## Monday, April 23, 2007

### Nabla

Did you know why nabla is called nabla? Well, I didn't know until today. It is named after an instrument somewhat similar to a harp, called by the Greeks the 'psaltery' and by the Hebrews the 'nabla'. Unlike the harp however, the shape of this early string instrument is very geometric: it is pretty much triangular:

The nabla symbol is used in maths (and physics of course) to denote a differential operator. It was introduced by Hamilton around 1837. Its name apparently goes back to a joke by Maxwell. According to Wikipedia, W. Thomson wrote in 1884:

"I took the liberty of asking Professor Bell whether he had a name for this symbol and he has mentioned to me nabla, a humorous suggestion of Maxwell's. It is the name of an Egyptian harp, which was of that shape"

I am kind of glad he didn't suggest to use the Greek name 'psaltery' as I admittedly have no idea how to pronounce it. You might be interested to hear though that it makes an appearance in the bible, Psalm 33:2

"Rejoice in the LORD, O ye righteous: for praise is comely for the upright.
Praise the LORD with harp: sing unto him with the psaltery and an instrument of ten strings. "

The word 'operator' is a very sophisticated expression for a thing that assigns things to things. The telephone operator for example, assigns incoming calls to the desired connection. Its correct mathematical notation is

[source]

An operator can be almost everything. Your kid who never tidies up is an operator that assigns toys to places in your living room. If you buy tickets for the opera, the online booking system is an operator that assigns seats to the audience.

A differential operators specifically acts on functions by differentiating them. The nabla for example, when applied to a scalar field, gives the gradient of that field. If you think about the scalar field as an altitude in a mountain range, then the gradient points towards the direction where the increase is the steepest.

Operators are the core concept of quantum mechanics. Quantities that in a classical theories are functions, like the position or energy of an object, become operators. To make something useful out of them, they now have to act on a function - that being the purpose of an operator. In quantum mechanics, it is the well-known wave-function that they act on.

But the usefulness of the operator concept is that one can deal with them on their own without applying them all the time. It's a bit like replacing 'classical' money with a credit card. If you want to see something 'real' you have to 'apply' it to an ATM to get cash. Most often the result is quantized, say, you can only get multiples of \$10 or so. You also typically have an offset, a smallest possible amount that you can get. But for most cases, you are fine dealing with the card itself. You have to be a bit careful though if you use it together with other cards, say the club card (payback card, member card, VIP card, whatever) from your local groceries. For your total, it matters in which order you present them at the register. We say that the operators don't commute: the result depends on the order of use.

The nabla is essentially the operator that, when acting on the wave-function, gives the momentum. That is, up to a constant - in this case a relevant constant. But this may be subject of another post.

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1. According to the original definition, the telephone operator produces a dial tone when acting on the vacuum. This is just for the readers in the younger generation which were born too late to realize that traditionally phones gave a dial tone when you picked up the receiver (other than your mobile or DECT phone).

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3. It's funny that Maxwell defines the use of nabla in the "Preliminary" section at the beginning of his two volume Treatise on Electricity and Magnetism (3rd ed., 1873), for "convergence" and "curl" operators, and in fact in the rest of the never uses them for his electromagnetic equations.

On page 28 of that Treatise, Maxwell defines the nabla operator as convergence instead of divergence and has it positive when the field line vectors are pointing inward: "I propose therefore to call the scalar part of nabla rho the convergence of rho ... I propose (with great diffidence) to call the vector part of nabla rho the curl, or the version of rho ..."

On page 29 he does gives almost the modern version of the Laplacian operator, but has the directional convention with the sign negative

nabla^2 = - (d^2 /dx^2 + d^2 /dy^2 + d^2 /dz^2)

It's weird that Maxwell makes no use of vector calculus in the rest of the book. The first publication of the vector calculus version of Maxwell's 20 long hand differential equations occurs twenty years later in Heaviside's book of 1893.

"Heaviside simplified and made useful for the sciences the original Maxwell's equations of electromagnetism. This innovation from the reformulation of Maxwell's original equations gives the four vector equations known today. Heaviside developed the Heaviside step function, which he used to model the flow of current in an electric circuit. Heaviside developed vectors (and vector calculus). Heaviside formed the operator method for linear differential equations. However, Heaviside's approach is short of rigorous mathematical basis. Thomas Bromwich supplemented Heaviside's operator method by providing a rigorous mathematics basis."

