Parametric Representation for Sphere Between Planes z = 1 & z = -1?

In summary, to determine a parametric representation for the part of the sphere x2 + y2 + z2 = 4 that lies between the planes z = 1 and z = -1, we can use spherical coordinates. The radius is 2 and the parametric equations are x = 2sin(\phi)cos(\theta), y = 2sin(\phi)sin(\theta), and z = 2cos(\phi), where -1 \leq z \leq 1.
  • #1
joemama69
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0

Homework Statement



Determine a parametric representation for the part of the sphere x2 + y2 + z2 = 4 that lies between the planes z = 1 & z = -1.

Homework Equations





The Attempt at a Solution



We never learned spherical coordinates in class so I am not sure if I am using this correctly.

radius = 2,

x = 2sin([tex]\phi[/tex])cos([tex]\theta[/tex])

y = 2cos([tex]\phi[/tex])sin([tex]\theta[/tex])

z = 2cos([tex]\phi[/tex]) where -1 [tex]\leq[/tex] z [tex]\leq[/tex] 1
 
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  • #3
oops that's a typo, should y be


y = 2sin[tex]\phi[/tex]sin[tex]\theta[/tex]
 

1. What is parametric representation?

Parametric representation is a mathematical technique used to describe a set of points or a curve in a multi-dimensional space. It involves expressing the coordinates of each point in terms of one or more parameters, rather than directly in terms of the coordinates themselves.

2. What are the advantages of using parametric representation?

Parametric representation allows for more flexibility and control in describing complex curves or shapes. It also allows for easier manipulation and analysis of the curve, as the parameters can be changed to alter the shape without having to recalculate every point.

3. How is parametric representation used in science?

Parametric representation is commonly used in fields such as physics, engineering, and computer graphics to describe and model complex phenomena. It is also used in data analysis and machine learning to represent and analyze data in a more efficient and flexible way.

4. Can any curve be represented parametrically?

Yes, any curve or shape can be represented using parametric equations. However, some curves may require more complex equations or multiple parameters to accurately describe them.

5. What are some common examples of parametric representation in everyday life?

Parametric representation can be seen in many everyday objects and phenomena, such as the motion of a pendulum, the trajectory of a thrown ball, or the shape of a rollercoaster track. It is also used in computer-generated images and animations to create realistic and detailed graphics.

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