The de rham mayer vietoris sequence is exact at the first two terms

Given: An open cover $M=U\cup V$ and the maps $r,s$ just defined.

Two open set de rham mayer vietoris cochain maps: For an open cover $M=U\cup V$, put $W=U\cap V$. The two-open-set de Rham maps are $r:{\mathrm{\Omega }}^{\bullet }(M)\to {\mathrm{\Omega }}^{\bullet }(U)\oplus {\mathrm{\Omega }}^{\bullet }(V)$, $r\omega =(\omega {\mathrm{\mid }}_U,\omega {\mathrm{\mid }}_V)$, and $s:{\mathrm{\Omega }}^{\bullet }(U)\oplus {\mathrm{\Omega }}^{\bullet }(V)\to {\mathrm{\Omega }}^{\bullet }(W)$, $s(\alpha ,\beta )=\beta {\mathrm{\mid }}_W-\alpha {\mathrm{\mid }}_W$. The complexes are def-de-rham-cochain-complex. Restrictions are pullbacks along open inclusions, so prop-pullback-is-a-morphism-of-de-rham-complexes gives $dr=rd$ and $ds=sd$. Both maps are real linear. The middle differential acts componentwise. Empty opens have zero form spaces. The order second minus first fixes the sign of every connecting map below.

A smooth differential $k$-form: Let $M$ be a smooth manifold and $k\ge 0$. A smooth differential $k$-form on $M$ is a smooth section of ${\bigwedge }^k{T}^{\ast }M\to M$. The space of such forms is denoted ${\mathrm{\Omega }}^k(M)$, and ${\mathrm{\Omega }}^0(M)=C^{\mathrm{\infty }}(M)$.