rate of the fluid through the upper hemisphere S is equal to the flux of the velocity vector through S ...

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 17.5, Problem 23E

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To calculate: The flow rate (in cubic meters per second) of a fluid through the upper hemisphere $(z \geq 0)$ of the sphere $x^{2} + y^{2} + z^{2} = 1$

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Chapter 17.5, Problem 22E

Chapter 17.5, Problem 24E

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The flow rate (in cubic meters per second) of a fluid through the upper hemisphere ( z ≥ 0 ) of the sphere x 2 + y 2 + z 2 = 1

Solution Summary: The author calculates the flow rate of a fluid through the upper hemisphere (z 0) of the sphere.

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To calculate: The flow rate (in cubic meters per second) of a fluid through the upper hemisphere $(z \geq 0)$ of the sphere $x^{2} + y^{2} + z^{2} = 1$

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Chapter 17 Solutions

The flow rate (in cubic meters per second) of a fluid through the upper hemisphere ( z ≥ 0 ) of the sphere x 2 + y 2 + z 2 = 1

Solution Summary: The author calculates the flow rate of a fluid through the upper hemisphere (z 0) of the sphere.

Concept explainers

Videos

To calculate: The flow rate (in cubic meters per second) of a fluid through the upper hemisphere (z≥0) of the sphere x2+y2+z2=1

Want to see the full answer?

Chapter 17 Solutions

To calculate: The flow rate (in cubic meters per second) of a fluid through the upper hemisphere $(z \geq 0)$ of the sphere $x^{2} + y^{2} + z^{2} = 1$