Using infinitesimals from $ A(x, y) = x * y $ I have $ dA = A_x * dx + A_y * dy $ solving it for $ dA $, I have $ dA = y * dx + x * dy $ which is a mistake. Where in my thinking way is that mistake? I have tried with $ A(x, y) = const $ but it is also wrong.
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$\begingroup$In a double integral $\iint_D f(x,y) \, dA = \iint_D f(x,y) \, dxdy$, the symbol $dA$ is just shorthand for the area element $dxdy$, “a little piece of area”.
That's something completely different from the differential $dA=y\,dx+x\,dy$ of the function $A(x,y)=xy$!
So there's no contradiction, just the same symbol used for two different things.
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