Talk:Del in cylindrical and spherical coordinates

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The pdf version

Hi, I think this page is a very useful resource. I would like to know what happened to pdf version of it. —Preceding unsigned comment added by (talk) 16:03, 13 January 2008 (UTC)

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)

The reference "Spherical coordinates (r,θ,φ)" on the top of the rightmost column to the article Spherical_coordinates is bad as the definition of θ and φ is inconsistent with this page. One would think that the link would explain the angles but it doesn't - θ and φ are the opposite! Confusing. Would be great with a standard wiki-nomenclature

Betonarbejder 14:40, 17 September 2007 (UTC)

I concur that the angle symbols seem to be correct, but that the top reference should be given as "Spherical coordinates (r,φ,θ)" ACielecki (talk) 04:06, 14 April 2008 (UTC)

The present notation is consistent with the (new) definition at the top of the page, and with the spherical coordinates article. PAR (talk) 05:26, 14 April 2008 (UTC)

I'm not sure this is an American/British thing like, say, the definition of a ring [once was - they may have reached a compromise there] or the many (but minor) differences in ordinary language - my understanding is it's a mathematician/physicist thing; every book and paper from the mathematician's front uses theta azimuthally, but every physics paper/book I've read that uses them has it switched around in context - I remember in my first and second years at university we had the maths and the physics lecturers using opposite conventions and taking pains to point this out, and many people had issues in exams remembering which way round it was meant to be. —Preceding unsigned comment added by (talk) 20:31, 30 July 2009 (UTC)

Correct. It is a mathematician/physicist thing, and a source of great confusion for people taking a course in both, no matter where. Not every conflict of convention comes down to the US vs. the UK - maths (American: 'math') is swarming with differences of convention which ultimately depend on which book the writer first learnt from. — Preceding unsigned comment added by Harsimaja (talkcontribs) 22:07, 23 August 2011 (UTC)

Great Report

This page is great report; Number of turns used in industry; Faraday LAW

\phi or \varphi?

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

It's the same thing, the only difference is the size of the letters. Admiral Norton 13:29, 14 October 2007 (UTC)
That may depend on your font. In TeX's Computer Modern, \phi and \varphi are decidedly different. —Preceding unsigned comment added by (talk) 19:03, 23 December 2008 (UTC)

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.

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,, 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

This apparently opposite (but correct, I think) expression might come from the fact that is the co-latitude (not latitude). This turns everything up-down. If you take a field in the vicinity of the point i.e. a field in the vicinity of the point , then, Curl(A) at this point is:

but precisely, we have , hence everything is OK. Correct me if I'm wrong, I'm not a mathematician --PBenard 11 September 2007 —Preceding unsigned comment added by PBenard (talkcontribs) 14:34, 11 September 2007 (UTC)

Directional derivative

What about the directional derivatives, i.e., ? 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:
Similarly for spherical coordinates this is:
Klaas van Aarsen, 23 februari 2006.

vectors in curvlinear coordinates

Please correct the definition of a vector in cylindrical and spherical coordinates. In both cases the angular local basis vectors are orthogonal to itself. For sperical coordinates this means that . The radial basis vector is parallel to and is itself a function of the angular components :

is a vector field and we're interested in its properties at some arbitrary point in spherical coordinates. The immediate implication is that . There is no relation between and however. Your function is actually: , whereas in spherical coordinates.
Klaas van Aarsen 17:14, 12 May 2006 (UTC)

Vector Laplacian - spherical coordinates

I changed a sin^2 term to a sin term in the vector laplacian, spherical coordinates. The only other statement of the vector laplacian I can find is Weissteins math world, and it agrees with the change. I have a mathematica program that gives faulty answers using the old version, correct answers using the new (and Weisstein) version. If anyone has a reference please check, just to make sure. This is one page that should have no errors. PAR (talk) 22:31, 9 December 2007 (UTC)

Question about notation

I´ve got a Question about the notation what means the wedge over the coordinates at  ? I searched at the linked pages , but i didn´t find a explanation. The German version of the article use also this notation and doesn´t linked it too... Perhaps you can link the definition in the article or explain it. (sorry for my english...) A2r4e1 (talk) 23:06, 9 December 2007 (UTC)

Hi - it means a unit vector in that direction (e.g. is a vector pointing in the direction of increasing r and its length is unity. PAR (talk) 06:27, 10 December 2007 (UTC)
Ah ok. Thanks. A2r4e1 (talk) 20:15, 10 December 2007 (UTC)


I've stumbled upon this page - it's the reason for which Wikipedia was created! I'd nominate it for featured article if I thought they would let it pass. (talk) 06:11, 16 December 2007 (UTC)


Please someone add the basic rules on how to transform the various formulae from one another. Bh3u4m (talk) 12:17, 18 December 2007 (UTC)

Cylindrical coordinates: S as the radial vector?

