Total derivative

[Sjoerd]

Hi

Can xAct recognize total derivatives?

And can xAct move the epsilon tensor inside the derivative and recognize this as a total derivative?
epsilong[a, b, c] CD[-a]@v[-b, -c]

Cheers,
Sjoerd

[Alfonso García-Parrado]

Hi Sjoerd,

A possible way of moving the epsilon tensor inside the derivative is
through the definition of the following inert head

DefInertHead[HoldDerivative]

Then you apply the rule

epsilong[a,b,c]CD[-a]@v[-b,-c]->CD[-a]@HoldDerivative[epsilong[a,b,c]v[-b,-c]]

Note that you can use (all ?) the xAct commands (ToCanonical, MakeRule,
etc) on the expression with HoldDerivative.

Best regards,

Alfonso.

[Sjoerd]

Hi Alfonso,

This works. Thank you for your answer!

Best,
Sjoerd

[Leo Stein]

Hi Alfonso,

This is a very nice idea! I think it might be usefully expanded as
follows. If you know that there is a total derivative in your
expression, you can use MapAt to apply a similar rule to one term, and
get the rest to cancel:
ToHoldDerivativeRule = {outside_ cd_?CovDQ[a_]@inside_ :>
cd[a]@HoldDerivative[outside inside] - inside cd[a]@outside};
ToHoldDerivative[expr_] := expr /. ToHoldDerivativeRule
Then for example, on some expanded expression,
In[] := CD[-a][v[b] w[c]]
Out[] := w[c] CD[-a]@v[b] + v[b] CD[-a]@[w[c]
Using MapAt,
In[] := MapAt[ToHoldDerivative, %, 1]
Out[] := CD[-a]@HoldDerivative[v[b] w[c]]

Unfortunately, I don't think there's any way to know in advance to
which term to apply this rule. After all, and expression may not have
any total derivatives.

Best
Leo

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