Is the second equation a sphere?

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Discussion Overview

The discussion revolves around the characterization of a specific equation in spherical coordinates and whether it represents a sphere. Participants explore the transformation of spherical coordinate equations into Cartesian form and the implications of their structure.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents an equation in spherical coordinates, ρ = 4sinθcosφ, and questions whether it represents a sphere, noting difficulties in converting it to Cartesian form.
  • Another participant challenges the assumption that the second equation would be a sphere, highlighting the asymmetry in the spherical coordinates θ and φ.
  • A participant reflects on their initial misunderstanding, admitting to confusion regarding the nature of the equation and recognizing that the terms sinφ and cosθ are "out of phase," suggesting it likely does not represent a sphere.
  • One participant suggests that the equation may be a quartic, indicating that it could be transformed into a fourth-degree equation upon clearing square roots and fractions.

Areas of Agreement / Disagreement

Participants express uncertainty about whether the second equation represents a sphere, with some suggesting it does not while others explore its characteristics without reaching a consensus.

Contextual Notes

Participants note the complexity of transforming the equation and the potential for misunderstanding due to the nature of spherical coordinates. The discussion includes unresolved mathematical steps and assumptions about the behavior of the equation.

Who May Find This Useful

This discussion may be useful for students or individuals interested in spherical coordinates, transformations between coordinate systems, and the geometric interpretation of equations in different forms.

gnome
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This equation of a sphere in spherical coordinate form:
ρ = 4sinφcosθ converts very readily to (x-2)2 + y2 + z2 = 4 with very little effort.

Now this similar equation looks to me like it should also be a sphere, but I can't seem to get anywhere with it:
ρ = 4sinθcosφ

I just end up with a very ugly
x2 + y2 + z2 = 4yz/(√(x2+y2)
and I have no idea what to do with that.

Is this a dead end? Is the second equation not a sphere?
 
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I can only ask what makes you think the second equation would be a sphere. Spherical coordinates are not "symmetric" in θ and φ.
 
I guess I can only plead insanity on this one.

When it comes to spherical coordinates I'm an absolute greenhorn. The only reason I thought it might be a sphere is that I thought that was the equation that was given on my calc 3 exam last night, and I guess I was "mis-remembering".

But actually, I did wake up this morning thinking that the 2nd equation probably isn't a sphere; I realized that the sinφ and cosθ are "out of phase", i.e. φ is approaching its max when θ is approaching its min, and I was going to post that as a "supplementary" question. So thanks for answering my second question before I even asked it.

And thanks for pointing out so succinctly what characterizes a sphere's equation in spherical coordinates.

So, do you have any idea what "my" equation looks like on a graph?
 
It's a quartic of some kind. Clear the square roots and fractions and I believe you will have a fourth degree equation.
 

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