What Is the Symbol \nabla^2? Definition & Explanation

[TheDestroyer]

What's that symbol?

Today we've studied in the electrodynamique an affector named dalamperes affector defined as:

[tex]\nabla^2 - \frac{1}{c^2}\cdot\frac{\partial^2}{\partial t^2}[/tex]

c is the speed of light in vacuum, t is time, [tex]\nabla[/tex] is hameltons affector,

HERE IS THE QUESTION:

What's the name of the symbol used in that affector, the symbol is like a square and has the second degree, and does it have a definition for the first degree? and what is it? can some one explain everything about it?

 

[dextercioby]

God,u mean the d'Alembertian,a.k.a.BOX...
Defined in SR as:
[tex] \Box =:\partial^{\mu}\partial_{\mu} [/tex]
,its form depends on the metric chosen...In your case the metric is:
[tex] \eta_{\mu\nu}=diag \ (+,+,+,-) [/tex] (rather uncharacteristic)

Nabla is no longer called Hamilton's...It's called simply nabla.

Daniel.

 

[TheDestroyer]

Dextercioboy thank you for the specific answer, but i didn't understand:

1- What's the name of that symbol, is it aka box?
2- Does it have a first degree definition?
3- and what's the meaning of what's after Eta symbol you've written above?

Please try being more simple and specific with me, The language is causing me to not understand

 

[dextercioby]

D'ALEMBERT-IAN after the french mathematician Jean Le Rond d'Alembert,the one which discovered the waves' equation...

No.It's a second order linear differential operator...

You mean "diag"...?It's a shorthand notation for "diagonal".It means the matrix [tex] \hat{\eta} [/tex] is diagonal...

On normal basis i should have written it:
[tex] (\hat{\eta})_{\mu\nu}=\left(\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\end{array}\right)[/tex]

Daniel.

 

[TheDestroyer]

Oh my god, calm down, why you're getting nervous so quickly,

...


......

It's better for me to not understand, thanks

 

[dextercioby]

Who said i wasn't calm...?I took it as u didn't see the name very clearly & that's why i wrote it bigger,nothing else...

Daniel.

 

[TheDestroyer]

Thank you anyway dextercioboy, you're a genius in maths and physics and that doesn't help you to teach a university boy like me, i'll try finding the solution in our library and internet,

 

[HallsofIvy]

? He answered precisely your question : the symbol you asked about is called, informally, "box", similar to "del" for the upside down triangle symbol, and, more formally, the "D'Alembertian". It is an extension of the LaPlacian: where the LaPlacian, in 3 dim space, is the sum of the second derivatives wrt each coordinate, the D'Alembertian includes subtracting the second derivative wrt time.

"box" f= [itex]\frac{\partial^2 f}{/partial x^2}+ \frac{\partial^2f}{/partial y^2}+ \frac{\partial^2 f}{/partial z}- \frac{\partial^2f}{/partial t^2}[/itex]