Cylindrical and spherical coordinates

[Sean Fitzpatrick]

I've just finished a first pass through the chapter on multiple integration in the "videos" branch.

Next step is to compare with master (I guess this will soon be renamed "main" as per the change in GitHub policy?) and then send the pull request.

Before proceeding, there was an email thread involving John Bowman suggesting that we change the naming of variables for cylindrical and spherical coordinates.

Should I move forward on this?

There is the small change; namely, use the order (rho, phi, theta) to have a positively-oriented coordinate system.

There is the larger change, of changing things to conform to ISO standards.

I realized one reason why I always use theta as the angle in the xy-plane: the way I typically introduce cylindrical coordinates is via iterated integrals.

Given a triple integral, converting to cylindrical -- as it is now presented -- is the same as writing \(\iiint_E f(x,y,z)\,dV\) as \(\int_a^b (\iint_R f(x,y,z)\,dA)\,dz\) (that is, iterating with the z integral on the outside) and then changing the inner integral with respect to x and y from rectangular to polar.

In any case, I will use whatever notation Greg prefers in this section, and now is a good time to make the change.

[Sean Fitzpatrick]

To summarize:

We have the following options for cylindrical and spherical coordinates:

1. Leave cylindrical as is; use the order (\rho,\phi,\theta) in spherical to have a right-handed coordinate system.

2. Leave cylindrical as is; keep order (\rho,\theta,\phi) but measure \phi from the xy plane, so \phi is in [-pi/2,\pi/2], rather than measuring from the z-axis with \phi in [0,\pi].

3. ISO standards (similar to Physics conventions):

     Cylindrical becomes (\rho, \varphi, z) where \rho and \varphi are the polar coordinates in the xy plane, but renamed

     Spherical becomes (r, \vartheta,\varphi), where r is radial distance, \vartheta is the polar angle (measured from the positive z axis) and \varphi is the azimuthal angle in the xy plane.

[Alex Jordan]

In that other thread, Greg concluded with the following, edited for brevity:

This is where I think I land:

    a. Let's leave polar alone; still use r & theta. 

    b. Let's change cylindrical and spherical to align with ISO. 

        i. That means "(rho, phi, z)" and "(r, theta, phi)", where theta is the polar angle and phi is the azimuth, although I'm with Alex in saying I use those terms backwards. 

        ii. We will change the current marginal note to say we are using the ISO standards that are common with physicists, though math texts may use different notation.

        iii. We'll have to add language to show that in cylindrical, "(rho, phi, 0) ~ (r, theta)." Not super clean, but I can live with that.

In item b.i., it's confusing which meaning of "polar" and "azimuth" is being
used. I think it's like this:

cylindrical: phi is the angle that you would call theta in 2D polar coordinates;
    rho is the length of the projection of the spherical radius onto the xy-plane

spherical: phi is that same angle, now in the third coordinate; and theta is the
    angle measured from the north pole; r is the spherical radius

As noted early in that thread, there are certain consequences that concern me,
but we have a sort of Impossibilty Theorem wrt making it a "perfect" situation.
I think these are mostly resolved if you do not talk about cylindrical coordinates
as a sort of embedding of 2D polar coordinates. Or if you stress that
theta -> phi and r -> rho, and it's for a good reason just trust us.

 

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[Sean Fitzpatrick]

This is what I put as option 3 above. (Option 2 was the one Alex came up with. I like it, although it changes the spherical coordinate Jacobian to use cosine instead of sine, and there are probably follow through implications for surface integrals parametrized using spherical coordinates.)

I think we all thought the angle measured from the z axis was the one called the azimuthal angle and then discovered we were wrong:

https://en.m.wikipedia.org/wiki/Azimuth

[Gregory Hartman]

Thanks, Alex, for digging up my previous post. And you interpreted my comments right. I'm also glad, per Sean's remarks, to find that I wasn't the only one using Azimuth incorrectly.

I agree that no matter what we choose to do, we'll disappoint someone. Sticking to the status quo is desirable, in my opinion, only because it requires the least amount of work. Once I realized that in spherical, the current system was left-handed it seemed wrong to stick with it. Finding from Mathworld that were 7+ published systems for spherical, it seems wise to choose a system that conforms to a standard of some sort. If the AMS had a standard, we could choose that. Aligning with ISO is defensible.

So if you two are good with the proposed system, let's go with it. And yes, work will have to be done on surface integrals. I won't be able to get to it soon, but don't mind giving it a shot in August or something. And sorry about the Jacobian.

