---- Let $M$ be a [[smooth manifold]] of dimension $n$. Recall that $\Omega^{r}(M)$ denotes the [[vector space]] of [[differential form|differential]] $r$-[[differential form|forms]] on $M$. > [!definition] Definition. ([[de Rham algebra of differential forms on a smooth manifold]]) > The **algebra of differential forms on $M$** is the skew-commutative [[differential graded algebra|differential]] [[graded algebra|graded]] [[algebra]] $\Omega^{\bullet}(M):=\bigoplus_{r=0}^{n} \Omega^{r}(M)$ where multiplication is given by the pointwise [[algebra of alternating multilinear forms|wedge product of alternating tensors]] $(\omega \wedge \eta)_{p}:=\omega_{p} \wedge \eta_{p}$, called thus the **wedge product of differential forms**. ^definition $\frac{1}{(4\pi t)^{n/2}}\exp(-\|x-y\|^{2} / 4t)$ ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```