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 ∧ dxj 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 · dxj 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.)