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 d*y*/d*x* as a literal ratio of the differentials d*y* and d*x* (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 d^{2}*y*/d*x*^{2} as a literal ratio.

I can't do this with the exterior differential, since both d^{2}*y* and d*x* ∧ d*x* 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 d*x*^{i} ∧ d*x*^{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 d^{2}*x* and d*x*^{i} · d*x*^{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 d^{2}, much less one that gives and proves its basic properties. I've even had to make up the notation ‘d^{2}’ (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 d^{2}*y*/d*x*^{2}; the correct formula is d^{2}*y*/d*x*^{2} − (d*y*/d*x*)(d^{2}*x*/d*x*^{2}); we cannot let d^{2}*x*/d*x*^{2} vanish and retain the simplicity of the algebraic rules. It would be better to write ∂^{2}*y*/∂*x*^{2}; the point is that this is the coefficient on d*x*^{2} in an expansion of d^{2}*y*, just as ∂y/∂x^{i} is the coefficient of d*y* on *x*^{i} when *y* is a function on a higher-dimensional space. The coefficient of d^{2}*y* on d^{2}*x*, which would be ∂^{2}*y*/∂^{2}*x*, is simply d*y*/d*x* again.)