Editing Multilinear form (section)

==== 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 <math>f:\R^n\to\R^m</math> and <math>k</math>-form <math>\eta\in\Omega^k(\R^m)</math>, we call <math>f^*\eta\in\Omega^k(\R^n)</math> the [[Pullback (differential geometry)|pullback]] of <math>\eta</math> by <math>f</math> and define it as the <math>k</math>-form such that

: <math>(f^*\eta)_p(v_{1p},\ldots, v_{kp}):=\eta_{f(p)}(f_*(v_{1p}),\ldots,f_*(v_{kp})),</math>

for <math>v_{1p},\ldots,v_{kp}\in\R^n_p</math>, where <math>f_*:\R^n_p\to\R^m_{f(p)}</math> is the map <math>v_p\mapsto(Df|_p(v))_{f(p)}</math>.

If <math>\omega=f\, dx^1\wedge\cdots\wedge dx^n</math> is an <math>n</math>-form on <math>\R^n</math> (i.e., <math>\omega\in\Omega^n(\R^n)</math>), we define its integral over the unit <math>n</math>-cell as the iterated Riemann integral of <math>f</math>:

: <math>\int_{[0,1]^n} \omega = \int_{[0,1]^n} f\,dx^1\wedge\cdots \wedge dx^n:= \int_0^1\cdots\int_0^1 f\, dx^1\cdots dx^n.</math>

Next, we consider a domain of integration parameterized by a differentiable function <math>c:[0,1]^n\to A\subset\R^m</math>, known as an '''''n''-cube'''.  To define the integral of <math>\omega\in\Omega^n(A)</math> over <math>c</math>, we "pull back" from <math>A</math> to the unit ''n''-cell:

: <math>\int_c \omega :=\int_{[0,1]^n}c^*\omega.</math>

To integrate over more general domains, we define an '''<math>\boldsymbol{n}</math>-chain <math display="inline">C=\sum_i n_ic_i</math>''' as the formal sum of <math>n</math>-cubes and set

: <math>\int_C \omega :=\sum_i n_i\int_{c_i} \omega.</math>

An appropriate definition of the <math>(n-1)</math>-[[Chain (algebraic topology)|chain]] <math>\partial C</math>, known as the boundary of <math>C</math>,<ref>The formal definition of the boundary of a chain is somewhat involved and is omitted here (''see {{harvnb|Spivak|1965|pp=98–99}} for a discussion'').  Intuitively, if <math>C</math> maps to a square, then <math>\partial C</math> is a linear combination of functions that maps to its edges in a counterclockwise manner.  The boundary of a chain is distinct from the notion of a boundary in point-set topology.</ref> 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., [[Germ (mathematics)|germs]] and [[Derivation (differential algebra)|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'').