Editing Differential form (section)

== Concept ==

Differential forms provide an approach to [[multivariable calculus]] that is independent of [[coordinate]]s.

=== 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.

=== Multi-index notation ===
A common notation for the wedge product of elementary {{mvar|k}}-forms is so called [[multi-index notation]]: in an {{mvar|n}}-dimensional context, for {{nowrap|<math>I = (i_1, i_2,\ldots , i_k), 1 \leq i_1 < i_2 < \cdots < i_k \leq n</math>,}} we define {{nowrap|<math display="inline">dx^I := dx^{i_1} \wedge \cdots \wedge dx^{i_k} = \bigwedge_{i\in I} dx^i</math>.}}<ref>{{Cite book|title=An introduction to manifolds|last=Tu, Loring W.|date=2011|publisher=Springer|isbn=9781441974006|edition= 2nd|location=New York|oclc=682907530}}</ref> Another useful notation is obtained by defining the set of all strictly increasing multi-indices of length {{mvar|k}}, in a space of dimension {{mvar|n}}, denoted {{nowrap|<math>\mathcal{J}_{k,n} := \{I=(i_1,\ldots,i_k):1\leq i_1<i_2<\cdots<i_k\leq n\}</math>.}} Then locally (wherever the coordinates apply), <math>\{dx^I\}_{I \in \mathcal{J}_{k,n}}</math> spans the space of differential {{mvar|k}}-forms in a manifold {{mvar|M}} of dimension {{mvar|n}}, when viewed as a module over the ring {{math|''C''{{sup|∞}}(''M'')}} of smooth functions on {{mvar|M}}. By calculating the size of <math>\mathcal{J}_{k,n}</math> combinatorially, the module of {{mvar|k}}-forms on an {{mvar|n}}-dimensional manifold, and in general space of {{mvar|k}}-covectors on an {{mvar|n}}-dimensional vector space, is {{mvar|n}}&nbsp;choose&nbsp;{{mvar|k}}: {{nowrap|<math display="inline"> |\mathcal{J}_{k,n}| = \binom{n}{k}</math>.}} This also demonstrates that there are no nonzero differential forms of degree greater than the dimension of the underlying manifold.

=== The exterior derivative ===
In addition to the exterior product, there is also the [[exterior derivative]] operator {{math|''d''}}.  The exterior derivative of a differential form is a generalization of the [[differential of a function]], in the sense that the exterior derivative of {{math|1=''f'' ∈ ''C''{{sup|∞}}(''M'') = Ω{{sup|0}}(''M'')}} is exactly the differential of {{mvar|f}}. When generalized to higher forms, if {{math|1=''ω'' = ''f'' ''dx''{{i sup|''I''}}}} is a simple {{mvar|k}}-form, then its exterior derivative {{math|''dω''}} is a {{math|(''k'' + 1)}}-form defined by taking the differential of the coefficient functions:
<math display="block">d\omega = \sum_{i=1}^n \frac{\partial f}{\partial x^i} \, dx^i \wedge dx^I.</math>
with extension to general {{mvar|k}}-forms through linearity: if {{nowrap|<math display="inline">\tau = \sum_{I \in \mathcal{J}_{k,n}} a_I \, dx^I \in \Omega^k(M)</math>,}} then its exterior derivative is
<math display="block">d\tau = \sum_{I \in \mathcal{J}_{k,n}}\left(\sum_{j=1}^n \frac{\partial a_I}{\partial x^j} \, dx^j\right)\wedge dx^I \in \Omega^{k+1}(M)</math>

In {{math|'''R'''<sup>3</sup>}}, with the [[Hodge star operator]], the exterior derivative corresponds to [[gradient]], [[Curl (mathematics)|curl]], and [[divergence]], although this correspondence, like the cross product, does not generalize to higher dimensions, and should be treated with some caution. 

The exterior derivative itself applies in an arbitrary finite number of dimensions, and is a flexible and powerful tool with wide application in [[differential geometry]], [[differential topology]], and many areas in physics. Of note, although the above definition of the exterior derivative was defined with respect to local coordinates, it can be defined in an entirely coordinate-free manner, as an [[Derivation (differential algebra)|antiderivation]] of degree 1 on the [[exterior algebra]] of differential forms. The benefit of this more general approach is that it allows for a natural coordinate-free approach to integrate on [[manifold]]s. It also allows for a natural generalization of the [[fundamental theorem of calculus]], called the (generalized) [[Stokes' theorem]], which is a central result in the theory of integration on manifolds.

=== Differential calculus ===
Let {{mvar|U}} be an [[open set]] in {{math|'''R'''<sup>''n''</sup>}}. A differential {{math|0}}-form ("zero-form") is defined to be a [[smooth function]] {{mvar|f}} on {{mvar|U}} – the set of which is denoted {{math|''C''{{sup|∞}}(''U'')}}. If {{math|''v''}} is any vector in {{math|'''R'''<sup>''n''</sup>}}, then {{math|''f''}} has a [[directional derivative]] {{math|∂<sub>'''v'''</sub> ''f''}}, which is another function on {{mvar|U}} whose value at a point {{math|''p'' ∈ ''U''}} is the rate of change (at {{mvar|p}}) of {{mvar|f}} in the {{math|'''v'''}} direction:
<math display="block"> (\partial_\mathbf{v} f)(p) = \left. \frac{d}{dt} f(p+t\mathbf{v})\right|_{t=0} .</math>
(This notion can be extended pointwise to the case that {{math|'''v'''}} is a [[vector field]] on {{mvar|U}} by evaluating {{math|'''v'''}} at the point {{mvar|p}} in the definition.)

