Parallel shift in polar coordinates and Christoffel Symbols

I’ve been looking at the Christoffel Symbols for parallel shift in polar coordinates.

Therefore I used the general basis vectors:

$vec{q}_r=vec{e}_r$; $vec{q}_varphi=rcdotvec{e}_varphi$

I looked at a carthesian vector $vec{A}=A^xvec{e}_x+A^yvec{e}_y$ and calculated $A^r$ and $A^varphi$ for $A=A^rvec{q}_r+A^varphivec{q}_varphi$ which gives me

$A^r=A^xcos(varphi)+A^ysin(varphi)$
; $A^varphi=frac{1}{r}(-A^xsin(varphi)+A^ycos(varphi))$.

Then I calculated the Christoffel Symbols using the total differentials

$delta A^r=frac{partial A^r}{partial r}delta r+frac{partial A^r}{partial varphi}delta varphi$ and $delta A^varphi=frac{partial A^varphi}{partial r}delta r+frac{partial A^varphi}{partial varphi}delta varphi$

and the known identity $delta A^i=-Gamma^i_{ml}A^mdelta x^l$ from ART, by comparing the coefficients.

So all this gives me the Christoffel Symbols which tell me how the parallel shift is carried out (since Christoffel Symbols do express that as far as my teachers tell me).

At this point I should as well figure out "how $A^x$ and $A^y$ behave during / due to that parallel shift". At this point I don’t understand what I need to do. Do I have to calculate $delta A^x$ and $delta A^y$? If so, how do I get this?