[Sean Fitzpatrick]

Hi Greg,

Just to clarify, is your preference at the moment to do nothing?

In my section on general change of variables (this is included in my version of the book but omitted from the official one), I've addressed the left-handedness. See

http://www.cs.uleth.ca/~fitzpat/apex-calculus/sec_transformations.html#ex_jacobian4 and the paragraph immediately after.

[Gregory Hartman]

It’s not that I prefer that nothing be done at the moment, it’s just that *I* can’t do anything at the moment. Just bought a house today that is a dump... will be spending the next few weeks fixing it up. Once it is good to go I’ll be happy to look at that content, though others can tackle it if need be.

[Sean Fitzpatrick]

Sounds like fun. So if I were to tackle it, which convention is your preference?

On Fri., Jun. 19, 2020, 7:50 p.m. Gregory Hartman wrote:

It’s not that I prefer that nothing be done at the moment, it’s just that *I* can’t do anything at the moment. Just bought a house today that is a dump... will be spending the next few weeks fixing it up. Once it is good to go I’ll be happy to look at that content, though others can tackle it if need be.

--

[Gregory Hartman]

Alex had it right when he quoted & interpreted:

    a. Let's leave polar alone; still use r & theta. 

    b. Let's change cylindrical and spherical to align with ISO. 

        i. That means "(rho, phi, z)" and "(r, theta, phi)", where theta is the polar angle and phi is the azimuth, although I'm with Alex in saying I use those terms backwards. 

        ii. We will change the current marginal note to say we are using the ISO standards that are common with physicists, though math texts may use different notation.

        iii. We'll have to add language to show that in cylindrical, "(rho, phi, 0) ~ (r, theta)." Not super clean, but I can live with that.

cylindrical: phi is the angle that you would call theta in 2D polar coordinates;

    rho is the length of the projection of the spherical radius onto the xy-plane

spherical: phi is that same angle, now in the third coordinate; and theta is the
    angle measured from the north pole; r is the spherical radius

Does that make sense? 

And do you guys agree that this is a fine way to go? 

Alex: you said there were consequences that concerned you. What are they? 

[David Farmer]

I will never forgive the idiots who decided that the x-y plane
in polar coordinates does not embed, notationally, in 3-D
cylindrical or spherical coordinates.

When I taught this, a long time ago, I used the coordinates the way
I thought it should be done, and warned them that in other classes
they would have to use different notation, and they should see if
they could get a sensible explanation from their other instructors.

Actually, I seem to recall cylindrical being reasonable and spherical
being the problem, but I could be mistaken.

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[Alex Jordan]

Alex: you said there were consequences that concerned you. What are they?

I think David just touched on them. I will boldface each thing that is a "bad consequence".

Forgive me if I made errors in what follows.

First there are issues with the 2D polar system "embedding" into 3D.
In the obvious way, (x,y) goes to (x,y,0) in 3D.

----

In the proposal, the corresponding (r, theta) becomes (rho,phi,0) for cylindrical
coordinates.

Symbols that represent "the same thing" change:

r -> rho

theta -> phi

At least numbers in coordinates stay "natural":

(9,pi/3) -> (9,pi/3,0)

----

In the proposal, the corresponding (r, theta) becomes (r, pi/2, phi) for cylindrical

coordinates.

Symbols that represent "the same thing" change:

theta -> phi

and now there is a theta that means something completely new

And with numbers, their 2nd coordinate becomes the 3rd.

At the same time, the newly introduced coordinate is pi/2, not 0:

(9, pi/3) -> (9, pi/2, pi/3)

====

Then there are issues with converting between the  coordinate systems:

Cartesian 3D <-> Cylindrical

1. x = rho cos(phi)

2. y = rho sin(phi)

3. z = z

4. rho^2 = x^2 + y^2

5. tan(phi) = y/x

6. z = z

Cartesian 3D <-> Spherical

7. x = r cos(phi) sin(theta)

8. y = r sin(phi) sin(theta)

9. z = r cos(theta)

10. r^2 = x^2 + y^2 + z^2

11. cos(theta) = z / sqrt(x^2 + y^2 + z^2)

12. tan(phi) = y/x

Cylindrical <-> Spherical

13. rho = r sin(theta)

14. phi = phi       (but the cylindrical phi is coord 2, and the spherical phi is coord 3)

15. z = r cos(theta)

16. r^2 = rho^2 + z^2

17. cos(theta) = z / sqrt(rho^2 + z^2)

18. phi = phi        (but the cylindrical phi is coord 2, and the spherical phi is coord 3)

Cartesian 2D <-> Polar

19. x = r cos(theta)

20. y = r sin(theta)

21. r^2 = x^2 + y^2

22. tan(theta) = y/x

It's nice that 3 and 6 are the same. And 5 and 12 are the same.

