Talk:Poincaré lemma

In the direct proof we integrate the expression ${\displaystyle \alpha _{i}(\cdot ,t)\,dt}$ . Originally reading through this page, I was under the impression that ${\displaystyle \alpha _{i}}$  was a constant coefficient, and not a function of two variables - is it possible to clarify this? Also, are we holding whatever ${\displaystyle \cdot }$  represents constant while integrating over ${\displaystyle dt}$  to get a function in ${\displaystyle \cdot }$ ? Or is something else happening? (Sorry if these are silly questions, but I think that clarification wouldn't be harmful and help people with a little less background - such as myself - understand this proof.) Justanothermathstudent (talk) 15:21, 31 October 2024 (UTC)Reply

No, they are not constants but functions (coefficients in differential forms are functions, not numbers). The notation ${\displaystyle \alpha _{i}(\cdot ,t)}$  means you plug in t in the second variable but leave the first variable intact. But maybe it should be written more clearly. —— Taku (talk) 17:57, 1 November 2024 (UTC)Reply

Thanks for the clarification. Justanothermathstudent (talk) 22:42, 8 November 2024 (UTC)Reply