I know that formula $∂^2=0$ and $d^2=0$ very important in the homology and cohomology theory. And I understand that this formula was generated from the process of finding a solution to the partial differential equation. So, I want to know the origin and motivation of the formula $∂^2=0$ and $d^2=0$.

(In this question, $∂$'s meaning is that the process of finding the boundary of a simplex. Like below picture. i.e. $∂^2=0$ means that boundaries have no boundary. )

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  • $\begingroup$ mathoverflow.net/questions/39597/… $\endgroup$ Oct 9, 2023 at 14:41
  • $\begingroup$ A remark since we don't have an answer yet: I always associated these formulas with integral theorems, like Gauss + Stokes, or Green's + path independence — that is, the formulas are generalizations. (Perhaps that's what you had in mind in your PDE remark.) That may have to do with the order I learned things. And the FTC for single-variable functions can be subsumed under the generalization. However, I don't have any historical evidence that this was the motivation. $\endgroup$
    – Michael E2
    Oct 11, 2023 at 11:00
  • $\begingroup$ @MichaelE2 Hello, I'm thanks to your comment! But I have some questions. What do you mean 'Gauss+Stokes' and 'FTC'? Could you explain more precisely about this comment? $\endgroup$
    – pokssin
    Oct 11, 2023 at 13:15
  • $\begingroup$ Gauss + Stokes, or Green's ... and the FTC means Gauss' Theorem plus Stokes' Theorem, or Green's Theorem ... and the Fundamental Theorem of Calculus $\endgroup$
    – Lee Mosher
    Oct 11, 2023 at 18:48
  • $\begingroup$ Isn't this just formalizing the geometric observation that the boundary of the boundary of a set is empty? $\endgroup$
    – quarague
    Oct 12, 2023 at 7:05


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