# How to apply Partial differentiation w.r.t. tensors?

Let's say I have an expression like $$\,a^{I}=2b^{I}+3c^{I}$$ where $$I$$ stands for an arbitrarily large set of indices. It's known that $$\,\frac{\partial a^K}{\partial c^{L}}=3\,\delta^K_L$$ (equals a product of Kronecker deltas) but, how can I do that in Mathematica using packages like XAct, Xpert and any else related to XTensor.

(i) How could I achieve that?.

(ii) Also, notice that I asumed that $$\,b \neq b(c)$$, how can I tell Mathematica to do so when computing the partial derivative?.

Try something like this:

<< xActxTensor

DefManifold[M, 4, {i, j, k, l}]

DefTensor[b[i], M]
DefTensor[c[i], M]


Then define

a[i_] := 2 b[i] + 3 c[i]


Finally you can compute

VarD[c[j]][a[i]]
(* 3  delta[-j, i] *)


Recall that -j is a covariant index in xAct.

There will be an error warning about differentiation of a non-scalar. For simple cases like this , you can ignore that message.

• That works well, thanks. Now, what would happen if $a_i$ is not a tensor anymore, $c_i=\partial_i\phi$ and I want to compute something like $\frac{\partial a_i}{\partial(\,\partial_j\phi\,)}$. How could I do it in that case? – JuanC97 Jan 24 at 5:21