Explanation Given: S be the ellipsoid ( x 4 ) 2 + ( y 4 ) 2 + ( z 4 ) 2 = 1 Calculation: Step 1: We parametrize the ellipsoid by a modified form of spherical coordinates. That is, Φ ( θ , ϕ ) = ( 4 cos θ sin ϕ , 3 sin θ sin ϕ , 2 cos ϕ ) with the parameter domain: D = { ( θ , ϕ ) : π ≤ θ ≤ 3 π 2 , π 2 ≤ ϕ < π } Step 2: One can easily verify that Φ ( u , v ) satisfies the equation of the ellipsoid. Compute the tangent and normal vectors. We have, T θ = ∂ Φ ∂ θ = 〈 -4 sin θ sin ϕ , 3 cos θ sin ϕ , 0 〉 T ϕ = ∂ Φ ∂ ϕ = 〈 4 cos θ cos ϕ , 3 sin θ cos ϕ , − 2 sin ϕ 〉 T θ × T ϕ = ( i j k -4 sin θ sin ϕ 3 cos θ sin ϕ 0 4 cos θ cos ϕ 3 sin θ cos ϕ − 2 sin ϕ ) = ( − 6 cos θ sin 2 ϕ ) i − ( 8 sin θ sin 2 ϕ ) j − ( 12 sin 2 θ cos ϕ sin ϕ + 12 cos 2 θ cos ϕ sin ϕ ) k = 〈 − 6 cos θ sin 2 ϕ , 8 sin θ sin 2 ϕ , 12 cos ϕ sin ϕ 〉 In D , π 2 ≤ ϕ < π hence − 12 cos ϕ sin ϕ ≥ 0 .This is the upward pointing normal on this portion of the ellipse since the z -coordinate is positive...

4th Edition

ISBN: 9781319055905

Author: FRANZOSA

Publisher: VST

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Question

Chapter 16.5, Problem 22E

To determine

To calculate: The flux of F = zi over the portion of S where $x, y, z \leq 0$ with upward-pointing normal.

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Chapter 16.5, Problem 21E

Chapter 16.5, Problem 23E

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Chapter 16 Solutions

EBK CALCULUS: EARLY TRANSCENDENTALS

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E Ch. 16.1 - Prob. 5E Ch. 16.1 - Prob. 6E

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Knowledge Booster

The flux of F = z i over the portion of S where x , y , z ≤ 0 with upward-pointing normal.

To calculate: The flux of F = zi over the portion of S where x,y,z≤0 with upward-pointing normal.

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Chapter 16 Solutions

To calculate: The flux of F = zi over the portion of S where $x, y, z \leq 0$ with upward-pointing normal.