Except 5 and 12 use phis in different coordinates.

14 and 18 are weird because the phis are in different coordinates.

19--22 use letters that would make sense for Cartesian 3D <-> Spherical, but
are entirely wrong when converting Cartesian 3D <-> Spherical. The student

whose memory returns the Cartesian 2D <-> Polar equations when they needed
Cartesian 3D <-> Spherical is in trouble.

In a sense, accounting for repetition and ignoring the identities z=z and
phi=phi, there are 17 distinct equations there.

Contrast this with the "Stewart" convention, where the 4 equations from
"Cartesian 2D <-> Polar" still work perfectly as a subset of the 6 equations for
"Cartesian 3D <-> Cylindrical". In that convention, there are only 13 distinct equations.

This is a lot of "bad consequence" to weigh. And I'm back on the fence about using ISO.

It would be partly mitigated if 2D polar used phi instead of theta. But that will turn off

math teachers, where theta is in use for pre-college trig courses.

Or as has been mentioned, almost everything is nice if you use the Stewart convention

but change the meaning of phi to be "angle from the equator" instead of "angle from
the north pole". The only not nice thing about that is that it is not the ISO standard, and

it's not even a convention listed in that catalog at MathWorld. So I guess it is not used

anywhere.

I suppose making phi be "angle from the south pole" is another one to consider.

 

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[Sean Fitzpatrick]

(My quick reply before my kids get out of bed and we get to work seeing if we can fit our over-packed camping supplies into our car.)

I do have reservations about ISO standards, from the perspective of students being able to learn them. (Spherical coordinates are already hard for many of them.)

Where all this trouble probably begins: there is a long-standing convention in physics (nearly universal in that discipline) to use \vec{r} for displacement vector, and r for the magnitude of that vector. So it becomes unnatural to use rho for the radial distance when r is already doing that job (and has been all the way back to when you first learn inverse square force laws in high school).

In math we want to be able to embed the xy plane with polar coordinates.

For cylindrical that's easy; just add z.

For spherical you're pretty much forced to drop the polar r in favour of the radial distance. We give this a new name since r is "taken". At least there is a simple relationship between r and rho.

Stewart takes the "student-friendly" approach: just tack phi on the end. But this leaves you with a left-handed coordinate system. (Certainly not the only place where Stewart chose something not quite right because doing it right was harder.)

The two resolutions that have minimal impact on students are:

1. Use (rho, phi, theta). Right handed, everything is defined the same as it is now. There will be a lot of coordinate swapping to do but that's not too hard.

2. Use Alex's idea of changing the way phi is defined. At first this seems like an innocent change, but note that it results in needed to interchange sin(phi) and cos(phi) everywhere. So all the examples and exercises need to be rewritten, since we're not just changing the symbols. Actually we're leaving the symbols, but the integrals will all change: we've shifted everything by pi/2 and replaced sine by cosine.

Option 2 takes more work to implement. But it might be the most student friendly. To wit: I know from grading student work that a lot of them want phi to be measured from the xy plane! Also, the Jacobian changes from rho^2 sin(phi) to rho^2 cos(phi).

Why is this easier for students? Because the integral of sin(phi) from 0 to pi involves a multiple minus signs that have to be carefully tracked! Integrating cos(phi) from -pi/2 to pi/2 is easier (and you can exploit symmetry). :-)

[Alex Jordan]

Actually since (x,y) is (independent, dependent), polar should be (theta, r).

And then spherical can be (phi, theta, r/rho), tacking the new guy on to the

front.

Back in polar, we just have to make clockwise the positive direction to make

(theta, r) a right-handed coordinate system. Then redefine sine and cosine

accordingly. No big deal.

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[Gregory Hartman]

In a previous post, I stated that I honestly don't care about ISO standards, but if we had to pick a convention, that seems like the one to pick. But then David (kind of) called the creators of those standards idiots, and I smiled.

