an important example of an alternating multilinear form. Exterior product The tensor product of alternating multilinear forms is, in general, no longer alternating. However, by summing over all permutations of the tensor product, taking into account the parity of each term, the exterior product (, also known as the wedge product) of multicovectors can be defined, so that if and , then : where the sum is taken over the set of all permutations over elements, . The exterior product is bilinear, associative, and graded-alternating: if and then . Given a basis for and dual basis for , the exterior products , with form a basis for . Hence, the dimensionality of for n-dimensional is . Differential forms Differential forms are mathematical objects constructed via tangent spaces and multilinear forms that behave, in many ways, like differentials in the classical sense. Though conceptually and computationally useful, differentials are founded on ill-defined notions of infinitesimal quantities developed early in the history of calculus. Differential forms provide a mathematically rigorous and precise framework to modernize this long-standing idea. Differential forms are especially useful in multivariable calculus (analysis) and differential geometry because they possess transformation properties that allow them be integrated on curves, surfaces, and their higher-dimensional analogues (differentiable manifolds). One far-reaching application is the modern statement of Stokes' theorem, a sweeping generalization of the fundamental theorem of calculus to higher dimensions. The synopsis below is primarily based on Spivak (1965) and Tu (2011). Definition of differential k-forms and construction of 1-forms To define differential forms on open subsets , we first need the notion of the tangent space of at , usually denoted or . The vector space can be defined most conveniently as the set of elements (, with fixed) with vector addition and scalar multiplication defined by and , respectively. Moreover, if is the standard basis for , then is the analogous standard basis for . In other words, each tangent space can simply be regarded as a copy of (a set of tangent vectors) based at the point . The collection (disjoint union) of tangent spaces of at all is known as the tangent bundle of and is usually denoted . While the definition given here provides a simple description of the tangent space of , there are other, more sophisticated constructions that are better suited for defining the tangent spaces of smooth manifolds in general (see the article