# The origin of $∂^2=0$ and $d^2=0$

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. )

• mathoverflow.net/questions/39597/… Oct 9, 2023 at 14:41
• 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. Oct 11, 2023 at 11:00
• @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? Oct 11, 2023 at 13:15
• Gauss + Stokes, or Green's ... and the FTC means Gauss' Theorem plus Stokes' Theorem, or Green's Theorem ... and the Fundamental Theorem of Calculus Oct 11, 2023 at 18:48
• Isn't this just formalizing the geometric observation that the boundary of the boundary of a set is empty? Oct 12, 2023 at 7:05