on differential k-forms The exterior product () and exterior derivative () are two fundamental operations on differential forms. The exterior product of a -form and an -form is a -form, while the exterior derivative of a -form is a -form. Thus, both operations generate differential forms of higher degree from those of lower degree. The exterior product of differential forms is a special case of the exterior product of multicovectors in general (see above). As is true in general for the exterior product, the exterior product of differential forms is bilinear, associative, and is graded-alternating. More concretely, if and , then Furthermore, for any set of indices , If , , and , then the indices of can be arranged in ascending order by a (finite) sequence of such swaps. Since , implies that . Finally, as a consequence of bilinearity, if and are the sums of several terms, their exterior product obeys distributivity with respect to each of these terms. The collection of the exterior products of basic 1-forms constitutes a basis for the space of differential k-forms. Thus, any can be written in the form where are smooth functions. With each set of indices placed in ascending order, (*) is said to be the standard presentation of . In the previous section, the 1-form was defined by taking the exterior derivative of the 0-form (continuous function) . We now extend this by defining the exterior derivative operator for . If the standard presentation of -form is given by (*), the -form is defined by A property of that holds for all smooth forms is that the second exterior derivative of any vanishes identically: . This can be established directly from the definition of and the equality of mixed second-order partial derivatives of functions (see the article on closed and exact forms for details). Integration of differential forms and Stokes' theorem for chains To integrate a differential form over a parameterized domain, we first need to introduce the notion of the pullback of a differential form. Roughly speaking, when a differential form is integrated, applying the pullback transforms it in a way that correctly accounts for a change-of-coordinates. Given a differentiable function and -form , we call the pullback of by and define it as the -form such that for , where is the map . If is an -form on (i.e., ), we define its integral over the