Multilinear form

In abstract algebra and multilinear algebra, a multilinear form on a vector space <math>V</math> over a field <math>K</math> is a map

:<math>f\colon V^k \to K</math>

that is separately <math>K</math>-linear in each of its <math>k</math> 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.

Multilinear forms <math>V^k \to K</math> are naturally identified with linear forms on the tensor product <math>V^{\otimes k}</math>. Therefore, a multilinear <math>k</math>-form on <math>V</math> over <math>\R</math> is called a (covariant) <math>\boldsymbol{k}</math>-tensor, and the vector space of such forms is usually denoted <math>\mathcal{T}^k(V)</math> or <math>\mathcal{L}^k(V)</math>.

Tensor product of multilinear forms

Given a <math>k</math>-tensor <math>f\in\mathcal{T}^k(V)</math> and an <math>\ell</math>-tensor <math>g\in\mathcal{T}^\ell(V)</math>, a product <math>f\otimes g\in\mathcal{T}^{k+\ell}(V)</math>, known as the tensor product, can be defined by the property

: <math>(f\otimes g)(v_1,\ldots,v_k,v_{k+1},\ldots, v_{k+\ell})=f(v_1,\ldots,v_k)g(v_{k+1},\ldots, v_{k+\ell}),</math>

for all <math>v_1,\ldots,v_{k+\ell}\in V</math>. The tensor product of multilinear forms is not commutative; however, it is bilinear and associative:

: <math>f\otimes(ag_1+bg_2)=a(f\otimes g_1)+b(f\otimes g_2)</math>, <math>(af_1+bf_2)\otimes g=a(f_1\otimes g)+b(f_2\otimes g),</math>

and

: <math>(f\otimes g)\otimes h=f\otimes (g\otimes h).</math>

If <math>(v_1,\ldots, v_n)</math> forms a basis for an <math>n</math>-dimensional vector space <math>V</math> and <math>(\phi^1,\ldots,\phi^n)</math> is the corresponding dual basis for the dual space <math>V^*=\mathcal{T}^1(V)</math>, then the products <math>\phi^{i_1}\otimes\cdots\otimes\phi^{i_k}</math>, with <math>1\le i_1,\ldots,i_k\le n</math> form a basis for <math>\mathcal{T}^k(V)</math>. Consequently, <math>\mathcal{T}^k(V)</math> has dimension <math>n^k</math>.

Examples

Bilinear forms

If <math>k=2</math>, <math>f:V\times V\to K</math> 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

: <math>f(x_{\sigma(1)},\ldots, x_{\sigma(k)}) = \sgn(\sigma)f(x_1,\ldots, x_k), </math>

where <math>\sigma:\mathbf{N}_k\to\mathbf{N}_k</math> is a permutation and <math>\sgn(\sigma)</math> 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., <math>\sigma(p)=q,\sigma(q)=p </math> and <math>\sigma(i)=i, 1\le i\le k, i\neq p,q </math>):

: <math>f(x_1,\ldots, x_p,\ldots, x_q,\ldots, x_k) = -f(x_1,\ldots, x_q,\ldots, x_p,\ldots, x_k). </math>

With the additional hypothesis that the characteristic of the field <math>K</math> is not 2, setting <math>x_p=x_q=x </math> implies as a corollary that <math>f(x_1,\ldots, x,\ldots, x,\ldots, x_k) = 0 </math>; 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 <math>\operatorname{char}(K)\neq 2 </math>.

An alternating multilinear <math>k</math>-form on <math>V</math> over <math>\R</math> is called a multicovector of degree <math>\boldsymbol{k}</math> or <math>\boldsymbol{k}</math>-covector, and the vector space of such alternating forms, a subspace of <math>\mathcal{T}^k(V)</math>, is generally denoted <math>\mathcal{A}^k(V)</math>, or, using the notation for the isomorphic kth exterior power of <math>V^*</math>(the dual space of <math>V</math>), <math display="inline">\bigwedge^k V^*</math>. Note that linear functionals (multilinear 1-forms over <math>\R</math>) are trivially alternating, so that <math>\mathcal{A}^1(V)=\mathcal{T}^1(V)=V^*</math>, while, by convention, 0-forms are defined to be scalars: <math>\mathcal{A}^0(V)=\mathcal{T}^0(V)=\R</math>.

The determinant on <math>n\times n</math> matrices, viewed as an <math>n</math> 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 (<math>\wedge</math>, also known as the wedge product) of multicovectors can be defined, so that if <math>f\in\mathcal{A}^k(V)</math> and <math>g\in\mathcal{A}^\ell(V)</math>, then <math>f\wedge g\in\mathcal{A}^{k+\ell}(V)</math>:

: <math>(f\wedge g)(v_1,\ldots, v_{k+\ell})=\frac{1}{k!\ell!}\sum_{\sigma\in S_{k+\ell (\sgn(\sigma)) f(v_{\sigma(1)}, \ldots, v_{\sigma(k)})g(v_{\sigma(k+1)}

,\ldots,v_{\sigma(k+\ell)}),</math>

where the sum is taken over the set of all permutations over <math>k+\ell</math> elements, <math>S_{k+\ell}</math>. The exterior product is bilinear, associative, and graded-alternating: if <math>f\in\mathcal{A}^k(V)</math> and <math>g\in\mathcal{A}^\ell(V)</math> then <math>f\wedge g=(-1)^{k\ell}g\wedge f</math>.

Given a basis <math>(v_1,\ldots, v_n)</math> for <math>V</math> and dual basis <math>(\phi^1,\ldots,\phi^n)</math> for <math>V^*=\mathcal{A}^1(V)</math>, the exterior products <math>\phi^{i_1}\wedge\cdots\wedge\phi^{i_k}</math>, with <math>1\leq i_1<\cdots<i_k\leq n</math> form a basis for <math>\mathcal{A}^k(V)</math>. Hence, the dimension of <math>\mathcal{A}^k(V)</math> for n-dimensional <math>V</math> is <math display="inline">\tbinom{n}{k}=\frac{n!}{(n-k)!\,k!}</math>.

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). Thus, as the dual of the standard basis for <math>\R^n_p</math>, <math>(dx^1_p,\ldots,dx^n_p)</math> forms a basis for <math>\mathcal{A}^1(\R^n_p)=(\R^n_p)^*</math>. As a consequence, if <math>\omega</math> is a 1-form on <math>U</math>, then <math>\omega</math> can be written as <math display="inline">\sum a_i\,dx^i</math> for smooth functions <math>a_i:U\to\R</math>. Furthermore, we can derive an expression for <math>df</math> that coincides with the classical expression for a total differential:

: <math>df=\sum_{i=1}^n D_i f\; dx^i={\partial f\over\partial x^1} \, dx^1+\cdots+{\partial f\over\partial x^n} \, dx^n.</math>

[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. allows us to state the celebrated Stokes' theorem (Stokes–Cartan theorem) for chains in a subset of <math>\R^m</math>: <blockquote>If <math>\omega</math> is a smooth <math>(n-1)</math>-form on an open set <math>A\subset\R^m</math> and <math>C</math> is a smooth <math>n</math>-chain in <math>A</math>, then<math>\int_C d\omega=\int_{\partial C} \omega</math>.</blockquote>Using more sophisticated machinery (e.g., germs and derivations), the tangent space <math>T_p M</math> of any smooth manifold <math>M</math> (not necessarily embedded in <math>\R^m</math>) can be defined. Analogously, a differential form <math>\omega\in\Omega^k(M)</math> on a general smooth manifold is a map <math>\omega:p\in M\mapsto\omega_p\in \mathcal{A}^k(T_pM)</math>. 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).