Cylindrical vs. spherical coordinates

In summary, the conversation discusses the difference between cylindrical and spherical coordinate systems and how it affects the differential operators (div and Laplacian). The question arises as to why there is a difference in the operators when both coordinate systems are describing the same plane. The conversation also mentions the use of spherical coordinates in solving physical problems, specifically the vibrating circular membrane, and how it may not be appropriate due to the incorrect application of the divergence and Laplacian operators.
  • #1
Marin
193
0
Hi everyone!

There's a question bothering me about the two coordinate systems - cylindrical and spherical:

Consider the two systems, i.e. [tex](r, \theta, \phi)\rightarrow\left(\begin{array}{c}r\sin\theta\cos\phi\\r\sin\theta\sin\phi\\r\cos\theta\end{array}\right)[/tex] and [tex](r,\phi,z)\rightarrow\left(\begin{array}{c}r\cos\phi\\r\sin\phi\\z\end{array}\right)[/tex]

Now, restricting us to only 2 dimensions and setting z=0 it automatically follows that [tex]\theta=\frac{\pi}{2}[/tex]

The big question is: WHY are the differential operators (div and Laplacian) different within this restriction. I mean, we are describing the plane via the usual polar coordinates. Thus the operators have to produce the same results. Here are some defs:

Let U be a scalar and F be a vector field:

[tex]\nabla U = \frac{\partial f}{\partial r}\vec r+\frac{1}{r}\frac{\partial f}{\partial\phi}\vec\phi[/tex] for cylindrical coordinates and

[tex]\nabla U=\frac{\partial f}{\partial r}\vec r+\frac{1}{r\sin\theta}\frac{\partial f}{\partial\phi}=\frac{\partial f}{\partial r}\vec r+\frac{1}{r}\frac{\partial f}{\partial\phi}\vec\phi[/tex] for spherical coordinates, since z=0 and [tex]\theta=\pi/2[/tex]

It's ok for now, just as it should be. But here comes the interesting part, considering the divergence:

[tex]\nabla\vec F= \frac{1}{r}\frac{ \partial (r F_r)}{\partial r}+\frac{1}{r}\frac{\partial F_{\phi}}{\partial\phi}[/tex] for cylindrical coordinates and

[tex]\nabla\vec F=\frac{1}{r^2}\frac{\partial (r^2F_r)}{\partial r}+\frac{1}{r\sin\theta}\frac{\partial F_{\phi}}{\partial\phi}=\frac{1}{r^2}\frac{\partial (r^2F_r)}{\partial r}+\frac{1}{r}\frac{\partial F_{\phi}}{\partial\phi}[/tex] for spherical coordinates

(all the \theta-terms do not contribute since the fields U and F have no \theta-coordinate)

Now I calculated that:

[tex]\frac{1}{r}\frac{ \partial (r F_r)}{\partial r}\neq\frac{1}{r^2}\frac{\partial (r^2F_r)}{\partial r} [/tex]

but How am I supposed to explain this to me?

I thought that the two coordinate systems are equivalent, when reduced to the 2-dim. polar coordinate form, but apperantly they're not.

What is more, the Laplacians are also different, which leads to different equations for the radial part of, let's say, the 2D wave equation. But the coordinate system has (physically) not been changed - just the angle \phi and the radius r. This would suggest that I could describe a 2-dimensional problem with a point-symmetry via both coordinate system restrictions and obtain different differential equations respectively, which have to describe the same problem in the same coordinate system?!
(it is the circular membrane wave problem (drum) described both ways via the cylindrical and spherical Bessel functions that is meant here in particular)Any help is much appreciated!

Thanks in advance!

marin
 
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  • #2
Quite simply in that what you regard as the radial component of the divergence from the spherical coordinates DO involve theta-differentiations of the radial unit vector.
 
  • #3
Ok, so it follows that spherical coordinates are inappropriate for dealing with 2D point-symmetry physical problems (like the vibrating cicular membrane), as the formulae for div and thus for the Laplacian do not correctly describe the plane using the angle phi and the radius, as they do in cylindrical?

But if this is so, then there must be a mistake in what is stated in this Wikipedia article, section 'Vibrating Membrane':

http://en.wikipedia.org/wiki/Helmholtz_equation

since they are applying spherical coordinates there, and not cylindrical!

What is more, the resulting equation is then NOT the original Bessel equation (as they call it), since the factor 2 in front of the first derivative, coming form the spherical divergence, should not be there!

...or I didn't quite get your idea?
 

1. What is the difference between cylindrical and spherical coordinates?

Cylindrical and spherical coordinates are two different systems used to describe the location of a point in three-dimensional space. The main difference between the two is the shape of the coordinate system. Cylindrical coordinates use a cylindrical shape, while spherical coordinates use a spherical shape.

2. Which coordinate system is better for describing points in 3D space?

Neither coordinate system is inherently better than the other. They both have their own advantages and disadvantages. Cylindrical coordinates are better for describing points in cylindrical or rotational systems, while spherical coordinates are better for describing points in spherical or radial systems. It ultimately depends on the context and application of the coordinates.

3. How do you convert between cylindrical and spherical coordinates?

To convert from cylindrical coordinates (ρ, θ, z) to spherical coordinates (r, θ, φ), you can use the following formulas:r = √(ρ^2 + z^2)φ = arctan(ρ/z)θ = θTo convert from spherical coordinates (r, θ, φ) to cylindrical coordinates (ρ, θ, z), use the following formulas:ρ = r*sin(φ)z = r*cos(φ)θ = θ

4. What are the applications of cylindrical and spherical coordinates?

Cylindrical coordinates are commonly used in engineering and physics to describe rotational systems, such as in cylindrical tanks or turbines. Spherical coordinates are used in astronomy and navigation to describe locations in space or on the surface of a sphere.

5. Can you use both cylindrical and spherical coordinates at the same time?

Yes, it is possible to use both cylindrical and spherical coordinates simultaneously. This is often done in more complex systems where different parts may require different coordinate systems. However, it is important to keep track of which coordinates are being used in each part to avoid confusion.

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