An unexpected behavior from @prodrule

Question

Hi everyone!

Please take a look at the first term of the result of the second @eliminate_metric and the first term of the @prodsort, in the following code. They must be equal but due an automatic "integral by part" in @prodsort all of the derivatives were moved to just one of the fields without any minus sign.

{\alpha, \beta, \rho, \sigma, \mu, \nu, \gamma, \lambda}::Indices.
    \eta^{\mu \nu}::Metric.
    \eta_{\mu \nu}::Symmetric.
    \eta^\mu_\nu::KroneckerDelta.
    h^{\mu \nu}::Symmetric.
    \partial_{#}::PartialDerivative.
    \Gamma^{\mu}_{\nu \rho}::Depends(\partial).
    \Gamma^{\mu}_{\nu \rho}::TableauSymmetry(shape={2}, indices={1,2}).
    \phi::Depends(\partial).
    h^{\mu \nu}::Depends(\partial).
    h::Depends(\partial).
    term:=(\eta^{\mu \nu} \eta^{\alpha \beta}\partial_{\mu}\phi
    -k h^{\mu \nu} \eta^{\alpha \beta}\partial_{\mu}\phi )
    (\partial_{\alpha}\partial_{\beta}\partial_{\nu}\phi 
    - \partial_{\alpha}(\Gamma^{\rho}_{\beta \nu}\partial_{\rho}\phi));
    @prodrule!(%);
    @distribute!(%);
    @distribute!(%);
    @eliminate_metric!(%);
    @eliminate_metric!(%);
    @prodsort!(%);

As another question, in the above code I used @eliminate_metric and @distribute twice sequentially to get the desired effect. Is there any way to use just once with the same effect?

Answer 1

If you write

\partial_{\mu}A

this may look like the right thing on your screen, but actually represents a partial derivative acting on nothing, multiplied with A.

What you need to write is

\partial_{\mu}{A}

to explicitly tell Cadabra on which object the derivative acts.

Comments

Oops! You are right! This issue is mentioned in the manual!

Would you please give me a hint for the second question that I made in the final.

---

You can use the 'repeat until expression no longer changes' form

 @distribute!!(%)

and ditto for eliminate_metric.

By the way there are much more powerful ways to do this in 2.x; I'd recommend that you switch to 2.x for new projects as that is where all the exciting new things happen. See http://cadabra.science.

---

But what should I do for three derivates like

\partial_{\alpha}\partial_{\beta}\partial_{\nu}{\phi}

The above form is correct?

---

Thanks for answer. For sure, I am in a phase transition to the second version. Now I am trying to learn Sympy to can use Cadabra2 in a more efficient way.

---

Either

\partial_{\alpha}{ \partial_{\beta}{ \partial_{\nu}{\phi} } }

or

\partial_{\alpha \beta \nu}{\phi}