Multilinear form

In abstract algebra and multilinear algebra, a multilinear form on a vector space  over a field  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  over  is called a (covariant) k-tensor, and the vector space of such forms is usually denoted  or .

 Tensor product 
Given a k-tensor  and an ℓ-tensor , a product , known as the tensor product, can be defined by the property

 

for all .  The tensor product of multilinear forms is not commutative; however it is bilinear and associative:

 , 

and

 

If  forms a basis for an n-dimensional vector space  and  is the corresponding dual basis for the dual space , then the products , with  form a basis for .  Consequently,  has dimensionality .

Examples

Bilinear forms

Main article: Bilinear form

If ,  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

Main article: Alternating multilinear map

An important class of multilinear forms are the alternating multilinear forms, which have the additional property that

 

where  is a permutation and  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.,  and ):

 

With the additional hypothesis that the characteristic of the field  is not 2, setting  implies as a corollary that ; 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 .

An alternating multilinear k-form on  over  is called a  multicovector of degree k or k-covector, and the vector space of such alternating forms, a subspace of , is generally denoted , or, using the notation for the isomorphic kth exterior power of (the dual space of ), .  Note that linear functionals (multilinear 1-forms over ) are trivially alternating, so that , while, by convention, 0-forms are defined to be scalars: .

The determinant on  matrices, viewed as an  argument function of the column vectors, is an important example of an alternating multilinear form.

Wedge 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 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 wedge product is bilinear, associative, and anticommutative: if  and  then . 

Given a basis  for  and dual basis  for , the wedge products , with  form a basis for .  Hence, the dimensionality of  for n-dimensional  is .

Differential forms
Main article: Differential form

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 on tangent spaces for details).      

A differential k-form on  is defined as a function  that assigns to every  a k-covector on the tangent space of at , usually denoted .  In brief, a differential k-form is a k-covector field.  The space of k-forms on  is usually denoted ; thus if  is a differential k-form, we write .  By convention, a continuous function on  is a differential 0-form: .

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 () functions.  Let  be a smooth function.  We define the 1-form  on  for  and  by , where  is the total derivative of  at .  (Recall that the total derivative is a linear transformation.)  Of particular interest are the projection maps (also known as coordinate functions) , defined by , where  is the ith standard coordinate of .  The 1-forms  are known as the basic 1-forms; they are conventionally denoted .  If the standard coordinates of  are , then application of the definition of  yields , so that , where  is the Kronecker delta.  Thus, as the dual of the standard basis for ,  forms a basis for .  As a consequence, if  is a 1-form on , then  can be written as  for smooth functions .  Furthermore, we can derive an expression for  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  as , so that  in terms of the standard basis .  In addition, superscripts appearing in the denominator of an expression (as in ) 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 wedge product () and exterior differentiation () are two fundamental operations on differential forms.  The wedge product of a k-form and an ℓ-form is a -form, while the exterior derivative of a k-form is a -form.  Thus, both operations generate differential forms of higher degree from those of lower degree.

The wedge product  of differential forms is a special case of the wedge product of multicovectors in general (see above).  As is true in general for the wedge product, the wedge product of differential forms is bilinear, associative, and anticommutative.

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 wedge product obeys distributivity with respect to each of these terms.  

The collection of the wedge 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 k-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 k-form , we call  the pullback of  by  and define it as the k-form such that

 

for , where  is the map .

If  is an n-form on  (i.e., ), we define its integral over the unit n-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 n-chain  as the formal sum of n-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
Multilinear map

References

Category:Abstract algebra
Category:Linear algebra
Category:Multilinear algebra