In this page 's' is used to represent the radial vector, which is a convention I have never seen. Wikipedia's article on Cylindrical_coordinates specifies using , which is apparently in agreement with ISO_31-11. All of the math and electromagnetics books I've used (in the US), use simply 'r'. It seems to me that it should be changed either to '' or 'r'. Mattskee (talk) 01:03, 1 October 2008 (UTC)

Biharmonic operator

It would be great if the biharmonic operator is there as well. Otherwise, it is a great page. JiriVejrazka (talk) 15:38, 26 August 2009 (UTC)

Cylindrical del and div don't match

It looks like the extra and just appear out of nowhere in the term. Why don't they also appear in the plain del operator?

Inconsistent variables

The notation isn't consistent when looking from the top of the table downwards..

What exactly is the inconsistency? PAR (talk) 20:58, 4 February 2011 (UTC)

Automatic generation of this table


Wondering whether this page could be generated automatically by a computer algebra software such as ginac or maxima. This would allow to generate it for different coordinate system conventions. Of course, this is a project somewhat different from wikipedia per se. Anyone interested?

Josce (talk) 14:42, 25 July 2011 (UTC)

This page lists a few maxima scripts that generate some of the differential operators. I add it as an external link.Josce (talk) 10:29, 26 July 2011 (UTC)

Vector Laplacian for Cylindrical

Looks like it's wrong. Used it for homework and got the wrong answer. Wolfram suggests this. — Preceding unsigned comment added by (talk) 18:56, 28 September 2011 (UTC)

It most certainly is wrong

I was deriving all of these formulae myself and just got to doing the "r" component of the vector laplacian. I arrived at the same formula shown in the Wolfram article. (Incidentally, their article also lacks proper rigour: the index for a vector component should be up; not down. Further, if the covariant derivative is defined using the Christoffel Connection, as you need to to derive these formulae, then the covariant derivative of the metric is zero.) — Preceding unsigned comment added by (talk) 18:00, 14 September 2013 (UTC)

I just took a quick look and as far as I can tell the result given by Wolfram is the same as the one given by Wikipedia. Yes they have expressed it a bit differently, but that doesn't make it wrong. The two are completely equivalent. If you disagree, perhaps you can point out which term you think is wrong? TDL (talk) 18:31, 14 September 2013 (UTC)
You are absolutely right! I was careless and didn't notice that they are neatly written in terms of the scalar laplacian. — Preceding unsigned comment added by (talk) 22:48, 25 September 2013 (UTC)

General Orthogonal Coordinates

Hi, I think that this link will prove very useful: Such a wonderful simplicity must not be ignored! Should a new section have to be included?

Thanks for the good work, -Aa. — Preceding unsigned comment added by (talk) 18:49, 21 October 2011 (UTC)

Hi. Just wanted to say thanks for sharing this. Helped me a lot. Dalba (talk) 16:30, 12 July 2015 (UTC)

Curl in cylindrical coordinates

Taken from the article :

I wanted to ask if the given formula was correct. I personally came upon the following :

ê_rho and ê_phi parts were correct, however ê_z part became as following :

(del A_phi / del rho) - (1/rho )* (del A_rho / del phi)

That is the difference I end up with. Basically the only difference I end up with is

(del A_phi / del rho) vs. (from the article) [ del (rho*A_phi) / del rho ] .

I think what has happened is that the rho should not have been multiplied within the differential operator, but from outside, hence the result would have been exactly as mine.

In fact looking at

(1/rho)* [ del (rho*A_phi) / del rho ],

we see that it becomes :

(1/rho)* [ del (rho*A_phi) / del rho ] = (1/rho)* [ A_phi + rho * (del A_phi / del rho) ] = A_phi / rho + (del A_phi / del rho)

Which makes no sense mathematically speaking(with regards to the curl of a del operator with a vector). Can anyone therefore please try to find out of the given formula is incorrect ?

(Sorry about the format I wrote this in) — Preceding unsigned comment added by (talk) 13:48, 30 October 2011 (UTC)

The correct z term is, as written in the article:
PAR (talk) 02:35, 31 October 2011 (UTC)

Change of name?

Since this also includes parabolic coordinates, perhaps the name should be changed to "Del in curvilinear coordinates" or something similar. — Preceding unsigned comment added by Xooll (talkcontribs) 23:30, 6 May 2012 (UTC)

Divergence in parabolic cylindrical coordinates is incorrect

The expression given for parabolic cylindrical coordinates is incorrect. If we think about and as having units of square root distance, then the units are inconsistent. There should be a term of appearing between the and derivatives and the and components of A. The correct formula appears in the Schaum's Outline on Vector Analysis by Murray R. Spiegel.

