Re: How to calculate covariant derivative by Mathematica?

*To*: mathgroup at smc.vnet.net*Subject*: [mg106850] Re: How to calculate covariant derivative by Mathematica?*From*: Simon <simonjtyler at gmail.com>*Date*: Sun, 24 Jan 2010 05:47:18 -0500 (EST)*References*: <hjeq9u$fr1$1@smc.vnet.net>

Hi Shen, It depends on the context in which you're working, as a covariant derivatives can _look_ quite different. But maybe what you basically need is an operator of the type In[1]:= DD[t_]:=(D[#,t]+Con[#,t])& so that In[2]:= DD[x]@f[x] Out[3]= Con[f[x],x]+(f^\[Prime])[x] Then you need to make your connection, Con act properly. For example, it should return 0 when acting on scalars, and if you're acting on explicit Tensors and don't distinguish between contravariant and covariant, then maybe something like this would work: In[5]:= Con[expr_?ArrayQ,t_]:=Module[{dim=Dimensions[expr],rep,perms}, rep=Array[Subscript[r, ##][t]&,{dim[[1]],dim[[1]]}]; perms=Table[Range[Length@dim]/.{1->i,i->1},{i,Length@dim}]; Sum[Transpose[rep.Transpose[expr,perm],perm],{perm,perms}] ] we can test that this works properly on a (square) matrix: In[6]:= rep=Array[Subscript[r, ##][t]&,{2,2}]; m=Array[Subscript[z, ##] &,{2,2}]; In[7]:= Con[m,t]==rep.m+m.rep\[Transpose]//Expand Out[7]= True The above can be extended to vector derivatives and associated connections. Symbolic covariant derivatives are a bit more tricky... There are some packages out there... a google search for "mathematica covariant derivative" brings up a few. The Wolfram pages to look at are http://library.wolfram.com/infocenter/BySubject/Mathematics/CalculusAnalysis/DifferentialGeometry/ http://library.wolfram.com/infocenter/BySubject/Science/Physics/Relativity/ Finally, if you want to do index / field theory style calculations, then maybe you could try Cadabra. Hope some of that helps, Simon On Jan 23, 8:33 pm, Shen <zshen2... at yahoo.com> wrote: > I need to calculate covariant derivative by Mathematica. I noticed > that there is no such a function in Mathematica. Can we define such a > funcation? I don't know how to do it. Who can tell me how to define > and calculate covariant derivative with Mathematica?