unit -cell as the iterated Riemann integral of : Next, we consider a domain of integration parameterized by a differentiable function , known as an n-cube. To define the integral of over , we "pull back" from to the unit n-cell: To integrate over more general domains, we define an -chain as the formal sum of -cubes and set An appropriate definition of the -chain , known as the boundary of , allows us to state the celebrated Stokes' theorem (Stokes–Cartan theorem) for chains in a subset of : If is a smooth -form on an open set and is a smooth -chain in , then.Using more sophisticated machinery (e.g., germs and derivations), the tangent space of any smooth manifold (not necessarily embedded in ) can be defined. Analogously, a differential form on a general smooth manifold is a map . Stokes' theorem can be further generalized to arbitrary smooth manifolds-with-boundary and even certain "rough" domains (see the article on Stokes' theorem for details). See also Bilinear map Exterior algebra Homogeneous polynomial Linear form Multilinear map References Abstract algebra Linear algebra Multilinear algebra