Please don't make the table too wide

This is a very useful reference page and contains a lot of information, but the table is extremely wide and will span beyond the typical screen width. The height of the table makes it very annoying to scroll horizontally, so I've made an effort to "condense" things a bit without loss of information. Some especially big formulas have been placed in Template:Collapsible sections to shrink the size a bit more (presumably, fewer people care about the bigger formulas as they would not be working such problems out by hand). If you have any additional ideas on how to improve the usability of this page, please go ahead!

As a side note, I think it is preferable, when breaking long equations, to put the sign at the beginning rather than at the end (and also avoid the "matrix" environment). Someone who is only looking for a specific term (vector component) might accidentally miss the minus sign if it were on a separate line. --Freiddie (talk) 01:56, 20 October 2013 (UTC)

I just removed the embedded scrollbars on the table. This causes the table to run out of the screen on narrow (or zoomed in) screens. I think the alternative was worse - when you wanted to scroll horizontally, you had to first scroll down to the table scrollbars, scroll horizontally, then scroll back up.
Anyway, as @Freiddie says, the table is too wide. The condensed formulas seem to help, but here's another idea. I assume that the majority of people landing on this page are interested in the first 3 coordinate systems, and not parabolic cylindrical coordinates. In fact, I'm surprised these made it onto this page (and why not others like paraboloidal coordinates, elliptic cylindrical coordinates, prolate spheroidal coordinates, etc.). We could just scrap this column... Alternatively, perhaps there's a way to collapse the column by default. Monsterman222 (talk) 04:53, 25 March 2014 (UTC)
I agree - just use the first 3, and another article for the less-used coordinate systems. PAR (talk) 06:31, 25 March 2014 (UTC)
@PAR, so I guess we should just delete the column for parabolic cylindrical coordinates. After all, the article is called "Del in cylindrical and spherical coordinates". What would we call the page with the less-used coordinate systems? — Preceding unsigned comment added by Monsterman222 (talkcontribs) 10:01, 31 March 2014 (UTC)
Moved the equations to parabolic cylindrical coordinates. --Fylwind (talk) 19:22, 14 September 2015 (UTC)

How about a row for the Jacobian of each coordinate system (wrt Cartesians)

Just an idea, it's something I came here to look for (this time). (This is an amazing page otherwise, we all know that!) — Preceding unsigned comment added by (talk) 00:27, 6 November 2013 (UTC)

The determinant of the Jacobian is effectively already listed as the differential volume dV. Klaas van Aarsen (talk) 19:26, 31 March 2014 (UTC)

Links for "domain" and "image"

In the Notes section, I made the references to "domain" and "image" into links to their respective pages. (See the diff.) These terms have multiple meanings, however, and although I'm pretty sure I linked to the correct ones, I request one of the main authors to verify that I have.
Christopher.ursich (talk) 05:59, 3 March 2014 (UTC)

Looks fine to me.--Salix alba (talk): 07:16, 3 March 2014 (UTC)

Qualms about the area elements

I edited and then reverted the area element as a sum of three differentials, for example,


My first edit removed the plus signs because this is not how the surface integral is usually set up. I had second thoughts when I realized that nevertheless, , as shown above, is a vector area element. The usual way is to use a cross product and a surface defined in parametric form:


We can now "force" (1) to occur with


In this context, writing (1) is putting the proverbial round peg into a square hole. I don't correct WP pages unless I am 100% sure I am right, but seeing (1) makes me uneasy. Students need know only the most basic surface integrals to learn Gauss' and Maxwell/Ampere's laws of electromagnetism. For that population, at least, it is better to state that:

are all vector area elements. With this one possible blemish, this seems to be an excellent page. --guyvan52 (talk) 20:53, 7 June 2014 (UTC)

The wikiversity link,

Verification of all the equations on this table is a massive problem. Not only must we find sources, but those sources are meaningless unless someone carefully verifies that each equation is correct. Then there is the problem of the surface element described above. Another problem is that each piece of information is inside a table, and putting footnotes inside tables is tedious -- we need to check every single formula, and hence every single box in the grid. So I thought it would be fun to make this a Wikiversity learning project. If you go to v:Coordinate systems/Derivation of formulas, you will see that I pulled everything out of the table and made it sequential (took me hours). Red links indicate unverified sections, and all but this one section is currently redlinked. I put my concern about the surface element there. Please come to Wikiversity and help sort this out. This page gets over 10,000 hits a month. So we must be certain that it is right. I will pull out my NRC and verify some of the more well known identities in a few days. The Wikiversity project asks students to either find references or derive the equations (both is best).--guyvan52 (talk) 05:37, 11 June 2014 (UTC)

lack of notation consistency with other articles

I noticed that the azimuthal angle symbol here is different than the one in the page Spherical coordinate system, It would probably take a bit of work to edit (I am probably not qualified as I struggle with Latex or wiki's version of it) It should be consistent, especially for those just learning the subject and may become confused Pjbeierle (talk) 05:56, 8 September 2014 (UTC)

