View source for Integration on manifolds

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i0517701.png" /> be a finite-dimensional smooth manifold. Tangent spaces and such provide the global analogues of differential calculus. There is also an  "integral calculus on manifolds" . Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i0517702.png" /> be the standard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i0517703.png" />-cube. A singular cube in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i0517704.png" /> is a smooth mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i0517705.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i0517706.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i0517707.png" />-form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i0517708.png" /> (cf. [[Differential form|Differential form]]). Then the integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i0517709.png" /> over a singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177010.png" />-cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177011.png" /> is defined as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177013.png" /> is the unique smooth function such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177014.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177015.png" /> and where on the right-hand side the ordinary Lebesgue integral is taken. A singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177017.png" />-chain is a formal finite sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177018.png" /> of singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177019.png" />-cubes with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177020.png" />. One defines

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177022.png" /> be oriented and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177024.png" /> be two singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177025.png" />-chains such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177026.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177027.png" /> and such that all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177028.png" /> are orientation preserving. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177029.png" />. In particular, if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177030.png" /> fit together to define a piecewise-smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177031.png" />-dimensional submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177033.png" />, then the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177034.png" /> is well-defined.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177035.png" /> denote the exterior derivative on exterior forms (cf. [[Exterior form|Exterior form]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177036.png" /> the (obvious) boundary operator on oriented (singular) chains. Then one has Stokes' theorem

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177038.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177039.png" />-form and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177040.png" /> is a singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177041.png" />-chain. This is the analogue of the fundamental theorem of calculus.

A particular consequence is Green's theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177042.png" /> be a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177043.png" />-dimensional manifold with boundary and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177044.png" /> be differentiable. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177045.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177046.png" /> now be an oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177047.png" />-dimensional Riemannian manifold, i.e. for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177048.png" /> an [[Orientation|orientation]] has been given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177049.png" />. The volume form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177050.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177051.png" /> is now defined by requiring that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177052.png" /> for one (and hence each) orthonormal basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177053.png" /> in the given orientation class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177054.png" />. Another consequence of the general Stokes' theorem (a3) is the divergence theorem:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177056.png" /> is a vector field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177058.png" /> is a three-dimensional oriented manifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177060.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177062.png" /> is an outward normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177063.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177065.png" /> are, respectively, the volume and area elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177067.png" />. The [[Inner product|inner product]] is induced from the standard one in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177068.png" />.

Finally there is the classical Stokes' formula: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177069.png" /> be an oriented two-dimensional submanifold with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177070.png" />. Give <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177071.png" /> an orientation such that together with the outward normal it gives back the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177072.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177073.png" /> parametrize <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177074.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177075.png" /> be the vector field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177076.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177077.png" /> everywhere. One then has the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177078.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>

where the curl of a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177079.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177080.png" /> is defined by:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177081.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051770/i05177082.png" /></td> </tr></table>

All these theorems have higher-dimensional analogues.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,   "Calculus on manifolds" , Benjamin  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Hazewinkel,   "A tutorial introduction to differentiable manifolds and calculus on manifolds"  W. Schiehlen (ed.)  W. Wedig (ed.) , ''Analysis and estimation of stochastic mechanical systems'' , Springer (Wien)  (1988)  pp. 316–340</TD></TR></table>