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Math Help - Want to clarify polar, spherical coordinates.

  1. #1
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    Want to clarify polar, spherical coordinates.

    I am always a little confuse in polar, cylindrical and spherical coordinates in vector calculus vs cylindrical and spherical coordinates in vector fields used in Electromagnetics. I want to clarify what my finding and feel free to correct me and add to it.

    A) Vector calculus:

    We use  x = r cos(\phi)sin(\theta),\; y = r sin(\phi)sin(\theta),\; z = rcos(\theta)

    \vec{r} = r cos(\phi)sin(\theta) \hat{x} + r sin(\phi)sin(\theta) \hat{y} + rcos(\theta)\hat{z}

    \hbox { Surface intergal }\; \int_{\Gamma} f(a,\phi, \theta) dA \hbox { where }\; dA = a^2sin(\theta)d\theta d\phi

    All these are just simply rectangular coordinate presented in polar and spherical coordinate value.

    B) Vector field in Spherical and cylindrical coordinates:

    This is "true" Spherical or cylindrical coordinates in  (r,\phi,\theta) represents a vector ( vector field ) at a single point P. P can be  P(x,y,z) \hbox { or }\; P(r,\phi,\theta) respect to the origin.

    So in conclusion, I think it is very very different between the two, where in the first case A), it is very much like the \; r,\phi,\theta \; representation of (x,y,z). The second case B) really about vector fields where you set up a coordinate system at a point P and use the coordinate system to represent the direction and magnitude of the vector field at that point. Therefore the two are not the same.
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  2. #2
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    Probably the most confusing aspect of these coordinate systems is the difference between math and physics with respect to the polar angle and the azimuthal angle. The polar angle is the angle measured down from the z axis. The azimuthal angle is measured around the z axis.

    Physics: \langle\text{radius},\text{polar angle},\text{azimuthal angle}\rangle=\langle r,\theta,\varphi\rangle.

    Some Mathematicians: \langle\text{radius},\text{polar angle},\text{azimuthal angle}\rangle=\langle r,\varphi,\theta\rangle.

    You can tell which system an author is using by which angle shows up in the z-coordinate. The z-coordinate, in spherical coordinates, always contains the cosine of the polar angle, and is independent of the azimuthal angle.

    Does this clear some things up?
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  3. #3
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    Quote Originally Posted by Ackbeet View Post
    Probably the most confusing aspect of these coordinate systems is the difference between math and physics with respect to the polar angle and the azimuthal angle. The polar angle is the angle measured down from the z axis. The azimuthal angle is measured around the z axis.

    Physics: \langle\text{radius},\text{polar angle},\text{azimuthal angle}\rangle=\langle r,\theta,\varphi\rangle.

    Some Mathematicians: \langle\text{radius},\text{polar angle},\text{azimuthal angle}\rangle=\langle r,\varphi,\theta\rangle.

    You can tell which system an author is using by which angle shows up in the z-coordinate. The z-coordinate, in spherical coordinates, always contains the cosine of the polar angle, and is independent of the azimuthal angle.

    Does this clear some things up?
    Thanks for you reply.

    So you agree with my presentation in my post that the two are different, that A is more the expansion of the xyz coordinates for non rectangular shape. AND B is mainly for representing a vector field at a given point P. This is very important for my understanding.

    Sincerely

    Alan
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  4. #4
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    I would say that in A, you're describing spherical coordinates. Your vector \vec{r} is actually a vector field: the vector at any point in space is pointed directly away from the origin. That is, \vec{r}=\vec{r}(r,\theta,\varphi). It's a function from a subset of \mathbb{R}^{n} to \mathbb{R}^{n}. Your equations for x, y, and z are the equations for spherical coordinates. They are scalars. But once you put in unit vectors, you've got yourself a vector, and in this case a vector field.
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  5. #5
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    Thanks for you reply, I'll study more and I'll come back tomorrow.

    Sincerely

    Alan
    Last edited by yungman; August 25th 2010 at 10:26 PM.
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  6. #6
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    Ok. Let me know if you have more questions.

    Cheers.
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  7. #7
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    I was totally wrong. Dis-regard my post all together. The three coordinate system is just three different ways to represent a vector or a point. They have nothing to do with the vector field or position vector.

    Thanks

    Alan
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