Timeline for Is there any way to rewrite a partial differential equation using language of differential forms, tensors, etc?
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| S Oct 22, 2023 at 12:28 | history | suggested | The Amplitwist | CC BY-SA 4.0 |
fixed broken link to Wikipedia
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| Oct 22, 2023 at 7:31 | review | Suggested edits | |||
| S Oct 22, 2023 at 12:28 | |||||
| Jun 12, 2020 at 10:11 | comment | added | Igor Khavkine | @Yai0Phah I think it is best to articulate this as a new question. | |
| Jun 12, 2020 at 10:09 | comment | added | user20948 | Sorry that I failed to articulate the question. I was asking the relation between $\mathcal D$-modules and exterior differential systems (it could be related through PDEs, but unsatisfactory). What I have in my mind is that, given a smooth complex variety $X$, we have the Spencer resolution $D_X\otimes_{\mathcal O_X}\bigwedge^*T_X\to\mathcal O_X$. Does that have something to do with a differential ideal of the de Rham complex? | |
| Jun 11, 2020 at 21:48 | comment | added | Igor Khavkine | @Yai0Phah No, any relation on derivatives can be expressed in terms of jets (linear, non-linear, inequality, etc.). In the PDE--$\mathcal{D}$-module dictionary, given a linear PDE (represented as a linear sub-bundle of the infinite jet bundle) the smooth functions on this sub-bundle (linear on the fibers) constitute a $\mathcal{D}$-module. It is a module over the ring of smooth functions of the independent variables, and the $\mathcal{D}$ operator is just the total coordinate derivative (extended to jets). | |
| Jun 11, 2020 at 17:58 | comment | added | user20948 | Is the jet description usually about linear PDEs? What I know is about $\mathcal D$-modules, which is essentially describing linear systems. I am struggling to understand the relation between this and exterior differential systems. | |
| May 24, 2013 at 13:33 | vote | accept | HYYY | ||
| May 23, 2013 at 12:26 | history | answered | Igor Khavkine | CC BY-SA 3.0 |