Explanation Given : The surface is the part of sphere x 2 + y 2 + z 2 = 1 where 1 2 ≤ z ≤ 3 2 , inward pointing normal Calculation : Step 1: We parametrize S by the following parametrization. Φ ( θ , ϕ ) = ( cos θ sin ϕ , sin θ sin ϕ , cos ϕ ) D : 0 ≤ θ ≤ 2 π , ϕ 0 ≤ ϕ ≤ ϕ 1 Step 2: The angles ϕ 0 and ϕ 1 are determined by cos ϕ 0 = 3 2 ⇒ ϕ 0 = π 6 cos ϕ 1 = 1 2 ⇒ ϕ 1 = π 3 Step 3: The normal vector pointing to the inside of the sphere is n = T θ × T ϕ = − sin ϕ ( c o s θ sin ϕ , sin θ sin ϕ , c o s ϕ ) . Also, that for π 6 ≤ ϕ ≤ π 3 , − sin ϕ cos ϕ < 0 therefore the z-component is negative and the normal points to the inside of the sphere. Step 3: Evaluate the dot product F • n ...

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 17.5, Problem 8E

To determine

To calculate: The value of $\int \int_{S} F . d s$ for $F = 〈 x, y, z 〉,$ over the part of sphere $x^{2} + y^{2} + z^{2} = 1$ where $\frac{1}{2} \leq z \leq \frac{\sqrt{3}}{2}$ , inward pointing normal.

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Chapter 17.5, Problem 7E

Chapter 17.5, Problem 9E

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Chapter 17 Solutions

CALCULUS W/SAPLING ACCESS >IC<

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The value of ∫ ∫ S F . d s for F = 〈 x , y , z 〉 , over the part of sphere x 2 + y 2 + z 2 = 1 where 1 2 ≤ z ≤ 3 2 , inward pointing normal.

Solution Summary: The author explains how to calculate the value of S F. d s for x, y, and z over the part of sphere.

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To calculate: The value of $\int \int_{S} F . d s$ for $F = 〈 x, y, z 〉,$ over the part of sphere $x^{2} + y^{2} + z^{2} = 1$ where $\frac{1}{2} \leq z \leq \frac{\sqrt{3}}{2}$ , inward pointing normal.

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Chapter 17 Solutions

The value of ∫ ∫ S F . d s for F = 〈 x , y , z 〉 , over the part of sphere x 2 + y 2 + z 2 = 1 where 1 2 ≤ z ≤ 3 2 , inward pointing normal.

Solution Summary: The author explains how to calculate the value of S F. d s for x, y, and z over the part of sphere.

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To calculate: The value of ∫∫SF.ds for F=〈x,y,z〉, over the part of sphere x2+y2+z2=1 where 12≤z≤32, inward pointing normal.

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Chapter 17 Solutions

To calculate: The value of $\int \int_{S} F . d s$ for $F = 〈 x, y, z 〉,$ over the part of sphere $x^{2} + y^{2} + z^{2} = 1$ where $\frac{1}{2} \leq z \leq \frac{\sqrt{3}}{2}$ , inward pointing normal.