Favorite Equation?

[Norman Albers]

For another fave, I too love the magic contained in exponentiating pi-i. So much handy accounting. We are accountants of the arcane.

[In My Memory]

My favorite equation:

 

(4MovAvg > 9MovAvg > 21MovAvg > 50MovAvg) AND (Today's_High > Yesterday's_HIGH) AND (Today's_Low < Yesterday's_Low) = Bullish

 

Its one of the most simple equations I use for stocks between US$10 and US$100, very fast too, and very reliable.

 

I also like e^(pi*i) + 1 = 0

[Norman Albers]

How are you at the horse races?

[Cloud]

Favourite equations -

I would say 'solving quadratic simultaneous equations

through substituing and completing the square.' Its the first time I have had a glimspe at proper pure mathematics. And the degree of operations is overwhemling at first but fun. Next stop - calculus.

 

As for the FORMULAS -

I would say e=mc2 since it probably has the greatest scientific importance (i think).

[Apeofman]

My favourite equation is the addition of velocities in accordance with the Lorentz Transforms of Albert Einstein s Special Relativity.

 

To illustrate the transformational magic that occurs due to the constant speed of light c, i put a Java based relativity calculator on the web to celebrate the centenary of Special Relativity. The calculator use's geometry, to set and calculate all velocities and times. The geometry echoes the magic of the Lorentz Transforms, and provides an interesting way of understanding what is going on.

 

Relativity Calculator. Interpretation.

[pHoToN_gUrL]

F=ma

i luv that equation...Newton's laws of motion...it's my favorite <3

[ajb]

My fav has to be maxwells equations written using differential forms. It goes straight to the heart of modern physics; physics does not care about the coordinates.

[Norman Albers]

[math] [F_{uv\vline\gamma}]=0[/math], and [math]\nabla_vF^{uv}= s^u[/math] laTeX queries: the bracket on the first term should be a brace, for antisymmetrization or generalized curl. How do we print that, and also a double vertical line symbol for covariant diffs? Also, the 'evaluated at' vline?

[reor]

Could somebody tell me the formula for pi and explain it?

[Norman Albers]

There have to be cool ways but one is to figure a limit with polygons inscribed, taking the ratio of circumference as sum of chords, to radius. This seems less than profound, though: [math] \lim_{n\to\infty} n\sin{\pi/n}[/math]

[Severian]

laTeX queries: the bracket on the first term should be a brace

You need \{ giving [math] \{F_{uv\vline\gamma}\}=0[/math]

 

and also a double vertical line symbol for covariant diffs?

That one is easy' date=' it is just ||, eg. [math']||A||[/math] for a norm. I am a bit confused by why you would use it for a coviant derivative though (maybe this is not what you meant?). I would write a covaraient derivative as [math] {\cal D}_\mu[/math].

 

Also, the 'evaluated at' vline?

This is a bit more complicated because you need to use \left and \right e.g. [math]\left. F \right|_{x=0} = 0[/math]

[Norman Albers]

Thanks! Your covariant derivative symbol is nice and clear. My text uses two vertical (small) lines between indices to distinguish from ordinary differentiation. This is economic notation when you are in the thick of it, tensorially speaking. (Ah'll be damned, I just found that symbol on the keyboard. Awesome. Not.) . . .(Speaking 'ex post post', [math]F_{uv||\gamma}[/math]).

[Severian]

You mean things like [math]F_{\mu \nu || \rho ||}[/math]? I have never seen that notation before.

[ajb]

relativists use commas.

[Norman Albers]

How so commas, for ordinary differentiation? I'm used to: . . . [math] \partial/\partial r= \partial_r= F_{|r}[/math], and I am just seeing Sweetser using nabla for covariant differentiation, as Severian offers . . .[math]{\cal D}_r=\nabla_r=F_{||r}[/math]. . . . . . . . . I am pretty much usually thirty years behind the times, but then too the times are always a-changin'.

[ajb]

was is it comma for partial derivative and ; for covariant? Something like that. I tend not to use that notation myself.

 

Nabla and D are common, it may depend on the exact context to which you use.

 

I have never seen | or || to refer to a derivative. What branch of mathematics uses this notation?

[Norman Albers]

Adler, Bazin, and Schiffer, "Introduction to General Relativity". Great mathematician's book, ca. 1975.

[ajb]

I think it is old notation and not in standard use today.

[Norman Albers]

Thanks, I'll check out current expressions before casting my work in 4-D relativistic notation! (How was that, "klattu niktu barrata" or something in a sci-fi story?)

[Severian]

How so commas, for ordinary differentiation? I'm used to: . . . [math] \delta/\delta r= \delta_r= F_{|r}[/math], and I am just seeing Sweetser using nabla for covariant differentiation, as Severian offers . . .[math]{\cal D}_r=\nabla_r=F_{||r}[/math']. . . . . . . . . I am pretty much usually thirty years behind the times, but then too the times are always a-changin'.

For me [math]\vec{\nabla}[/math] (or I suppose [math]\nabla_i, \, i=1...3[/math] to be consistant with the component notation I will use for the others) is an ordinary three-derivative, while [math]\partial_\mu, \, \mu=0...3[/math] is a four-derivative, and finally [math]{\cal D}_\mu,\, \mu=0...3[/math] is a covariant derivative.

[Norman Albers]

Good linguistics and laTeX discussion, thanks.

[ajb]

[math]

\nabla

[/math] to me would usually be the Levi-Civita connection on the tangent bundle of a Riemannian manifold. I would also use it to represent a connection on a complex vector bundle, unless I am thinking about it as an an associated bundle.

 

[math]

D

[/math] would be a connection in a principle bundle or on its associated vector bundle.

[Norman Albers]

In Reimannian geometry is it then the same as the Christoffel symbol of the second kind?

[Norman Albers]

[math] \Gamma^r_{ik}= -\{^r_{ik} \}[/math]

[ajb]

In Reimannian geometry is it then the same as the Christoffel symbol: [math']\Gamma^r_{ik} [/math]

 

Yes, if [math]e_{j}[/math] is a local basis for the tangent vectors then

 

 

[math]\nabla e_{j} = \Gamma^k_{ij} dx^{i} e_{k}[/math].