Let $S^n(r)$ be the sphere of radius $r$ , $x_1^2 + … + x_{n+1}^2 = r^2$ and let $$w = \frac{1}{r} \sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \cdots\hat{dx_i},\cdots dx_{n+1} $$
Write $S^n$ for the unit sphere $S^n(1)$. Compute the integral $$\int\limits_{S^n}w$$ and conclude that $w$ is not exact.
For me this is a hard exercise. I understand that from (a) we obtain an explicit formula for the generator of the top cohomology of $S^n$ (although not as a bump form).
Best Answer
We have
$$w = \frac{1}{r} \sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \cdots\hat{dx_i},\cdots dx_{n+1} \ ,$$
thus
$$dw = \frac{1}{r} \sum_{i=1}^{n+1} (-1)^{i-1} dx_i dx_1 \cdots\hat{dx_i},\cdots dx_{n+1} = \frac{1}{r} \sum_{i=1}^{n+1} dx_1\cdots dx_{n+1} = \frac{n+1}{r} dx_1 \cdots dx_{n+1}\ .$$
Note that the second equality holds because in commuting those $dx_j$ we have the rule
$$dx_i dx_j = - dx_j dx_i\ .$$
Now by Stokes theorem, as $\mathbb S^n = \partial B^{n+1}$,
$$\int_{\mathbb S^n} w = \int_{\partial B^{n+1}} w = \int_{\mathbb B^{n+1}} dw = (n+1) V(B^{n+1})\neq 0$$
Thus $w$ when restricted to $\mathbb S^n$ is not exact (that it is not exact in $\mathbb R^{n+1}$ is obvious as $dw \neq 0$). The reason is again by Stokes theorem: if $w = d\alpha$, then
$$\int_{\mathbb S^n} w = \int_{\mathbb S^n} d\alpha = 0\ .$$
But we have seen that this is not zero.