It states that for a $k$-dimensional manifold (with boundary) $M$ and $k-1$-differential form $f$ we have $$\int_{\partial M}f = \int_M df.$$
Most of textbooks spend significant effort to prove the general version and then derive the classical low-dimensional version, which are known for their applications in classical physics. However, I was curious whether the generalized version has some interesting applications, including in mathematics, except proving Brouwer-fixed point theorem.