Let $x_i$ such that $i=1,2,\ldots,n$, and $\vec{x}=(x_1,\ldots,x_n)$
Define $A:= M_{ij}(\vec{x})\dot{x}^i\dot{x}^j$ where Einstein summation applies.
Also, $M$ is symmetric and invertible -- a metric.
What then is ${\partial \over \partial \vec{x}}A$ and ${\partial \over \partial \dot{\vec{x}}}A$?
I can't remember how this sort of index notation work, could someone please help?
Also, if anyone has any good references on the subject, and would not mind sharing, that would be greatly appreciated.