Editing Multilinear form (section)

==== Definition of differential k-forms and construction of 1-forms ====
To define differential forms on open subsets <math>U\subset\R^n</math>, we first need the notion of the '''tangent space''' of <math>\R^n</math>at <math>p</math>, usually denoted <math>T_p\R^n</math> or <math>\R^n_p</math>. The vector space <math>\R^n_p</math> can be defined most conveniently as the set of elements <math>v_p</math> (<math>v\in\R^n</math>, with <math>p\in\R^n</math> fixed) with vector addition and scalar multiplication defined by <math>v_p+w_p:=(v+w)_p</math> and <math>a\cdot(v_p):=(a\cdot v)_p</math>, respectively.  Moreover, if <math>(e_1,\ldots,e_n)</math> is the standard basis for <math>\R^n</math>, then <math>((e_1)_p,\ldots,(e_n)_p)</math> is the analogous standard basis for <math>\R^n_p</math>.  In other words, each tangent space <math>\R^n_p</math> can simply be regarded as a copy of <math>\R^n</math> (a set of tangent vectors) based at the point <math>p</math>. The collection (disjoint union) of tangent spaces of <math>\R^n</math> at all <math>p\in\R^n</math> is known as the '''tangent bundle''' of <math>\R^n</math> and is usually denoted <math display="inline">T\R^n:=\bigcup_{p\in\R^n}\R^n_p</math>.  While the definition given here provides a simple description of the tangent space of <math>\R^n</math>, there are other, more sophisticated constructions that are better suited for defining the tangent spaces of [[Differentiable manifold|smooth manifolds]] in general (''see the article on [[Tangent space|tangent spaces]] for details'').      

A '''differential <math>\boldsymbol{k}</math>-form''' on <math>U\subset\R^n</math> is defined as a function <math>\omega</math> that assigns to every <math>p\in U</math> a <math>k</math>-covector on the tangent space of <math>\R^n</math>at <math>p</math>, usually denoted <math>\omega_p:=\omega(p)\in\mathcal{A}^k(\R^n_p)</math>.  In brief, a differential <math>k</math>-form is a <math>k</math>-covector field.  The space of <math>k</math>-forms on <math>U</math> is usually denoted <math>\Omega^k(U)</math>; thus if <math>\omega</math> is a differential <math>k</math>-form, we write <math>\omega\in\Omega^k(U)</math>.  By convention, a continuous function on <math>U</math> is a differential 0-form: <math>f\in C^0(U)=\Omega^0(U)</math>.

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 [[Smoothness|smooth]] differential forms constructed from smooth (<math>C^\infty</math>) functions.  Let <math>f:\R^n\to\R</math> be a smooth function.  We define the 1-form <math>df</math> on <math>U</math> for <math>p\in U</math> and <math>v_p\in\R^n_p</math> by <math>(df)_p(v_p):=Df|_p(v)</math>, where <math>Df|_p:\R^n\to\R</math> is the [[total derivative]] of <math>f</math> at <math>p</math>.  (Recall that the total derivative is a linear transformation.)  Of particular interest are the projection maps (also known as coordinate functions) <math>\pi^i:\R^n\to\R</math>, defined by <math>x\mapsto x^i</math>, where <math>x^i</math> is the ''i''th standard coordinate of <math>x\in\R^n</math>.  The 1-forms <math>d\pi^i</math> are known as the '''basic 1-forms'''; they are conventionally denoted <math>dx^i</math>.  If the standard coordinates of <math>v_p\in\R^n_p</math> are <math>(v^1,\ldots, v^n)</math>, then application of the definition of <math>df</math> yields <math>dx^i_p(v_p)=v^i</math>, so that <math>dx^i_p((e_j)_p)=\delta_j^i</math>, where <math>\delta^i_j</math> is the [[Kronecker delta]].<ref>The Kronecker delta is usually denoted by <math>\delta_{ij}=\delta(i,j)</math> and defined as <math display="inline">\delta:X\times X\to\{0,1\},\ (i,j)\mapsto \begin{cases} 1, & i=j \\ 0, & i\neq j \end{cases}</math>.  Here, the notation <math>\delta^i_j</math> is used to conform to the tensor calculus convention on the use of upper and lower indices. </ref>  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.<ref name=":0" />  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 <math>v\in\R^n</math> as <math>(v^1,\ldots,v^n)</math>, so that <math display="inline">v=\sum_{i=1}^n v^ie_i</math> in terms of the standard basis <math>(e_1,\ldots,e_n)</math>.  In addition, superscripts appearing in the ''denominator'' of an expression (as in <math display="inline">\frac{\partial f}{\partial x^i}</math>) 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.]