Surface integral finding the $u$ and $v$ vectors

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While calculating a surface integral, how do you know which variable to use as $u$ and $v$ in the calculation?

For example in Cartesian coordinates you have $x$, $y$ and $z$.

In cylindrical $\theta$, $r$ and $z$.

And in polar $\theta$, $\rho$ and $\phi$.

I’m confused as to which two should you choose as $u$ and $v$ to work out the question as I end up with different answers.