It's interesting that Heaviside, besides writing the Maxwell equations, also came up with crucial ideas which preceded the Lorentz-FitzGerald contraction:

"In two papers of 1888 and 1889, Heaviside calculated the deformations of electric and magnetic fields surrounding a moving charge, as well as the effects of it entering a denser medium. This included a prediction of what is now known as Cherenkov radiation, and inspired Fitzgerald to suggest what now is known as the Lorentz-Fitzgerald contraction." - Wiki

What always makes me amazed is when someone makes a mathematical invention but doesn't use it. Newton didn't use any calculus whatsoever in Principia which is solely done with Euclidean geometry. (He didn't even write the inverse square law of gravity with algebra; G was invented long after by Laplace.) Maxwell put nabla "convergence" and curl operators in the Preliminary section of his Treatise and then failed to use them in the remainder for electromagnetism. Also, Tullio Levi-Civita published the tensor calculus (although Einstein came up with the tensor name) in 1900 with Gregorio Ricci-Curbastro, but they didn't use it to work out general relativity. Why? Riemann's non-Euclidean geometry was well known. If I came up with anything useful mathematically (which sadly has nil probability), I'd definitely apply it to every important problem there is ASAP.

4. Hi Nige,

thanks, this is interesting indeed. It might just be a matter of inertia. Like, my laptop has the great invention of a keyboard back-light that helps you find the keys in the dark. Just that I never use it because I constantly forget about it.

Anyway.

It just occurred to me that the comparison with the credit card and cash is kind of appropriate, as the latin word 'quantum' means 'how much' :-)

Best,

B.

5. The correct etymology of nabla is that it is a short form of Czech "na blastním vrcholu stojící trojúhelník" which translates as "a triangle standing on its own bertex". ;-)

6. Dear Bee, the notation I always liked most was the bra kets notation (Dirac?) :-)

best

7. Bee, have you "cracked" an egg WRT relevent observers?

http://en.wikipedia.org/wiki/Renormalization_group

If so then another thread is most definitely needed!

P.S interesting ∇ has a varied historical significance, it's origin in language/alphabetical terms, has its roots from Pheonicians?

8. Thanks for such a clear and understandable explanation!

9. Hi RaeAnn,

Hi Rafa,

I too always loved the Bra-Cats :-)

Best,

B.

10. Couldn't help think of Gauss's Mountain

But yes the introduction process is always nice here for us lay people.

11. Can you imagine the ensuing word play if one used instead the shape of the hurdy gurdy? Although in Italian it is called the 'gironda', so that is little bit easier to say and use for mathematical notation. Thanks for the nice post!

12. I'm just going to go ahead and assume you have a mac.

Go to "system preferences" and select "keyboard and mouse." There you can check the box that makes the keyboard illuminate automatically.

Hmm, I often wish I had a checkbox that would bring automatic illumination.

And I'm sure it's too late to offer this advice, but when googling "nabla," don't click on the "nambla" links.

13. Hi Garrett,

thanks :-) But sorry, I don't have a mac. I have an IBM thinkpad (thats the one with the backlight I keep forgetting), a Dell (that I am currently working with, but if it has a backlight, I never found it) and a work-laptop that has a logo hp (hewlett packard?), that I have only had for a week, and I don't even know how to wake it out of sleep mode without rebooting. Best,

B.

14. nige, I will have to disagree with your statement that "Newton didn't use any calculus whatsoever in Principia which is solely done with Euclidean geometry."

Newton mentions some of his kind of calculus in Principia, such as infinitesimals and he introduces his ugly notation which is not used today. But my point is that Newton did not use Euclidean geometry. Can you tell why? If you open the Principia and look at Newton's figures you will see why.

Also, G was not invented by Laplace. It was invented in the 19th century by hard core British Newtonists led by Sir Charles Vernon Boys to make celestial mechanics British by replacing German Kepler's constant K with G.

Thanks, to Bee for another great post. Did you write that paper using Ma Bell as the telephone operator? Real funny!

He's a smooth operator
Smooth operator, smooth operator
Smooth operator

Coast to coast, LA to Chicago, western male
Across the north and south, to Key Largo, love for sale

Face to face, each classic case
We shadow box and double cross
Yet need the chase

A license to love, insurance to hold
Melts all your memories and change into gold
His eyes are like angels but his heart is cold

He's a smooth operator
Smooth operator, smooth operator
Smooth operator .....

16. Hi Pioneer,

the paper is by Warren Siegel, you find more entertaining stuff on his homepage

http://insti.physics.sunysb.edu/~siegel/plan.html

E.g. the theory of the Super G-String

SUPER G-STRING FIELD THEORY*

ABSTRACT

In conclusion, this is a major new paper by the pioneers in the field, so you're going to have to read the rest of the paper whether you understand it or not, or at least convince other people that you did by memorizing all the catch phrases. If you're a faculty member, just get one of your students to read it for you. If you don't have any students, probably nobody talks to you anyway, so forget it. If you are a student, good luck.

------------------------------------
*Work not supported. In fact, they told us specifically not to do it.

17. Among other things, you can learn from this paper that NEW means 'Not Ed Witten' :-)

18. Hi Bee

I did not have read Bra-Cats. Lol!. I enjoyed it a lot.

best

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