Explanation Explanation Given : The sphere has radius 3 and is centered at origin. Formula Used: Parameterization of a sphere of radius r centered at origin is given as: r → ( θ , ϕ ) = r sin ϕ cos θ i → + r sin ϕ sin θ j → + r cos ϕ k → N → = r ϕ × r θ Calculations: Parameterization of a sphere of radius 3 centered at origin is given as: r → ( θ , ϕ ) = 3 sin ϕ cos θ i → + 3 sin ϕ sin θ j → + 3 cos ϕ k → Next, we need to determine r ϕ × r θ . Here are the two individual vectors : r θ → ( θ , ϕ ) = − 3 sin ϕ sin θ i → + 3 sin ϕ cos θ j → r ϕ → ( θ , ϕ ) = 3 cos ϕ cos θ i → + 3 cos ϕ sin θ j → − 3 sin ϕ k → Taking cross product we get: r ϕ × r θ = − 16 sin 2 ϕ cos θ i → − 16 sin ϕ cos ϕ sin 2 θ k → − 16 sin 2 ϕ sin θ j → − 16 sin ϕ cos ϕ cos 2 θ k → r ϕ × r θ = − 16 sin 2 ϕ cos θ i → − 16 sin 2 ϕ sin θ j → − 16 sin ϕ cos ϕ ( sin 2 θ + cos 2 θ ) k → r ϕ × r θ = − 16 sin 2 ϕ cos θ i → − 16 sin 2 ϕ sin θ j → − 16 sin ϕ cos ϕ k → N → = − r ϕ × r θ = 16 sin 2 ϕ cos θ i → + 16 sin 2 ϕ sin θ j → + 16 sin ϕ cos ϕ k → (1) The magnitude of the normal vector is given as: ‖ N → ‖ = ‖ r ϕ × r θ ‖ ‖ N → ‖ = 16 ( sin 2 ϕ cos θ ) 2 + ( sin 2 ϕ sin θ ) 2 + ( sin ϕ cos ϕ ) 2 ‖ N → ‖ = 16 sin 4...

4th Edition

ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

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Chapter 17.4, Problem 6PQ

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To determine:

What is the outward-pointing unit normal to the sphere of radius 3 centered at the origin at $P = (2, 1, 1)$ ?

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Chapter 17.4, Problem 5PQ

Chapter 17.4, Problem 1E

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Chapter 17 Solutions

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What is the outward-pointing unit normal to the sphere of radius 3 centered at the origin at P = ( 2 , 1 , 1 ) ?

Solution Summary: The author explains the formula used to determine the outward-pointing unit normal to the sphere of radius 3 centered at the origin.

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To determine:

What is the outward-pointing unit normal to the sphere of radius 3 centered at the origin at $P = (2, 1, 1)$ ?

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Chapter 17 Solutions

What is the outward-pointing unit normal to the sphere of radius 3 centered at the origin at P = ( 2 , 1 , 1 ) ?

Solution Summary: The author explains the formula used to determine the outward-pointing unit normal to the sphere of radius 3 centered at the origin.

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To determine: What is the outward-pointing unit normal to the sphere of radius 3 centered at the origin at P=(2,1,1)?

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Chapter 17 Solutions

To determine:

What is the outward-pointing unit normal to the sphere of radius 3 centered at the origin at $P = (2, 1, 1)$ ?