I understand the basics of exterior differential geometry and how to do calculus with exterior differential forms. I know how to use this to justify the notation dy/dx as a literal ratio of the differentials dy and dx (by treating x and y as scalar-valued functions on a 1-dimensional manifold and introducing division formally). I would like to extend this to second derivatives. Ideally, this would justify the notation d²y/dx² as a literal ratio.

I can't do this with the exterior differential, since both d²y and dx ∧ dx are zero in the exterior calculus. It occurs to me that this would work if, instead of exterior differential forms (sections of the exterior bundle), I used sections of the cojet bundle (cojet differential forms). In particular, while degree-2 exterior forms may be written in local coordinates as linear combinations of dxⁱ ∧ dx^j for i < j (so on a 1-dimensional manifold the only exterior 2-form is zero), degree-2 cojet forms may be written in local coordinates as linear combinations of d²x and dxⁱ · dx^j for i ≤ j (so on a 1-dimensional manifold the cojet 2-forms at a given point form a 2-dimensional space).

I know some places to read about cojets (and more so about jets) theoretically, but I don't know where to learn about practical calculations in a cojet calculus analogous to the exterior calculus. In particular, I don't know any reference that introduces the concept of the degree-2 differential operator d², much less one that gives and proves its basic properties. I've even had to make up the notation ‘d²’ (although you can see where I got it) and the term ‘cojet differential form’. I can work some things out for myself, but I'd rather have the confidence of seeing what others have done and subjected to peer review.

(Incidentally, I don't think that it is quite possible to justify d²y/dx²; the correct formula is d²y/dx² − (dy/dx)(d²x/dx²); we cannot let d²x/dx² vanish and retain the simplicity of the algebraic rules. It would be better to write ∂²y/∂x²; the point is that this is the coefficient on dx² in an expansion of d²y, just as ∂y/∂xⁱ is the coefficient of dy on xⁱ when y is a function on a higher-dimensional space. The coefficient of d²y on d²x, which would be ∂²y/∂²x, is simply dy/dx again.)