Edit : (I didn't intend this as an insult or a debate discussing which way is best or better for what, I'm just asking a question for my interest and I believe in the interest of science, at least for variety sake..
I do not not idealise any man or work, the only reason I brought up principia is to save myself the trouble of answering unending strings of questions on how will I practically calculate without a basis, so that's why I called upon the highest authority in this regard.


I know coordinates are useful when used right, I only have a problem when people say you must use them in practical calculations and it can't be done other way.
Invariant formulations are most useful in the long run, when it comes to unification of different areas, and attacking the deepest problems that almost always require some level of unification.
If someone is genuinely interested in the details especially for research purposes I can elaborate on this further
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Basically without pasting any non existant (non intrinsic) structure on an actual space, which for euclidian geometry is an euclidian affine space of points.
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The way they did geometry from the ancient Greeks to Descartes.


Coordinates and their maps are the foundation of standard differential geometry. The theory is coordinate free, but riddled with non geometric objects, and with the need to prove that geometrical objects are not just coordinate nonsense.


I am looking for a theory including differential operators that builds directly on the pre Descartes approach to geometry.


Newton developed the entire principia mathematica this way, and I believe he could have used calculus with that geometric approach.


Is there any such exposition that would deal with differential operators like like covariant derivative, vector fields and differential forms, without assuming any analytical (coordinate) geometry



Is it possible to do calculus and differential geometry the old school way, without any ortho frames or axis?
*The Geometry of Geodesics*, by Herbert Busemann, provides a purely intrinsic approach to a large part of differential geometry, through axioms on the metric.


* It does not define covariant derivatives — but it defines geodesics without them, as length-preserving maps from the real line.
* It does not define vector fields — but it analyzes motions, which are a finite analog to that infinitesimal notion.
* It does not define differential forms — but it defines scalar curvature synthetically.


Busemann then proved a whole book of impressive theorems on this basis. (I gave some examples at [Characterizations of Euclidean space](https://mathoverflow.net/questions/47882/characterizations-of-euclidean-space.)) If you want a result in Riemannian geometry that you can state without coordinate definitions, you’ll probably find a proof there.


