Multilinear form

In abstract algebra and multilinear algebra, a multilinear form on a vector space formula_1 over a field formula_2 is a map
that is separately "K"-linear in each of its "k" arguments. More generally, one can define multilinear forms on a module over a commutative ring. The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces.
A multilinear "k"-form on formula_1 over formula_5 is called a (covariant) "k"-tensor, and the vector space of such forms is usually denoted formula_6 or formula_7.
Tensor product.
Given a "k"-tensor formula_8 and an "ℓ"-tensor formula_9, a product formula_10, known as the tensor product, can be defined by the property
for all formula_12. The tensor product of multilinear forms is not commutative; however it is bilinear and associative:
and
If formula_16 forms a basis for an "n"-dimensional vector space formula_1 and formula_18 is the corresponding dual basis for the dual space formula_19, then the products formula_20, with formula_21 form a basis for formula_6. Consequently, formula_6 has dimensionality formula_24.
Examples.
Bilinear forms.
If formula_25, formula_26 is referred to as a bilinear form. A familiar and important example of a (symmetric) bilinear form is the standard inner product (dot product) of vectors.
Alternating multilinear forms.
An important class of multilinear forms are the alternating multilinear forms, which have the additional property that
where formula_28 is a permutation and formula_29 denotes its sign (+1 if even, –1 if odd). As a consequence, alternating multilinear forms are antisymmetric with respect to swapping of any two arguments (i.e., formula_30 and formula_31):
With the additional hypothesis that the characteristic of the field formula_2 is not 2, setting formula_34 implies as a corollary that formula_35; that is, the form has a value of 0 whenever two of its arguments are equal. Note, however, that some authors use this last condition as the defining property of alternating forms. This definition implies the property given at the beginning of the section, but as noted above, the converse implication holds only when formula_36.
An alternating multilinear "k"-form on formula_1 over formula_5 is called a multicovector of degree "k or k"-covector, and the vector space of such alternating forms, a subspace of formula_6, is generally denoted formula_40, or, using the notation for the isomorphic "k"th exterior power of formula_41(the dual space of formula_1), formula_43. Note that linear functionals (multilinear 1-forms over formula_5) are trivially alternating, so that formula_45, while, by convention, 0-forms are defined to be scalars: formula_46.
The determinant on formula_47 matrices, viewed as an formula_48 argument function of the column vectors, is 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" (formula_49, also known as the "wedge product") of multicovectors can be defined, so that if formula_50 and formula_51, then formula_52:
where the sum is taken over the set of all permutations over formula_54 elements, formula_55. The exterior product is bilinear, associative, and graded-alternating: if formula_50 and formula_51 then formula_58. 
Given a basis formula_16 for formula_1 and dual basis formula_18 for formula_62, the exterior products formula_63, with formula_64 form a basis for formula_40. Hence, the dimensionality of formula_40 for "n"-dimensional formula_1 is formula_68.
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 formula_69, we first need the notion of the tangent space of formula_70at formula_71, usually denoted formula_72 or formula_73. The vector space formula_73 can be defined most conveniently as the set of elements formula_75 (formula_76, with formula_77 fixed) with vector addition and scalar multiplication defined by formula_78 and formula_79, respectively. Moreover, if formula_80 is the standard basis for formula_70, then formula_82 is the analogous standard basis for formula_73. In other words, each tangent space formula_73 can simply be regarded as a copy of formula_70 (a set of tangent vectors) based at the point formula_71. The collection (disjoint union) of tangent spaces of formula_70 at all formula_77 is known as the tangent bundle of formula_70 and is usually denoted formula_90. While the definition given here provides a simple description of the tangent space of formula_70, there are other, more sophisticated constructions that are better suited for defining the tangent spaces of smooth manifolds in general ("see the article on tangent spaces for details"). 
A differential "k"-form on formula_69 is defined as a function formula_93 that assigns to every formula_94 a "k"-covector on the tangent space of formula_70at formula_71, usually denoted formula_97. In brief, a differential "k-"form is a "k"-covector field. The space of "k"-forms on formula_98 is usually denoted formula_99; thus if formula_93 is a differential "k"-form, we write formula_101. By convention, a continuous function on formula_98 is a differential 0-form: formula_103.
