Editing Differential form (section)

=== Integration and orientation ===
A differential {{mvar|k}}-form can be integrated over an oriented [[manifold (mathematics)|manifold]] of dimension {{mvar|k}}.  A differential {{math|1}}-form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density.  A differential {{math|2}}-form can be thought of as measuring an infinitesimal oriented area, or 2-dimensional oriented density.  And so on.

Integration of differential forms is well-defined only on [[Orientability|oriented]] [[manifold (mathematics)|manifolds]].  An example of a 1-dimensional manifold is an interval {{math|[''a'', ''b'']}}, and intervals can be given an orientation: they are positively oriented if {{math|''a'' < ''b''}}, and negatively oriented otherwise.  If {{math|''a'' < ''b''}} then the integral of the differential {{math|1}}-form {{math|''f''(''x'') ''dx''}} over the interval {{math|[''a'', ''b'']}} (with its natural positive orientation) is
<math display="block">\int_a^b f(x) \,dx</math>
which is the negative of the integral of the same differential form over the same interval, when equipped with the opposite orientation.  That is:
<math display="block">\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx.</math>
This gives a geometrical context to the [[Integral#Conventions|conventions]] for one-dimensional integrals, that the sign changes when the orientation of the interval is reversed.  A standard explanation of this in one-variable integration theory is that, when the limits of integration are in the opposite order ({{math|''b'' < ''a''}}), the increment {{math|''dx''}} is negative in the direction of integration.

More generally, an {{mvar|m}}-form is an oriented density that can be integrated over an {{mvar|m}}-dimensional oriented manifold.  (For example, a {{math|1}}-form can be integrated over an oriented curve, a {{math|2}}-form can be integrated over an oriented surface, etc.)  If {{mvar|M}} is an oriented {{mvar|m}}-dimensional manifold, and {{math|''M''{{′}}}} is the same manifold with opposite orientation and {{mvar|ω}} is an {{mvar|m}}-form, then one has:
<math display="block">\int_M \omega = - \int_{M'} \omega \,.</math>
These conventions correspond to interpreting the integrand as a differential form, integrated over a [[Chain (algebraic topology)|chain]]. In [[measure theory]], by contrast, one interprets the integrand as a function {{mvar|f}} with respect to a measure {{mvar|μ}} and integrates over a subset {{mvar|A}}, without any notion of orientation; one writes <math display="inline">\int_A f\,d\mu = \int_{[a,b]} f\,d\mu</math> to indicate integration over a subset {{mvar|A}}. This is a minor distinction in one dimension, but becomes subtler on higher-dimensional manifolds; see [[#Relation with measures|below]] for details.

Making the notion of an oriented density precise, and thus of a differential form, involves the [[exterior algebra]].  The differentials of a set of coordinates, {{math|''dx''{{sup|1}}}}, ..., {{math|''dx''{{i sup|''n''}}}} can be used as a basis for all {{math|1}}-forms.  Each of these represents a [[covector]] at each point on the manifold that may be thought of as measuring a small displacement in the corresponding coordinate direction.  A general {{math|1}}-form is a linear combination of these differentials at every point on the manifold:
<math display="block">f_1\,dx^1+\cdots+f_n\,dx^n ,</math>
where the {{math|1=''f''{{sub|''k''}} = ''f''{{sub|''k''}}(''x''{{sup|1}}, ... , ''x''{{sup|''n''}})}} are functions of all the coordinates.  A differential {{math|1}}-form is integrated along an oriented curve as a line integral.

The expressions {{math|''dx''{{i sup|''i''}} ∧ ''dx''{{i sup|''j''}}}}, where {{math|''i'' < ''j''}} can be used as a basis at every point on the manifold for all {{math|2}}-forms.  This may be thought of as an infinitesimal oriented square parallel to the {{math|''x''{{i sup|''i''}}}}–{{math|''x''{{i sup|''j''}}}}-plane.  A general {{math|2}}-form is a linear combination of these at every point on the manifold: {{nowrap|<math display="inline">\sum_{1 \leq i<j \leq n} f_{i,j} \, dx^i \wedge dx^j</math>,}} and it is integrated just like a surface integral.

A fundamental operation defined on differential forms is the [[exterior product]] (the symbol is the [[Wedge (symbol)|wedge]] {{math|∧}}).  This is similar to the [[cross product]] from vector calculus, in that it is an alternating product. For instance,
<math display="block">dx^1\wedge dx^2=-dx^2\wedge dx^1</math>

because the square whose first side is {{math|''dx''{{sup|1}}}} and second side is {{math|''dx''{{sup|2}}}} is to be regarded as having the opposite orientation as the square whose first side is {{math|''dx''<sup>2</sup>}} and whose second side is {{math|''dx''{{sup|1}}}}.  This is why we only need to sum over expressions {{math|''dx''{{i sup|''i''}} ∧ ''dx''{{i sup|''j''}}}}, with {{math|''i'' < ''j''}}; for example: {{math|1=''a''(''dx''{{i sup|''i''}} ∧ ''dx''{{i sup|''j''}}) + ''b''(''dx''{{i sup|''j''}} ∧ ''dx''{{i sup|''i''}}) = (''a'' − ''b'') ''dx''{{i sup|''i''}} ∧ ''dx''{{i sup|''j''}}}}.  The exterior product allows higher-degree differential forms to be built out of lower-degree ones, in much the same way that the [[cross product]] in vector calculus allows one to compute the area vector of a parallelogram from vectors pointing up the two sides. Alternating also implies that {{math|1=''dx''{{i sup|''i''}} ∧ ''dx''{{i sup|''i''}} = 0}}, in the same way that the cross product of parallel vectors, whose magnitude is the area of the parallelogram spanned by those vectors, is zero. In higher dimensions, {{math|1=''dx''{{i sup|''i''{{sub|1}}}} ∧ ⋅⋅⋅ ∧ ''dx''{{i sup|''i''{{sub|''m''}}}} = 0}} if any two of the indices {{math|''i''<sub>1</sub>}}, ..., {{math|''i''<sub>''m''</sub>}} are equal, in the same way that the "volume" enclosed by a [[Parallelepiped#Parallelotope|parallelotope]] whose edge vectors are [[Linear independence|linearly dependent]] is zero.