Re: ctensor - components as a matrix

Well, sure. For a mixed-index rank-2 tensor, the trace is just the sum 
of its diagonal components, and the representation of the tensor is 
invariant under a change of coordinates.

I just wanted to show you a simple, transparent example of what a 
component-wise representation of a tensor in itensor could be used for. 
But as I said, the very idea of assigning a component-wise 
representation is at odds with the core value of itensor, which is about 
the coordinate-free, abstract index formalism.

Since you mentioned ctensor, however, I should caution against careless 
use. For instance,

load(ctensor);
ct_coordsys(exteriorschwarzschild);
mat_trace(lg);

does NOT give you the trace of the Schwarzschild metric. That would be 
given by

cmetric();
mat_trace(lg.ug);

In other words, we need to form the "mixed index metric g^m_n (that, of 
course, is just the Kronecker-delta) and take its trace (which would of 
course be just the dimensionality of the manifold.)


Viktor



On 2026-04-13 18:27, Daniel Volinski via Maxima-discuss wrote:
> Hi Viktor,
>
> Thank you for your answer, nice example.
>
> Of course, you can obtain the same result with:
> *
> *
> *mat_trace(diag_matrix(ρ,-p,-p,-p));*
>
> and you don't need "ctensor" for that.
>
> Daniel Volinski.
>
> En martes, 7 de abril de 2026, 22:42:30 GMT+3, Viktor T. Toth 
> <[email protected]> escribió:
>
>
> Dear Daniel,
>
> On the face of it, the component-wise definition of a tensor is indeed 
> fundamentally at odds, philosophically, with the abstract index 
> formalism that itensor aims to represent.
>
> Having said that, I could think of a few legitimate use cases, 
> including, e.g., mixed-index rank-2 tensors in particular, which are 
> unchanged under a change of coordinates. Having said that, it's not a 
> construct I ever used much myself.
>
> Still, here's a silly but valid example (trace of the perfect fluid 
> stress-energy tensor):
>
> load(itensor);
> imetric(g);
> components(T([m],[n]),diag_matrix(rho,-p,-p,-p));
> ishow(g([m,n],[])*T([],[m,n]))$
> ishow(contract(%))$
> sum(%,n,1,dim),eval;
>
> Viktor
>
>
>
> On 2026-04-06 02:27, Daniel Volinski via Maxima-discuss wrote:
> Hi all,
>
> In the package "ctensor" I can define the components of a tensor in 4 
> ways,
> The second is to define the components as a matrix, like this:
> lg:-ident(4)$lg[1,1]:1$lg;
> components(g([i,j],[]),lg);
> ishow(g([i,j],[]))$
> I've never found a usage for that. Do you know of any usage of this 
> construct?
> i.e. something that will be useful for further calculations.
>
> Thank you,
>
> Daniel Volinski
>
>
>
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