De Rham complexAs a partucular example, let us consider , . In this case we get:This is the de Rham complex for the flat space .
Explanation:Our is an -dimensional vector space. Let us introduce a basis and call the basis elements :
(this is just a notation, just use letter instead of ). What is ? It is the space of polynomials of , i.e.:And what is the dual space ? It is the space of “Taylor series”:The dual space has the following basis:Finally, what is ? Here it is:The De Rham differential acts as usual:
This operator is nilpotent. Moreover, it provides the resolution of in the sense that the following complex:is exact.
Mathematically speaking, the de Rham complex provides: