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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.)

flag	asked Apr 3 2011 at 18:55 Toby Bartels 923●4●10

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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

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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

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Peter Michor · Answer 1 · 2012-09-19 09:12:10Z UTC

Maybe the following links help: Gerd Kainz, Peter W. Michor: Natural transformations in differential geometry. Czechoslovak Math. J. 37 (1987), 584-607, accessible as scanned paper under: http://www.mat.univie.ac.at/~michor/nat-transf.pdf. A slightly more extended version is in chapter 8 of: Ivan Kolár, Jan Slovák, Peter W. Michor: Natural operations in differential geometry. Springer-Verlag, Berlin, Heidelberg, New York, (1993), vi+434 pp. which is accessible via http://www.mat.univie.ac.at/~michor/kmsbookh.pdf

Is there a convenient differential calculus for cojets?

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