Ok great, I'm glad that you're getting somewhere. So lets try to get to the 
bottom of this indexing issue (as a side note, and in case anyone else is 
interested, in this issue <https://github.com/dealii/dealii/issues/3646> 
we've started discussing how to document the index-related functions in a 
bit more detail). 

Can I ask a quick question: Are you familiar with the step-8 
<https://dealii.org/8.4.1/doxygen/deal.II/step_8.html> tutorial? This is 
the tutorial that details how vector-valued shape functions work in 
deal.II, and also what it means to ask (without the help of an extractor) 
for the gradient of one (primitive) shape function associated with a vector 
field. Although the tutorial described an application in solid mechanics, 
exactly the same concepts apply in your scenario. I think that this should 
give more insight into how one might go from the weak general form to a 
explicit assembly pseudo-code for a vector valued field. Its also better 
explained than anything that I could fit into this text-box, but if you 
don't find that to be the case then I've also expanded on it in this PDF 
<https://github.com/jppelteret/code-gallery/blob/353c02d31d76fdd76851a7313681d4d8bd1c42f4/Linear_Elastic_Active_Skeletal_Muscle_Model/doc/theory/theory-linear_elastic_active_muscle_model.pdf>
 (from 
page 5 is relevant). 

Using the extractors means that you (typically) don't have to worry about 
the manual component indexing via calls to
const unsigned int component_i = fe.system_to_component_index 
<https://dealii.org/developer/doxygen/deal.II/classFiniteElement.html#a27220a135402b96c7e6eecbb04acda56>
(i).first;
This allows one to get, for a primitive vector shape function, its only 
spatial component that is non-zero. The extractors help by filtering out 
the shape functions associated with other fields and reinterprets and 
collects the shape function value/gradient information for the given field. 

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