mersenneforum.org  

Go Back   mersenneforum.org > Great Internet Mersenne Prime Search > Math

Reply
 
Thread Tools
Old 2007-06-04, 21:22   #1
Damian
 
Damian's Avatar
 
May 2005
Argentina

2×3×31 Posts
Default Covariant Derivation

Can someone explain me in simple terms what is the covariant derivation?
For example if we take as a Manifold the unit circle in R^2, what would be its covariant derivation?
Damian is offline   Reply With Quote
Old 2007-06-04, 23:56   #2
m_f_h
 
m_f_h's Avatar
 
Feb 2007

24×33 Posts
Default

Quote:
Originally Posted by Damian View Post
Can someone explain me in simple terms what is the covariant derivation?
For example if we take as a Manifold the unit circle in R^2, what would be its covariant derivation?
the explicit expr of the corvar deriv depends on the object : you have to add a term with the gauge potential (connection) for each index.
e.g. for R_ab^cd
DR_ab^cd = dR_ab^cd+R_eb^cd phi_a^e +R_ae^cd phi_a^e +R_ab^ed phi_e^c+R_ab^ce phi_e^d

sorry I may have not 100% mainstream conventions ,
also that expression might simplify (to zero of course, but suppose R was sth else than d phi + phi phi),
i dont remember well : I did that in an earlier life...

Last fiddled with by m_f_h on 2007-06-04 at 23:57
m_f_h is offline   Reply With Quote
Old 2007-06-05, 18:20   #3
Damian
 
Damian's Avatar
 
May 2005
Argentina

2×3×31 Posts
Default

I don't get it
Damian is offline   Reply With Quote
Old 2007-06-05, 20:45   #4
ewmayer
2ω=0
 
ewmayer's Avatar
 
Sep 2002
Rep├║blica de California

2·5·13·89 Posts
Default

The proper term is "covariant differentiation."

Any decent text or webpage on differential geometry should have an adequate description.
ewmayer is online now   Reply With Quote
Old 2007-06-06, 15:41   #5
Damian
 
Damian's Avatar
 
May 2005
Argentina

2·3·31 Posts
Default

Quote:
Originally Posted by ewmayer View Post
The proper term is "covariant differentiation."

Any decent text or webpage on differential geometry should have an adequate description.
I know, I wanted a less formal definition, maybe with some numerical example, to make it less abstract and more easy to understand it.
Damian is offline   Reply With Quote
Old 2007-06-07, 17:04   #6
m_f_h
 
m_f_h's Avatar
 
Feb 2007

1101100002 Posts
Default

Quote:
Originally Posted by Damian View Post
I know, I wanted a less formal definition, maybe with some numerical example, to make it less abstract and more easy to understand it.
I suggest you the very nice book by Nakahara: "Geometry, Topology and Physics" (if I recall correctly). It has lots of explicit examples etc. on this and related subjects (even if there are typos in several formulae, but usually just signs (+/-) or so.)

PS: it seems there is some explicit calculation on
http://en.wikipedia.org/wiki/Connection_(mathematics)
PPS: well, not much... I think you have to plug in those into the formulae on the page "covariant derivative"

In fact, there are different notions of covariant derivatives. In general, "covariant" is w.r.t. some local ("gauge") transformation. In general relativity, there are 2 such transformations to be considered : local Lorentz transformations (SU(2) or SO(3,1) acting on "Lorenz" indices), and local coordinate transformations (diffeomorphisms ; acting on "Einstein indices"). "of course", both are linked...

Last fiddled with by m_f_h on 2007-06-07 at 17:19
m_f_h is offline   Reply With Quote
Reply

Thread Tools


All times are UTC. The time now is 03:43.

Wed Dec 2 03:43:46 UTC 2020 up 83 days, 54 mins, 1 user, load averages: 1.36, 1.55, 1.61

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.