Talk:Del in cylindrical and spherical coordinates

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1 spherical coordinates formula wrong?
2 Great page!
3 \phi or \varphi?
4 mathbf vs. boldsymbol
5 curl in spherical coordinates
6 Directional derivative
7 vectors in curvlinear coordinates

[edit] spherical coordinates formula wrong?

uhh, I'm fairly sure you have theta and phi reversed from their standard usages (i.e., standard usage is that theta is the angle in the xy plane). certainly the main wikipedia article on coordinate systems indicates theta is in the xy plane (though the article also mentions inconsistency between American usage and (rest of the world?) regarding phi being latitude versus colatitude.

I am indeed an American, and I've tended to think that phi == colatitude is a mistake all in all, but has the theta/phi role reversal happened everywhere else?

--Ethelred 18:59, 30 August 2006 (UTC)

These nabla formulas are consistent with taking:

r ≥ 0 is the distance from the origin to a given point P.
0 ≤ θ ≤ 180° is the angle between the positive z-axis and the line formed between the origin and P.
0 ≤ φ ≤ 360° is the angle between the positive x-axis and the line from the origin to the P projected onto the xy-plane.

(I have checked them with the ones in the famous book by J.D.Jackson, Classical Electrodynamics)

This is the "non-american" convention. I think this should be stressed somewhere, since the wikipedia article on spherical coordinates uses the amercan one instead. I think it would be very useful to put some small image in the "Definition of coordinates" row.

Luca Naso 09:07, 19 March 2007 (UTC)

[edit] Great page!

This page is a phenomenal resource; I wish I had chanced upon it when I was struggling with my fluid mechanics class! 202.156.6.54 08:51, 2 January 2006 (UTC)

Thanks, I learned the formulas myself while struggling with my thermo dynamics class. When, much later, I was looking them up on the internet, I could not find them. That's when I decided to write this article. Klaas van Aarsen, 23 februari 2006.

[edit] \phi or \varphi?

I think that $\varphi$ is more of a norm in mathematical notation when writing in spherical coordinates than $φ$ is. Is there some decided policy about this? Has it been discussed?

[edit] mathbf vs. boldsymbol

why do x hat, y hat, z hat get mathbf, but rho hat, phi hat, z hat get boldsymbol?

\mathbf is the norm to indicate a vector. However, it does not work with greek symbols. To resolve this, I used \boldsymbol. Klaas van Aarsen, 20 februari 2006.

[edit] curl in spherical coordinates

Is the formula for the curl in spherical coordinates really correct? I don't have a good reference around at the moment, but for me it looks as if the given expression is the negative of the curl. If somebody could double-check this, this would be good. --Jochen 23:45, 6 November 2005 (UTC)

Checking this against, for example, http://mathworld.wolfram.com/SphericalCoordinates.html, gives an equivalent result. Ian Cairns 08:48, 7 November 2005 (UTC)

I checked all items on this page (vs a very good physics book) except for vector laplacian and non-trivial calculation rules. I found them to be correct except for a minor error in the curl expressed in cylindrical coordinates (which I already fixed). Luca Ermidoro, 19 December 2005

[edit] Directional derivative

What about the directional derivatives, i.e., $(\mathbf A\cdot\nabla)\ \mathbf B$ ? I can't find a decent reference anywhere on the web nor in any book, do they actually exist in curvilinear co-ordinates?

At any point in space you can define a local orthogonal basis of unit vectors related to curvilinear space. The directional vector A and the gradient can both be expressed in this local coordinate system and their inner product is found by simply replacing the unit vectors in the gradient formula by the corresponding components of A. For cylindrical coordinates this is:

$\mathbf A\cdot\nabla = A_\rho {\partial \over \partial \rho} + A_\phi {1 \over \rho}{\partial \over \partial \phi} + A_z {\partial \over \partial z}$

Similarly for spherical coordinates this is:

$\mathbf A\cdot\nabla = A_r {\partial \over \partial r} + A_\theta {1 \over r}{\partial \over \partial \theta}\boldsymbol + A_\phi {1 \over r\sin\theta}{\partial \over \partial \phi}$

Klaas van Aarsen, 23 februari 2006.

[edit] vectors in curvlinear coordinates

Please correct the definition of a vector $\mathbf{A}$ in cylindrical and spherical coordinates. In both cases the angular local basis vectors are orthogonal to $\mathbf{A}$ itself. For sperical coordinates this means that $\mathbf{A} = A_\rho \boldsymbol{\hat \rho}$ . The radial basis vector is parallel to $\mathbf{A}$ and is itself a function of the angular components : $\boldsymbol{\hat \rho} = \boldsymbol{\hat \rho}(A_\phi,A_\theta)$

$\mathbf{A}$ is a vector field and we're interested in its properties at some arbitrary point $\boldsymbol r = (r, \theta, \phi)$ in spherical coordinates. The immediate implication is that $\boldsymbol r = r \boldsymbol {\hat r}$ . There is no relation between $\mathbf{A}$ and $\boldsymbol {\hat r}$ however. Your function is actually: $\boldsymbol {\hat r} = \boldsymbol {\hat r}(\theta, \phi)$ , whereas $\mathbf{A}=(A_r, A_\theta, A_\phi)$ in spherical coordinates.