= Solution
The <exterior derivative> is an $\mathbb R$-linear map $d:\Omega^p(M)\to\Omega^{p+1}(M)$, agrees with the differential $df$ of a <smooth function>, satisfies the <graded Leibniz rule> $d(\alpha\wedge\beta)=d\alpha\wedge\beta+(-1)^p\alpha\wedge d\beta$ for $\alpha\in\Omega^p(M)$, and has $d^2=0$. These properties determine it locally and hence globally. To see locality directly, if a <differential form> $\alpha$ vanishes near $x$, choose a <smooth bump function> $\chi$ equal to one near $x$ and supported where $\alpha=0$. The identity $d(\chi\alpha)=d\chi\wedge\alpha+\chi d\alpha$ gives $(d\alpha)_x=0$. Thus global forms can be computed using local extensions in a <manifold chart>.
In coordinates write $\alpha=\sum_Ia_I\,dx^{i_1}\wedge\cdots\wedge dx^{i_p}$. Since $d(dx^i)=d^2x^i=0$, the <graded Leibniz rule> forces
$$
d\alpha=\sum_{I,j}\partial_ja_I\,dx^j\wedge dx^{i_1}\wedge\cdots\wedge dx^{i_p}.
$$
This is the <uniqueness of the exterior derivative from its axioms>. The coordinate formula also establishes existence: it has the stated properties, and the <chain rule> shows that coordinate changes give the same operator.
The <de Rham cohomology> is the real <quotient vector space>
$$
H^p_{\mathrm{dR}}(M)=\frac{\ker(d:\Omega^p(M)\to\Omega^{p+1}(M))}{\operatorname{im}(d:\Omega^{p-1}(M)\to\Omega^p(M))}.
$$
Thus a <cohomology class> records a <closed differential form> modulo an <exact differential form>. In degree zero there are no exact forms; closed functions are locally constant. The <Poincare lemma> says that every closed form of positive degree on a star-shaped open subset of $\mathbb R^n$ is exact, and hence that positive-degree closed forms are locally exact on a <smooth manifold>.
For the <first de Rham cohomology of the two-sphere>, let $U=S^2\setminus\{N\}$ and $V=S^2\setminus\{S\}$. <Stereographic projection> identifies each with $\mathbb R^2$, so a closed one-form $\alpha$ has primitives $f_U,f_V$. On the connected overlap $U\cap V$, the <derivative> of $f_U-f_V$ is zero, so this difference is a constant. Subtracting that constant from $f_U$ makes the primitives agree. They glue to a global smooth primitive. Therefore \b[$H^1_{\mathrm{dR}}(S^2)=0$.]
Let $q:S^2\to\mathbb{RP}^2$ be the double <covering map> and $a$ the <antipodal map>. The <pullback of a differential form> identifies forms downstairs with $a$-invariant forms upstairs. Averaging $(\eta+a^*\eta)/2$ commutes with $d$. If an invariant form is exact upstairs, averaging its primitive proves it exact downstairs. Conversely an invariant <cohomology class> has an invariant representative by the same averaging. This proves the <de Rham cohomology of a finite quotient> identification $H^p_{\mathrm{dR}}(\mathbb{RP}^2)\cong H^p_{\mathrm{dR}}(S^2)^{a^*}$. In degree zero the sphere is connected and $a^*$ fixes constants; degree one is zero; in degree two the granted action is multiplication by $-1$, whose invariant real subspace is zero. Forms of degree greater than two vanish. Hence
$$
\boxed{H^p_{\mathrm{dR}}(\mathbb{RP}^2)=\begin{cases}\mathbb R,&p=0,\\0,&p>0.\end{cases}}
$$
This is real <de Rham cohomology>; it does not detect the integral two-torsion of the <real projective plane>.
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