## Is there a convenient differential calculus for cojets?

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.)

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Could you provide an example of a calculation that you would like to be able to do or do more easily using such a calculus? – Deane Yang Apr 3 2011 at 19:17
If you want to see a modern approach to the formal theory of PDE's (i.e., a cohomological approach to the Cartan-Kahler theorem, which was developed originally using exterior differential systems), look at the work of Hubert Goldschmidt, which builds on work by Spencer, Quillen, Guillemin, and Guillemin-Sternberg. See, for example, Chapters IX and X of the book "Exterior differential systems" by Bryant, Chern, Goldschmidt, and Griffiths. – Deane Yang Apr 3 2011 at 22:54
I didn't want to get into the context, in case people started discussing that instead. But there is a place to discuss that, in [this old thread](mathoverflow.net/questions/40082/…). I want to understand the theory behind a differentials-based approach to teaching freshman calculus, [as advocated by Dray and Manogue](physics.orst.edu/bridge/papers/…) (pdf). For first derivatives, I know how to make everything that they write formally correct. But what about higher derivatives? – Toby Bartels Apr 3 2011 at 23:29
So here's a problem from freshman calculus: Given that $y = x^3 - 3x$, for which values of $x$ does $y$ reach a local maximum or minimum? We compute $\mathrm{d}{y} = (3x^2 - 3) \,\mathrm{d}{x}$, set this to $0$ and solve for $x$ to get two critical points, which we test using the second derivative. Starting from $\mathrm{d}y$ above, we compute $\mathrm{d}^2{y} = 6x \,(\mathrm{d}{x})^2 + (3x^2 - 3) \,\mathrm{d}^2{x}$. Plugging in $x = \pm{1}$, we get $\mathrm{d}^2{y} = \pm{6x^2} \,(\mathrm{d}x)^2$. So $y$ has a local minimum when $x = 1$ and a local maximum when $x = -1$. – Toby Bartels Apr 4 2011 at 0:13
Maybe I'm mistaken but if $\Gamma(N)$ is the module of sections of a vector bundle over $M$ then $\Gamma(J^K(N)^*$ is "the same" as the module of linear differential operators from $\Gamma(N)$ to $C^\infty(R)$. I.e. differential operators associating a function to a section. There is a differential d on these objects (if you allow the order k to change) called the Spencer complex (see for example the articles of Spencer on overdetermined system or the book cited by Dean or books by Vinogradov and Lychagin). But i don't know if this d is the one you are looking for. – Michael Apr 5 2011 at 6:23