Done. Changed \phi to \varphi. --Fylwind (talk) 18:35, 14 September 2015 (UTC)

Conversion of z unit vector into spherical coordinates

Perhaps this is an unfamiliar convention for the unit vectors, but it seems like the two terms in the numerator of the definition of z-hat in spherical coordinates should be subtracted not added? If θ goes from the positive z axis down to the negative axis, then wouldn't z-hat have a negative component in the theta-hat direction? Sorry for not putting in the proper symbols, new here. Edit: this is in the "source coordinates" table. And4e (talk) 02:22, 27 November 2015 (UTC)

Tensor divergence in spherical coordinates formula is wrong

I believe the term was wrong in the version of 3rd May 2016 and @Markweitzman: tried to fix it but I think it's still (version 27 May 2016‎) incorrect (but now with errors in and terms).

In my book this formula is given for second-order tensor divergence in spherical coordinates:

The book may also be wrong but if someone could check it, that would be great. (talk) 07:12, 1 June 2016 (UTC) -- D.Dmitriy

I am new to all this wiki stuff so please excuse any guideline violations. I am pretty sure my corrections are correct. I hand calculated them and verified with Maple 2015 (using both the differential geometry package, and the separate physics package). Many sources on the internet seem to have incorrect and differing results here, I suspect because of copying of prior results whether from books or other references. Also note (should be obvious) that the components (the T's) are physical components and not despite the notation covariant components. I would be happy to email my calculations if anyone would like to look at them. Markweitzman (talk) 12:49, 1 June 2016 (UTC)

I am newbie here too so I don't know all the rules either.
I'd like to look at your calculations. Do you mind e-mailing me to Nuvola apps email.svg mail DDimin  ?
I tried calculating by hand divergence of the tensor
( )
in three ways: 1) using directly your formula for divergence in spherical coordinates, 2) using formula in my book 3) converting to cartesian system and calculating divergence there. I evaluated them at point (or equivalently ). (1) gives while (2) and (3) evaluate to . It shows that at least term should have coefficient 2 (but surely I could have messed up my calculations). I'm worried that you might have made a wrong assumption somewhere.
/cc: @Markweitzman:
Dmitriy D. (talk) 11:13, 2 June 2016 (UTC)

I am humbled and stand corrected, your expressions above are correct, and mine were incorrect. I will make the appropriate changes on the wiki page. My confidence in my calculations was based on having done manual and software calculations and having them in agreement. Unfortunately for me there was a redundancy in my calculations. In both cases I used covariant calculations and those were done correctly. But the trivial part of converting to physical components was done manually in both cases, and it seems the more trivial a calculation, the more likely an error will be made (I neglected to differentiate some metric components that arose from converting from covariant components to physical components). This was my first attempt at a wiki correction, and I apologize to the community for my errors and misplaced confidence. Markweitzman (talk) 20:53, 2 June 2016 (UTC)


Truly an excellent page. Are there any suggestions now that it still contains errors? if it is agreed that it doesn't I suggest that the page be fully protected against vandalism like that that occurred on the Spherical Harmonics page where some 4s were changed to 6s, a change not readily identified. Xxanthippe (talk) 04:00, 11 August 2017 (UTC).

I agree with the above and would like to respond

Xxanthippe is correct on two accounts:

  1. This an excellent page that could almost become the "industry standard" for those seeking a quick and convenient, but also reliable access to these equations.
  2. ALL errors need to be removed. To understand my (and Xxanthippe's) concern, look at this recent IP edit:

This might be an appropriate project for the WikiJournal of Science. I am affiliated with that Journal but not very active at the moment. But I am active in a project that would greatly benefit from a wiki-version of a trusted source. This equation sheet for an OpenStax textbook might evolve into a formula students would use during exams. The current draft includes a permalink to a trusted wiki so that students could also use it as a study guide. I discovered the aforementioned problem with IP edit while working on Volume 2 of this project. I consider that IP edit to be both plausible and questionable, and had no idea which permalink to include in the current draft of the formula sheet.

Is there any way we could get a team of Wikipedia "Del" experts to fine-tooth comb these equations? It would be a large and time-consuming project that would require many people and many person-hours, something that would probably require a grant to fund. OpenStax is a large and successful effort. The WikiJournal of Science is a struggling "startup". I have only a little experience in grant seeking, but I am intrigued by what a joint Wikipedia/WikiJournal/OpenStax effort could accomplish. --Guy vandegrift (talk) 15:43, 17 February 2018 (UTC)

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