De Rham Cohomology of Sphere

Theorem
Let $S^n$ denote the n-Sphere.

Then the de Rham Cohomology of $S^n$ are:


 * $\map {H^0} {S^0} = \Z^2$


 * $\map {H^k} {S^n} = \begin {cases} \Z & : k = 0, n \\ 0 & : 0 < k < n \end {cases}$ for $n > 0$