Partial derivative of a vector

Question

Hello!

I am very a new Cadabra user and I am trying to expand partial derivative of a vector field A i.e. $\partial_{\mu} { A^{\mu} }$ with the following snippet of code:

{A^t,A^x, A^y, A^z}::Depends(\partial{#});
A := {A^{t}=A^t,A^{x}=A^x, A^{y}=A^y, A^{z}=A^z};
dA:=\partial_{\mu}{ A^{\mu} };
evaluate(dA,A);

which returns an error: RuntimeError: Dependencies on derivatives are not yet handled in the SymPy bridge

I ideally want it to give me the componenet expansion like: $ \partial_{\mu} { A^{\mu} } = - \partial_t A^t + \partial_x A^x + \partial_y A^y + \partial_z A^z $.

Also I want it to return a 2x2 matrix when I try to evaulate $ \partial_{\mu} { A^{\nu} } $.

Thanks in advance!

Answer 1

Whenever you evaluate, the expression should not contain derivatives, otherwise simplification by sympy will throw that error. But you can turn that off, so the following does what you want:

\partial{#}::PartialDerivative;
{t,x,y,z}::Coordinate;
{\mu, \nu}::Indices(values={t,x,y,z});
A^{\mu}::Depends(\partial{#});

dA:=\partial_{\mu}{ A^{\mu} }; 
evaluate(dA, simplify=False);

If you replace the last two lines with

dA:=\partial_{\mu}{ A^{\nu} }; 
evaluate(dA, simplify=False);

you get that 2x2 expression.