The essence of integration of a differential \(k\)-form \(\int_{\Omega} \omega\) is to discretize the domain into infinitesimal patches, project the \(k\)-form onto each of them, then take the sum. Since there is projection, there is the inner product operation, which naturally involves these dual pairs: vector and covector, \(k\)-vector and \(k\)-form, \(k\)-vector field and differential \(k\)-form.