Graphing Coordinate Systems in R3: Spherical Equations and Inequalities

In summary, to graph the surfaces in R3, you need to draw a big diagram and mark in the boundaries, using the inequalities as equalities to determine the boundaries.
  • #1
STLCards002
12
0

Homework Statement


Graph the surface in R3

Homework Equations


Spherical equation [tex]\rho[/tex] = 2asin([tex]\varphi[/tex])

The Attempt at a Solution


I think its just a sphere with a radius of 2
_______________________________________________

Homework Statement


Graph the solid whose given coordinates satisfy the inequalities

Homework Equations


a) 0 [tex]\leq[/tex] r [tex]\leq[/tex] 3, 0 [tex]\leq[/tex] [tex]\theta[/tex] [tex]\leq[/tex] [tex]\pi[/tex]/2, -1 [tex]\leq[/tex] z [tex]\leq[/tex] 2
b) 2r [tex]\leq[/tex] z [tex]\leq[/tex] 5 - 3r

The Attempt at a Solution


I figure these are cylinders because of variables used to describe the coordinates but besides that I'm having trouble know what to use the inequalities for
_______________________________________________

Homework Statement


Graph the solid whose given coordinates satisfy the inequalities

Homework Equations


a) 0 [tex]\leq[/tex] [tex]\rho[/tex] [tex]\leq[/tex] 1, 0 [tex]\leq[/tex] [tex]\theta[/tex] [tex]\leq[/tex] [tex]\pi[/tex]/2,
b) 0 [tex]\leq[/tex] [tex]\rho[/tex] [tex]\leq[/tex] 2/cos[tex]\varphi[/tex], 0 [tex]\leq[/tex] [tex]\varphi[/tex] [tex]\leq[/tex] [tex]\pi[/tex]/4

The Attempt at a Solution


Spheres, but same problem as with the others, I'm just not doing well at understanding how to use the inequalities to graph it
 
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  • #2
Hi STLCards002! :smile:

Sorry, but everything is wrong. :cry:

The only thing I can suggest is that you draw a nice big diagram (take up a whole page), with x y and z axes marked, and just draw in the boundaries.

If it helps, pretend the inequalities are equalities … that will give you the boundaries, and you can work out which parts to shade, later.
 

What is a spherical coordinate system?

A spherical coordinate system is a way of representing points in three-dimensional space using distance (r), inclination (θ), and azimuth (φ) coordinates. It is based on the concept of a sphere, where the distance from the center is the radius of the sphere, the inclination is the angle from the positive z-axis, and the azimuth is the angle from the positive x-axis in the xy-plane.

How does a spherical coordinate system differ from a Cartesian coordinate system?

Unlike a Cartesian coordinate system, where points are represented using x, y, and z coordinates, a spherical coordinate system uses r, θ, and φ coordinates. This means that instead of measuring distances along three perpendicular axes, spherical coordinates measure distances from a fixed point (the center of the sphere) and angles from a fixed reference plane (the xy-plane).

What is the equation for a sphere in spherical coordinates?

The equation for a sphere in spherical coordinates is r = a, where a is the radius of the sphere. This means that all points in space with a distance of a from the origin will lie on the surface of the sphere.

How do you graph spherical inequalities?

To graph spherical inequalities, you first need to rewrite the inequality in terms of r, θ, and φ. Then, you can use the r, θ, and φ values to plot points on a 3D graph and shade in the appropriate region. It is helpful to use a graphing calculator or computer software to visualize the inequality.

Can you use spherical coordinates in real-world applications?

Yes, spherical coordinates are commonly used in physics, engineering, and astronomy to describe the positions of objects in three-dimensional space. They are particularly useful when dealing with objects that have a spherical or cylindrical shape, such as planets, satellites, or antennas.

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