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: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 17.5, 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 17.5, Problem 21E

Chapter 17.5, Problem 23E

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Chapter 17 Solutions

CALCULUS (CLOTH)

Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

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Prob. 13E Ch. 17.3 - Prob. 14E Ch. 17.3 - Prob. 15E Ch. 17.3 - Prob. 16E Ch. 17.3 - Prob. 17E Ch. 17.3 - Prob. 18E Ch. 17.3 - Prob. 19E Ch. 17.3 - Prob. 20E Ch. 17.3 - Prob. 21E Ch. 17.3 - Prob. 22E Ch. 17.3 - Prob. 23E Ch. 17.3 - Prob. 24E Ch. 17.3 - Prob. 25E Ch. 17.3 - Prob. 26E Ch. 17.3 - Prob. 27E Ch. 17.3 - Prob. 28E Ch. 17.3 - Prob. 29E Ch. 17.3 - Prob. 30E Ch. 17.3 - Prob. 31E Ch. 17.3 - Prob. 32E Ch. 17.3 - Prob. 33E Ch. 17.3 - Prob. 34E Ch. 17.3 - Prob. 35E Ch. 17.4 - Prob. 1PQ Ch. 17.4 - Prob. 2PQ Ch. 17.4 - Prob. 3PQ Ch. 17.4 - Prob. 4PQ Ch. 17.4 - Prob. 5PQ Ch. 17.4 - Prob. 6PQ Ch. 17.4 - Prob. 1E Ch. 17.4 - Prob. 2E Ch. 17.4 - Prob. 3E Ch. 17.4 - Prob. 4E Ch. 17.4 - Prob. 5E Ch. 17.4 - Prob. 6E Ch. 17.4 - Prob. 7E Ch. 17.4 - Prob. 8E Ch. 17.4 - Prob. 9E Ch. 17.4 - Prob. 10E Ch. 17.4 - Prob. 11E Ch. 17.4 - Prob. 12E Ch. 17.4 - Prob. 13E Ch. 17.4 - Prob. 14E Ch. 17.4 - Prob. 15E Ch. 17.4 - Prob. 16E Ch. 17.4 - Prob. 17E Ch. 17.4 - Prob. 18E Ch. 17.4 - Prob. 19E Ch. 17.4 - Prob. 20E Ch. 17.4 - 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The flux of F = z i over the portion of S where x , y , z ≤ 0 with upward-pointing normal.

Solution Summary: The author explains how to calculate the flux of F = z i over the portion of S where x, y - 0 with upward-pointing normal is -4.

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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 17 Solutions

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

Solution Summary: The author explains how to calculate the flux of F = z i over the portion of S where x, y - 0 with upward-pointing normal is -4.

Videos

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

Want to see the full answer?

Chapter 17 Solutions

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