This is an example that is computed in Bott and Tu's differential forms. I do not get the obvious reasoning here.
Take $U,V$ covering $S^1$. Then one obtain compact support de Rham cohomology. $0\to H^0(U\cap V)=0\to H^0(U)\oplus H^0(V)=0\to H^0(S^1)\to H^1(U\cap V)=R^2\to H^1(U)\oplus H^1(V)= R^2\to H^1(S^1)\to H^2(U\cap V)=0$
Q: $H^1(U\cap V)\to H^1(U)\oplus H^1(V)$ is induced through extending the forms supported on $U\cap V$ by $0$ to obtain forms defined on $U$ and $V$. The book claim the kernel of this map is $R$. How do I compute the kernel and image of this map directly without resorting to the map $H^1(U)\oplus H^1(V)\to H^1(S^1)$?(i.e. I knew how to compute it this way but not directly through fomer map.)