Cohomology-Computations 1

Let $U,V$ be open cover of circle - $S^1$. Let $X,Y$ be arcs on the circle that correspond to the open cover with disjoint overlaps at top of the circle and at the bottom of the circle.

Each arc $X,Y$ is diffeomorphic to an interval and thus to $\mathcal{R}$.

Here instead of writing one forms and seeking existence of integral solutions, we can use Mayer-Vietoris sequence to make computations. The Mayer-Vietoris sequence for $S^1$ is as follows:
$$0\to H^0(M)\stackrel{i^{\ast }}{\to }{H}^0(U)\oplus H^0(V)\stackrel{j^{\ast }}{\to }{H}^0(U\cap V)\stackrel{d^{\ast }}{\to }{H}^1(M)\to 0.$$
Using dimensional formula for sequence of vector spaces $\sum_{k=0}^n(-1)^k d^k$, we can figure out dimension of $H^1(M)=d^1=1$ as follows:
$$1-2+2-d^1=0$$
Since $S^1$ is connected, $H^0(S^1)=\mathcal{R}$. As shown before $H^0(U)=H^0(V)=\mathcal{R}$. Since overlaps are disjoint we have $\mathcal{R} \oplus \mathcal{R}$. All this results in the following sequence.
$$0\to \mathcal{R}\stackrel{i^{\ast }}{\to }\mathcal{R}\oplus \mathcal{R}\stackrel{j^{\ast }}{\to }\mathcal{R}\oplus \mathcal{R}\to 0$$
Notice $j^*:H^{0}(U) \oplus H^{0}(V) \xrightarrow{j^*} H^{0}(U \cap V)$ is given as follows. Since, we are dealing with $0$ dimensional space, corresponding vectors are $0$ dimensionals - that is scalars or real numbers.
$$j^{\ast }(m,n)=(n-m,n-m)$$
That is these are elements of diagonal in $\mathcal{R}\times\mathcal{R}$. $d^*: H^{0}(U \cap V) \xrightarrow{d^*} H^{1}(M)$ sends this to one dimensional space which is isomorphic to $\mathcal{R}$, hence points $m \neq n$ will result in this element. Among many, we can choose one.