We first construct differential 1-forms from 0-forms and deduce some of their basic properties. To simplify the discussion below, we will only consider smooth differential forms constructed from smooth (formula_104) functions. Let formula_105 be a smooth function. We define the 1-form formula_106 on formula_98 for formula_94 and formula_109 by formula_110, where formula_111 is the total derivative of formula_112 at formula_71. (Recall that the total derivative is a linear transformation.) Of particular interest are the projection maps (also known as coordinate functions) formula_114, defined by formula_115, where formula_116 is the "i"th standard coordinate of formula_117. The 1-forms formula_118 are known as the basic 1-forms; they are conventionally denoted formula_119. If the standard coordinates of formula_109 are formula_121, then application of the definition of formula_106 yields formula_123, so that formula_124, where formula_125 is the Kronecker delta. Thus, as the dual of the standard basis for formula_73, formula_127 forms a basis for formula_128. As a consequence, if formula_93 is a 1-form on formula_98, then formula_93 can be written as formula_132 for smooth functions formula_133. Furthermore, we can derive an expression for formula_106 that coincides with the classical expression for a total differential:
["Comments on" "notation:" In this article, we follow the convention from tensor calculus and differential geometry in which multivectors and multicovectors are written with lower and upper indices, respectively. Since differential forms are multicovector fields, upper indices are employed to index them. The opposite rule applies to the "components" of multivectors and multicovectors, which instead are written with upper and lower indices, respectively. For instance, we represent the standard coordinates of vector formula_76 as formula_137, so that formula_138 in terms of the standard basis formula_80. In addition, superscripts appearing in the "denominator" of an expression (as in formula_140) are treated as lower indices in this convention. When indices are applied and interpreted in this manner, the number of upper indices minus the number of lower indices in each term of an expression is conserved, both within the sum and across an equal sign, a feature that serves as a useful mnemonic device and helps pinpoint errors made during manual computation.]
Basic operations on differential "k"-forms.
The "exterior product" (formula_49) and "exterior derivative" (formula_142) are two fundamental operations on differential forms. The exterior product of a "k"-form and an "ℓ"-form is a formula_143-form, while the exterior derivative of a "k"-form is a formula_144-form. Thus, both operations generate differential forms of higher degree from those of lower degree.
The exterior product formula_145 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 formula_146 and formula_147, then
Furthermore, for any set of indices formula_149,
If formula_151, formula_152, and formula_153, then the indices of formula_154 can be arranged in ascending order by a (finite) sequence of such swaps. Since formula_155, formula_156 implies that formula_157. Finally, as a consequence of bilinearity, if formula_93 and formula_159 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 formula_160 constitutes a basis for the space of differential "k"-forms. Thus, any formula_101 can be written in the form
where formula_163 are smooth functions. With each set of indices formula_164 placed in ascending order, (*) is said to be the standard presentation of formula_93. <br>
In the previous section, the 1-form formula_106 was defined by taking the exterior derivative of the 0-form (continuous function) formula_112. We now extend this by defining the exterior derivative operator formula_168 for formula_169. If the standard presentation of "k"-form formula_93 is given by (*), the formula_144-form formula_172 is defined by
A property of formula_142 that holds for all smooth forms is that the second exterior derivative of any formula_93 vanishes identically: formula_176. This can be established directly from the definition of formula_142 and the equality of mixed second-order partial derivatives of formula_178 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 formula_179 and "k"-form formula_180, we call formula_181 the pullback of formula_159 by formula_112 and define it as the "k"-form such that
for formula_185, where formula_186 is the map formula_187.
If formula_188 is an "n"-form on formula_70 (i.e., formula_190), we define its integral over the unit "n"-cell as the iterated Riemann integral of formula_112:
Next, we consider a domain of integration parameterized by a differentiable function formula_193, known as an "n"-cube. To define the integral of formula_194 over formula_195, we "pull back" from formula_196 to the unit "n"-cell:
To integrate over more general domains, we define an "n-"chain formula_198 as the formal sum of "n"-cubes and set
An appropriate definition of the formula_200-chain formula_201, known as the boundary of formula_202, allows us to state the celebrated Stokes' theorem (Stokes–Cartan theorem) for chains in a subset of formula_203: "If formula_93 is a" "smooth" formula_200"-form on an open set formula_206" "and formula_202" "is a smooth" formula_48"-chain in formula_196, thenformula_210."Using more sophisticated machinery (e.g., germs and derivations), the tangent space formula_211 of any smooth manifold formula_212 (not necessarily embedded in formula_203) can be defined. Analogously, a differential form formula_214 on a general smooth manifold is a map formula_215. 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").