What I am hearing from everyone else is that we really hate that standard! Certainly I don't like it, either. And think what you will of my mathematical background, but after not using spherical in years, I was surprised when I revisited it and found that phi wasn't the "angle of elevation" as I thought I remembered, but rather the angle looking down from the north pole. (And furthermore, phi wasn't what I thought it was. Phi was the angle I thought was theta, and theta was the angle I thought was phi.)

I'd much rather rather use what think is "natural" (which is certainly subjective):

Polar:

(r,theta) as currently done. (When drawn on the board, theta is an "angle of elevation".) Would still call this the "polar angle."

Cylindrical:

(r,theta,z)    Using these letters makes sense, and it's a right handed system. Theta is still the "polar angle".

Note: (r,theta,0) ~ (r,theta)

Spherical:

(rho, theta, phi), where theta is the same as in both polar and cylindrical; rho is the radius (but as it doesn't directly embed into polar/cylindrical, we use a different symbol), and phi is *another* angle of elevation, using the "natural" inclination to measure angles "up from the horizontal."

Note: (5, theta, 0) means the same thing in both cylindrical and spherical, and corresponds to (5,theta) in polar.

Theta is still the "polar angle." And we'd have to get a good term for phi. Not azimuth. "Angle of elevation"? "Elevation angle"?

I've been thinking more and more about my previous statement of not wanting to create new barriers to text adoption; I don't want to introduce a new coordinate system (that doesn't appear on Mathworld (yet!)) and have instructors turned off. But I think this system is so much better. Easier for students. And right-handed. And the Jacobian is somewhat easier? And in the last chapter so adopting instructors won't notice until they've already committed to the text!

I don't have time tonight to list all the conversion equations. But at least we are using fewer symbols. And meanings of symbols don't change. 

I should have stated all that at first, but figured I shouldn't venture away from the norms too much. I think I hear you all wishing we could.

So I throw it back to you all: what do you think?

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[David Farmer]

I talked with a very good friend of mine, who is a physicist.
She is aware that mathematicians and physicists do spherical
coordinates differently, having encountered the difference in
discussions with her intro physics students who are also taking
calculus. She agreed that the math way is logical and she does
not know the explanation for the physics ways of doing it.

She also felt that it is not bad to have "cultural differences"
in how it is done, as long as those differences are clearly
described. Maybe it even is a feature.

One thing Greg wrote, which I wish I had thought of, is having
phi measure up from the equator instead of down from the North
pole. That makes the system right-handed, and also arguably
more natural because it literally (littorally?) is more
natural: that is how we measure latitude on the Earth.

In my opinion,
the change of variables between coordinate systems is something
that should be looked up and not memorized. So, as long
as the diagrams and formulas are there, no confusion should
occur, and it is not a bad thing to have multiple conventions.
You could even have an exercise about the difference, asking
to set up integrals in different spherical coordinate systems.

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[Alex Jordan]

I've been thinking more and more about my previous statement of not wanting to create new barriers to text adoption; I don't want to introduce a new coordinate system (that doesn't appear on Mathworld (yet!)) and have instructors turned off. But I think this system is so much better. Easier for students. And right-handed. And the Jacobian is somewhat easier? And in the last chapter so adopting instructors won't notice until they've already committed to the text!

Consider that when a potential adopter is looking through APEX to
assess it, how likely is it they give scrutiny to the details of the spherical
coordinates section? I would not. If I hadn't taught it recently, I probably
wouldn't even have details in my head to compare it to. It's enough to
see that such a section exists in APEX and to browse the exercises. If
someone finds it unorthodox, it's more likely they will only discover that
once it's too late and they've been using APEX for a month or more.

I like the new proposal. It's only "down sides" are

• being "new" (not on that MathWorld list, so not a "standard")
  
• fractions in the parametrization ([-pi/2,pi/2] instead of [0,pi])
• editing work to swap a bunch of sine's and cosines.

But it feels like the best option pedagogically. Once students understand

it, you can talk about changing coordinates to a different system in the

appropriate section for that.

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[Rob Beezer]

Eventually we will have support for better "Instructors Resources". I'd suggest capturing the important parts of this decision once concluded. There is a "commentary" element right now, which would suffice in the meantime.

