(a) To Draw: The parametrized form of the given curve.

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Answer 1

Question

Chapter 17.5, Problem 37E

To determine

(a)

To Draw:

The parametrized form of the given curve.

To determine

(b)

To Prove:

That the intersection of the surface S with the xy-plane is the unit circle given by $G (u, 0) = (\cos u, \sin u, 0)$

And verify that the normal vector along this circle is given by $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ parametrized form of the given curve.

To determine

(c)

To Show:

As u varies from 0 to 2p, the point $G (u, 0) = (\cos u, \sin u, 0)$ moves once around the unit circle, beginning and ending at $G (0, 0) = G (2 π, 0) = (1, 0, 0)$ .

That $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ is a unit vector that varies continuously but that $N (2 π, 0) = - N (0, 0)$ which shows that S is not orientable.

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Students have asked these similar questions

Vector F is mathematically defined as F = M x N, where M = p 2p² cos + 2p2 sind while N is a vector normal to the surface S. Determine F as well as the area of the plane perpendicular to F if surface S = 2xy + 3z.

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Answer 2

Question

Chapter 17.5, Problem 37E

To determine

(a)

To Draw:

The parametrized form of the given curve.

To determine

(b)

To Prove:

That the intersection of the surface S with the xy-plane is the unit circle given by $G (u, 0) = (\cos u, \sin u, 0)$

And verify that the normal vector along this circle is given by $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ parametrized form of the given curve.

To determine

(c)

To Show:

As u varies from 0 to 2p, the point $G (u, 0) = (\cos u, \sin u, 0)$ moves once around the unit circle, beginning and ending at $G (0, 0) = G (2 π, 0) = (1, 0, 0)$ .

That $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ is a unit vector that varies continuously but that $N (2 π, 0) = - N (0, 0)$ which shows that S is not orientable.

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Answer 3

Question

Chapter 17.5, Problem 37E

To determine

(a)

To Draw:

The parametrized form of the given curve.

To determine

(b)

To Prove:

That the intersection of the surface S with the xy-plane is the unit circle given by $G (u, 0) = (\cos u, \sin u, 0)$

And verify that the normal vector along this circle is given by $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ parametrized form of the given curve.

To determine

(c)

To Show:

As u varies from 0 to 2p, the point $G (u, 0) = (\cos u, \sin u, 0)$ moves once around the unit circle, beginning and ending at $G (0, 0) = G (2 π, 0) = (1, 0, 0)$ .

That $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ is a unit vector that varies continuously but that $N (2 π, 0) = - N (0, 0)$ which shows that S is not orientable.

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Answer 4

Question

Chapter 17.5, Problem 37E

To determine

(a)

To Draw:

The parametrized form of the given curve.

To determine

(b)

To Prove:

That the intersection of the surface S with the xy-plane is the unit circle given by $G (u, 0) = (\cos u, \sin u, 0)$

And verify that the normal vector along this circle is given by $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ parametrized form of the given curve.

To determine

(c)

To Show:

As u varies from 0 to 2p, the point $G (u, 0) = (\cos u, \sin u, 0)$ moves once around the unit circle, beginning and ending at $G (0, 0) = G (2 π, 0) = (1, 0, 0)$ .

That $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ is a unit vector that varies continuously but that $N (2 π, 0) = - N (0, 0)$ which shows that S is not orientable.

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Answer 5

Chapter 17.5, Problem 37E

To determine

(a)

To Draw:

The parametrized form of the given curve.

To determine

(b)

To Prove:

That the intersection of the surface S with the xy-plane is the unit circle given by $G (u, 0) = (\cos u, \sin u, 0)$

And verify that the normal vector along this circle is given by $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ parametrized form of the given curve.

To determine

(c)

To Show:

As u varies from 0 to 2p, the point $G (u, 0) = (\cos u, \sin u, 0)$ moves once around the unit circle, beginning and ending at $G (0, 0) = G (2 π, 0) = (1, 0, 0)$ .

That $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ is a unit vector that varies continuously but that $N (2 π, 0) = - N (0, 0)$ which shows that S is not orientable.

Answer 6

Chapter 17.5, Problem 37E

To determine

(a)

To Draw:

The parametrized form of the given curve.

Answer 7

Chapter 17.5, Problem 37E

To determine

(a)

To Draw:

The parametrized form of the given curve.

Answer 8

To determine

(b)

To Prove:

That the intersection of the surface S with the xy-plane is the unit circle given by $G (u, 0) = (\cos u, \sin u, 0)$

And verify that the normal vector along this circle is given by $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ parametrized form of the given curve.

Answer 9

To determine

(b)

To Prove:

That the intersection of the surface S with the xy-plane is the unit circle given by $G (u, 0) = (\cos u, \sin u, 0)$

And verify that the normal vector along this circle is given by $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ parametrized form of the given curve.

Answer 10

To determine

(c)

To Show:

As u varies from 0 to 2p, the point $G (u, 0) = (\cos u, \sin u, 0)$ moves once around the unit circle, beginning and ending at $G (0, 0) = G (2 π, 0) = (1, 0, 0)$ .

That $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ is a unit vector that varies continuously but that $N (2 π, 0) = - N (0, 0)$ which shows that S is not orientable.

Answer 11

To determine

(c)

To Show:

As u varies from 0 to 2p, the point $G (u, 0) = (\cos u, \sin u, 0)$ moves once around the unit circle, beginning and ending at $G (0, 0) = G (2 π, 0) = (1, 0, 0)$ .

That $N (u, 0) = (\cos u \sin \frac{u}{2}, \sin u \sin \frac{u}{2}, - \cos \frac{u}{2})$ is a unit vector that varies continuously but that $N (2 π, 0) = - N (0, 0)$ which shows that S is not orientable.

Answer 12

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Answer 13

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Answer 14

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Answer 15

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

(a) To Draw: The parametrized form of the given curve.

Solution Summary: The author illustrates the parametrized form of the given curve by drawing the surface with parametrization G and verifying the normal vector along this circle.

Videos

(a)

To Draw:

The parametrized form of the given curve.

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Chapter 17 Solutions

(a) To Draw: The parametrized form of the given curve.

Solution Summary: The author illustrates the parametrized form of the given curve by drawing the surface with parametrization G and verifying the normal vector along this circle.

Videos

(a) To Draw: The parametrized form of the given curve.

(b) To Prove: That the intersection of the surface S with the xy-plane is the unit circle given by G(u,0)=(cosu,sinu,0) And verify that the normal vector along this circle is given by N(u,0)=(cosusinu2,sinusinu2,−cosu2) parametrized form of the given curve.

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Chapter 17 Solutions

(a)

To Draw:

The parametrized form of the given curve.