Is it possible to do calculus and differential geometry the old school way, without any ortho frames or axis? 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
 A: It's possible to do differential geometry in a purely intrinsic way, at least once you've gotten past the initial hurdle of defining what a manifold is. The standard definition of a manifold is a second countable, Hausdorff, locally-Euclidean space, so coordinate charts naturally show up (due to that last part). It might be possible to avoid charts entirely, but it almost requires a new definition for manifold. But once you've gotten past this issue you can do everything else in a coordinate-free way, if you so choose.
The real reason that most geometers fo not do this is that it makes explicit computations extremely difficult. Intrinsic approaches and notation have a philosophical appeal, but are ill-suited for many application,  where you might need to compute six or seven derivatives. Picking a convenient coordinate chart (or orthonormal frame) to make the analysis easier is absolutely worth the conceptual loss of simplicity. In fact, there are insights that can be found using a particular choice of coordinates that are nearly impossible to see (or fundamentally more difficult to prove) using a more abstract approach.
A: The Geometry of Geodesics, by Herbert Busemann, provides a purely intrinsic approach to a large part of differential geometry, through axioms on the metric.

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*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) If you want a result in Riemannian geometry that you can state without coordinate definitions, you’ll probably find a proof there.
A: I do think you're asking a reasonable question, but many do not like your way of asking it. It would be better received if you could express it more rigorously and mathematically and showed that you have thought about it more deeply than your wording indicates. After all, this is a research math forum. But let me make some comments.
The first thing is Newton versus Descartes. I have never read Newton's works, so I could be wrong. But since Descartes preceded Newton, I believe Newton must have embraced Cartesian coordinates and used them in his work on planetary motion and the shape of the earth. Is that not so?
As for developing differential geometry without coordinates, many mathematicians, including me, have tried. I'm not sure whether you're talking about surfaces in Euclidean space or abstract spaces known as manifolds. In either case, my impression is that the hardest steps are right at the beginning. First, you need to develop multivariable calculus without coordinates. This can be done but is it worth the pain? Not as far as I can tell, but you can see if you can do it. I definitely could be wrong about that. Second, it's defining what a surface or manifold is.
Some very abstract-minded mathematicians did manage to do this for manifolds, but you lose all geometric intuition and end up in a very algebraic world. Is it worth the pain? Also, not as far as I can tell. After you've defined a manifold, then you can work out the fundamentals of Riemannian geometry using only abstract vector fields. This is demonstrated both in Milnor's monograph Morse Theory and the book by Cheeger and Ebin, Comparison Theorems in Riemannnian Geometry.
As for a surface in Euclidean space, you could first define Euclidean space as an abstract vector space with an inner product. Then you could define a surface to be the level set of a function whose gradient is nonzero and work with derivatives of the function (without using coordinates). The geometry of the surface can now be derived from studying curves in the surface and their derivatives. Some of this is very nice, but some aspects are still easier to calculate and understand using coordinates. In particular, it's difficult to work out examples without using coordinates.
However, in the long run, what professional differential geometers discover is the following: Our main goal is to prove interesting new theorems as efficiently as possible. The most efficient approach depends on the specific circumstances. So we dump the ideology and pragmatically learn how to use all of them. We switch between them as needed. So the fact is that using coordinates is often the easiest way. The basic reason for that is partial derivatives commute. This fact is fundamental and used all the time. Without using coordinates or differential forms (as when using orthonormal frames), that fact is hard to use efficiently.
I do continue to think about all this in the context of teaching differential geometry. I do agree that coordinates can often obscure what's really going on. I don't like most textbooks on elementary differential geometry. So I do try to think of coordinate-free approaches that better elucidate the geometry. Sometimes I succeed. Otherwise, it's coordinates or orthonormal frames. Whatever works best.
