Editing Generalized Stokes theorem (section)

==Formulation for smooth manifolds with boundary==
Let <math>\Omega</math> be an [[oriented manifold|oriented]] [[smooth manifold]] of [[dimension]] <math>n</math> with boundary and let <math>\alpha</math> be a [[smooth function|smooth]] <math>n</math>-[[differential form]] that is [[Support (mathematics)#Compact support|compactly supported]] on <math>\Omega</math>. First, suppose that <math>\alpha</math> is compactly supported in the domain of a single, oriented [[coordinate chart]] <math>\{U,\varphi\}</math>. In this case, we define the integral of <math>\alpha</math> over <math>\Omega</math> as
<math display="block">\int_\Omega \alpha = \int_{\varphi(U)} (\varphi^{-1})^* \alpha\,,</math>
i.e., via the [[Pullback (differential geometry)|pullback]] of <math>\alpha</math> to <math>\R^n</math>.

More generally, the integral of <math>\alpha</math> over <math>\Omega</math> is defined as follows: Let <math>\{\psi_i\}</math> be a [[partition of unity]] associated with a [[locally finite collection|locally finite]] [[cover (topology)|cover]] <math>\{U_i,\varphi_i\}</math> of (consistently oriented) coordinate charts, then define the integral
<math display="block">\int_\Omega \alpha \equiv \sum_i \int_{U_i} \psi_i \alpha\,,</math>
where each term in the sum is evaluated by pulling back to <math>\R^n</math> as described above.  This quantity is well-defined; that is, it does not depend on the choice of the coordinate charts, nor the partition of unity.

The generalized Stokes theorem reads:
{{math theorem
| note = ''Stokes–Cartan''
| math_statement = Let <math>\omega</math> be a [[infinitely differentiable|smooth]] <math>(n-1)</math>-[[differential form|form]] with [[compact support]] on an [[oriented]], <math>n</math>-dimensional [[manifold|manifold-with-boundary]] <math>\Omega</math>, where <math>\partial \Omega</math> is given the induced orientation. Then <math display="block">\int_{\Omega} d\omega = \int_{\partial\Omega} \omega.</math>
}}

Here <math>d</math> is the [[exterior derivative]], which is defined using the manifold structure only.  The right-hand side is sometimes written as <math display="inline">\oint_{\partial\Omega} \omega</math> to stress the fact that the <math>(n-1)</math>-manifold <math>\partial\Omega</math> has no boundary.<ref group="note">For  mathematicians this fact is known, therefore the circle is redundant and often omitted.  However, one should keep in mind here that in [[thermodynamics]], where frequently expressions as <math>\oint_W\{\text{d}_\text{total}U\}</math> appear (wherein the total derivative, see below, should not be confused with the exterior one), the integration path <math>W</math> is a one-dimensional closed line on a much higher-dimensional manifold. That is, in a thermodynamic application, where <math>U</math> is a function of the temperature <math>\alpha_1=T</math>, the volume <math>\alpha_2=V</math>, and the electrical polarization <math>\alpha_3=P</math> of the sample, one has
<math display="block">\{d_\text{total}U\} = \sum_{i=1}^3\frac{\partial U}{\partial\alpha_i}\,d\alpha_i\,,</math>
and the circle is really necessary, e.g. if one considers the ''differential'' consequences of the ''integral'' postulate
<math display="block">\oint_W\,\{d_\text{total}U\}\, \stackrel{!}{=}\,0\,.</math></ref> (This fact is also an implication of Stokes' theorem, since for a given smooth <math>n</math>-dimensional manifold <math>\Omega</math>, application of the theorem twice gives <math display="inline">\int_{\partial(\partial \Omega)}\omega=\int_\Omega d(d\omega)=0</math> for any <math>(n-2)</math>-form <math>\omega</math>, which implies that <math>\partial(\partial\Omega)=\emptyset</math>.) The right-hand side of the equation is often used to formulate ''integral'' laws; the left-hand side then leads to equivalent ''differential'' formulations (see below).

The theorem is often used in situations where <math>\Omega</math> is an embedded oriented submanifold of some bigger manifold, often <math>\R^k</math>, on which the form <math>\omega</math> is defined.