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\omega = 3xz\;dx - 7y^2z\;dy + 2x^2y\;dz

$$ but this is only correct if we're in "flat" space, right?

In general, a differential ##1##-form associates a covector with each point of ##M##. If we use some coordinates ##(x^i)## on an open set ##U## of ##M##, then an expression like ##3xz## is correct and is valid for all points in ##U##, but shouldn't ##dx## represent a different basis vector for each point of ##U##? I guess if I write ##3xz\;dx## I'm assuming that ##(dx)_p = (dx)_q## for all ##p,q\in U##.

Sorry for the many edits.