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I want to calculate below matrix vector multiplication which involve differentiation.

$$\left( {\begin{array}{*{20}{c}} { - \mu \left( r \right)}&{\Delta {e^{ - i\phi }}\left( { - {\partial _r} + \frac{{i{\partial _\theta }}}{r}} \right)}\\ {\Delta {e^{i\phi }}\left( { - {\partial _r} + \frac{{i{\partial _\theta }}}{r}} \right)}&{\mu \left( r \right)} \end{array}} \right){e^{in\theta }}\left( {\begin{array}{*{20}{c}} {{e^{ - i\phi /2}}\left[ {f\left( r \right) + ig\left( r \right)} \right]}\\ {{e^{i\phi /2}}\left[ {f\left( r \right) - ig\left( r \right)} \right]} \end{array}} \right)$$

I use "basic math assitant" to write the following

enter image description here

However, it doesn't work. How to done it right?

PS. here you can download the notebook contains my expression

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    $\begingroup$ Please share the code in copyable form, so that other users can play with it. No one wants to retype all this code (and double check for correct transition). This will raise your chances for getting quick and competent help. $\endgroup$ – Mariusz Iwaniuk Jun 10 '18 at 8:55
  • $\begingroup$ @MariuszIwaniuk Hi, The code directly copied is not perfect. So I attached a link to my notebook directly $\endgroup$ – matheorem Jun 10 '18 at 9:03
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    $\begingroup$ we do not do links to notebooks at stackexchange. These could contains viruses. and when the link goes away, the question become useless. Code used should be posted in the question so it is self contained. $\endgroup$ – Nasser Jun 10 '18 at 9:06
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Just some toy example in order to give you an impression how this could be done.

A = Table[With[{t = t}, D[#, t] &], {t, {x, y}}, {i, 1, 2}];
U = {u[x, y], v[x, y]};
(Table[Sum[A[[i, j]]@U[[j]], {j, 1, 2}], {i, 1, 2}])//MatrixForm

$$\left( \begin{array}{c} u^{(1,0)}(x,y)+v^{(1,0)}(x,y) \\ u^{(0,1)}(x,y)+v^{(0,1)}(x,y) \\ \end{array} \right)$$

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