Rob

On June 22, 2020 8:29:52 PM PDT, Alex Jordan <jordanc...@gmail.com> wrote:
>>
>> I've been thinking more and more about my previous statement of not
>> wanting to create new barriers to text adoption; I don't want to
>introduce
>> a new coordinate system (that doesn't appear on Mathworld (yet!)) and
>have
>> instructors turned off. But I think this system is so much better.
>Easier
>> for students. And right-handed. And the Jacobian is somewhat easier?
>And in
>> the last chapter so adopting instructors won't notice until they've
>already
>> committed to the text!
>>
>
>Consider that when a potential adopter is looking through APEX to
>assess it, how likely is it they give scrutiny to the details of the
>spherical
>coordinates section? I would not. If I hadn't taught it recently, I
>probably
>wouldn't even have details in my head to compare it to. It's enough to
>see that such a section exists in APEX and to browse the exercises. If
>someone finds it unorthodox, it's more likely they will only discover
>that
>once it's too late and they've been using APEX for a month or more.
>
>I like the new proposal. It's only "down sides" are
>

> - being "new" (not on that MathWorld list, so not a "standard")
> - fractions in the parametrization ([-pi/2,pi/2] instead of [0,pi])
> - editing work to swap a bunch of sine's and cosines.

[Gregory Hartman]

I'm not particularly concerned with any of Alex's "down sides". I certainly *was* concerned with doing something non-standard, but this thread has eased those fears to nonexistence. I say we make the switch. Honestly, I'm actually quite excited about it.

I can tackle the rewriting of that section though I won't be able to get to it for about a month. I'd enjoy the process, though others can beat me to it if need be. Sounds like Sean is on holiday; he can weigh in with his thumbs up/down when he returns.

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[Sean Fitzpatrick]

OK -- I'm on board with this change too! (I think credit goes to Alex for suggesting it initially.)

We were camping last week just outside of cellular range so I'm just catching up on my emails now.

There are some other housekeeping tasks I want to get to before I'd look at this, so Greg has a chance to get to it first.

It should be relatively straightforward to rewrite. (If we had already written full solutions to the exercises there would be a fair bit of work there. But right now there are just answers, and those shouldn't change --- right?)

Students will like this because

\int_{-\pi/2}^{\pi/2}\cos(\phi)\,d\phi = 2\int_0^{\pi/2}\cos(\phi)\,d\phi (symmetry!) = 2(\sin(\pi/2)-\sin(0))=2

is easier than keeping track of minus signs in

\int_0^{\pi} \sin(\phi)\,d\phi = -\cos(\phi)|_0^\pi = -\cos(\pi)-(-\cos(0))= -(-1)-(-1)=2

[Sean Fitzpatrick]

Resurrecting this because: (a) we never got around to it, and (b) there are some exercises in this section that are very wrong:

Look at 19, 20, and 21 in particular; here's the link to 21: https://opentext.uleth.ca/apex-standard/sec_cylindrical_spherical.html#exercise-2828

All of them have the bounds for phi going from 0 to 2pi!!

Each of these exercises can be corrected if we use the order d\rho d\phi d\theta (which is how I've done it ever since I was a student).

But I think we didn't agree on this as the resolution.

That is, the easy way out is to use the order (rho, phi, theta) instead of (rho, theta, phi), since it gives a right-handed coordinate system, and aligns with other math (but not other physics/engineering) books.

But I think we were planning to move to a system where phi is last, but is measured from the equator rather than the north pole.

That's fine, but in the meantime, these exercises are problematic. Swap the bounds around, I guess?

[gregory...@gmail.com]

The only thing worse than *wrong* exercises are exercises that are *very wrong* ! I'm apparently very good at the latter. 

If we are making quick changes, let's not change the order of integration. Rather, change the bounds of integration to match the intent as explained in the answer.

In each, the left-most integral and the inner integral should be swapped. So in 19, it should read 

0 to pi/4, then 0 to 2pi, then 0 to 2.

We can certainly revisit the topic of how to explain spherical coordinates. I liked what we agreed to in the thread above, just haven't had time to fix anything. If/when we do make that switch, these problems will have to change. In the meantime, as all integrals appear in the order of rho - theta - phi, let's not change that order for a few exercises. Just change the bounds to match.

[Sean Fitzpatrick]

I'll make the following changes:

In 17, both angles should go from 0 to pi/2 (right now theta goes from 0 to pi) since the answer says the region is in the first octant.

In 19, swap pi/4 and 2pi.
Also the answer says "bounded above the cone and below the sphere", which I find a bit confusing. Changed to "bounded below *by* the cone, and above *by* the sphere".

In 20, 21, and 22: swapped the outer and middle integral limits.

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[Sean Fitzpatrick]

This change is now implemented. Pull request sent. (Don't merge yet -  I want to add the power series and other updates we still have in the to do list)