What Constitutes something being "coordinate free" People say that exterior calculus ie. differentiating and integrating differential forms, can be done without a metric, in without specifying a certain coordinate system. I don't really get what qualifies something to be 'coordinate free', I mean in the differential forms I do, one still references components ie. x1,x2, etc., yet I never specified a metric, so is this classified as 'coordinate free'. Also, how does one do differential geometry without a coordinate system, in my mind once you don't specify a coordinate system or a metric, and things become vague, it sort of turns into differential topology, is there a 'middle ground' I am missing, keep in mind I have never taken a coarse in differential geometry. Also, in differential geometry, it has always been pertinent to give specific parametrization in order to find tangent vectors, metrics, etc.