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I know how the unit vectors are defined in cylindrical coords. If I have a point P, how do I express it as a combination of the unit vectors uρ, uφ and uz. In the case of Cartesian coordinates this combination is linear. But what about cylindrical coords? Does such a combination exist for them? And, BTW, could you suggest a simple Physics problem where the use of cylindrical coordinates is convenient or, in general, the reason to choose them?

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skymandr has covered a useful application of cylindrical coordinates. There are many problems with this kind of symmetry.

However, I must differ on one thing: the $\hat \phi$ unit vector doesn't enter into a position vector at all. Any general position vector can be written as $\vec r = \rho \hat \rho + z \hat z$. This is because the direction from the origin to any given point goes along only $\hat \rho, \hat z$. However, the vector between two arbitrary points (that is, when one of the points is not the origin) may involve $\hat \phi$.

If you think about that statement even more closely, you might realize the expression for such a vector will be different depending on which point you choose to evaluate the basis vectors (because they change with $\rho, \phi$). This is a feature absent in Cartesian coordinates but most coordinate systems do exhibit this behavior, and it's something you should account for.

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This is a correct statement about the position vector. 'skymandr' is right about an arbitrary vector, which they appear to be concerned with. So there doesn't appear to be a disagreement. –  MarkWayne Oct 19 '12 at 18:14
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Cylindrical coordinates is appropriate in many physical situations, such as that of the electric field around a (very) long conductor along the $z$-axis. Polar coordinates is a special case of this, where the $z$ coordinate is neglected.

As for the use of unit vectors, a point is not uniquely defined in the $\phi$ direction ($\phi +n2\pi$ maps to the same point for any integer $n$), but otherwise any point can be uniquely written as $\vec{v} = A\vec{\hat{z}} + B\vec{\hat{r}} + C\vec{\hat{\phi}}$, just as in Cartesian coordinates.

In terms of the Cartesian unit vectors, the cylindrical can be written:

  • $\vec{\hat{r}}=\cos\phi \vec{\hat{x}} + \sin\phi \vec{\hat{y}}$
  • $\vec{\hat{\phi}} = -\sin\phi \vec{\hat{x}} + \cos\phi \vec{\hat{y}}$
  • $\vec{\hat{z}} = \vec{\hat{z}}$
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