In particular, if {{math|1='''v''' = '''e'''{{sub|''j''}}}} is the {{mvar|j}}th [[coordinate vector]] then {{math|∂{{sub|'''v'''}} ''f''}} is the [[partial derivative]] of {{mvar|f}} with respect to the {{mvar|j}}th coordinate vector, i.e., {{math|∂''f'' / ∂''x''{{i sup|''j''}}}}, where {{math|''x''{{i sup|1}}}}, {{math|''x''{{i sup|2}}}}, ..., {{math|''x''{{i sup|''n''}}}} are the coordinate vectors in {{mvar|U}}. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates {{math|''y''{{i sup|1}}}}, {{math|''y''{{i sup|2}}}}, ..., {{math|''y''{{i sup|''n''}}}} are introduced, then
<math display="block">\frac{\partial f}{\partial x^j} = \sum_{i=1}^n\frac{\partial y^i}{\partial x^j}\frac{\partial f}{\partial y^i} .</math>

The first idea leading to differential forms is the observation that {{math|∂<sub>'''v'''</sub> ''f'' (''p'')}} is a [[linear function]] of {{math|''v''}}:

<math display="block">\begin{align}
  (\partial_{\mathbf{v} + \mathbf{w}} f)(p) &= (\partial_\mathbf{v} f)(p) + (\partial_\mathbf{w} f)(p) \\
    (\partial_{c \mathbf{v}} f)(p) &= c (\partial_\mathbf{v} f)(p)
\end{align}</math>

for any vectors {{math|'''v'''}}, {{math|'''w'''}} and any real number {{mvar|c}}. At each point ''p'', this [[linear map]] from {{math|'''R'''<sup>''n''</sup>}} to {{math|'''R'''}} is denoted {{math|''df''<sub>''p''</sub>}} and called the [[derivative]] or [[Differential of a function|differential]] of {{mvar|f}} at {{mvar|p}}. Thus {{math|1=''df''<sub>''p''</sub>('''v''') = ∂<sub>'''v'''</sub> ''f'' (''p'')}}. Extended over the whole set, the object {{math|''df''}} can be viewed as a function that takes a vector field on {{mvar|U}}, and returns a real-valued function whose value at each point is the derivative along the vector field of the function {{mvar|f}}. Note that at each {{mvar|p}}, the differential {{math|''df''<sub>''p''</sub>}} is not a real number, but a linear functional on tangent vectors, and a prototypical example of a differential [[1-form|{{math|1}}-form]].

Since any vector {{math|'''v'''}} is a [[linear combination]] {{math|Σ ''v''{{i sup|''j''}}'''e'''{{sub|''j''}}}} of its [[Euclidean vector#Decomposition|components]], {{math|''df''}} is uniquely determined by {{math|''df''{{sub|''p''}}('''e'''{{sub|''j''}})}} for each {{math|''j''}} and each {{math|''p'' ∈ ''U''}}, which are just the partial derivatives of {{mvar|f}} on {{mvar|U}}. Thus {{math|''df''}} provides a way of encoding the partial derivatives of {{mvar|f}}. It can be decoded by noticing that the coordinates {{math|''x''{{i sup|1}}}}, {{math|''x''{{sup|2}}}}, ..., {{math|''x''{{i sup|''n''}}}} are themselves functions on {{mvar|U}}, and so define differential {{math|1}}-forms {{math|''dx''{{i sup|1}}}}, {{math|''dx''{{i sup|2}}}}, ..., {{math|''dx''{{i sup|''n''}}}}. Let {{math|1=''f'' = ''x''{{i sup|''i''}}}}. Since {{math|1=∂''x''{{i sup|''i''}} / ∂''x''{{i sup|''j''}} = ''δ''{{sub|''ij''}}}}, the [[Kronecker delta function]], it follows that

{{NumBlk|:|<math>df = \sum_{i=1}^n \frac{\partial f}{\partial x^i} \, dx^i .</math>|{{EquationRef|*|<nowiki>*</nowiki>}}}}

The meaning of this expression is given by evaluating both sides at an arbitrary point {{mvar|p}}: on the right hand side, the sum is defined "[[pointwise]]", so that
<math display="block">df_p = \sum_{i=1}^n \frac{\partial f}{\partial x^i}(p) (dx^i)_p .</math>
Applying both sides to {{math|''e''<sub>''j''</sub>}}, the result on each side is the {{mvar|j}}th partial derivative of {{mvar|f}} at {{mvar|p}}. Since {{mvar|p}} and {{mvar|j}} were arbitrary, this proves the formula {{EquationNote|*|(*)}}.

More generally, for any smooth functions {{math|''g''<sub>''i''</sub>}} and {{math|''h''<sub>''i''</sub>}} on {{mvar|U}}, we define the differential {{math|1}}-form {{math|1=''α'' = Σ<sub>''i''</sub> ''g''<sub>''i''</sub> ''dh''<sub>''i''</sub>}} pointwise by
<math display="block">\alpha_p = \sum_i g_i(p) (dh_i)_p</math>
for each {{math|''p'' ∈ ''U''}}. Any differential {{math|1}}-form arises this way, and by using {{EquationNote|*|(*)}} it follows that any differential {{math|1}}-form {{mvar|α}} on {{mvar|U}} may be expressed in coordinates as
<math display="block"> \alpha = \sum_{i=1}^n f_i\, dx^i</math>
for some smooth functions {{math|''f''<sub>''i''</sub>}} on {{mvar|U}}.

The second idea leading to differential forms arises from the following question: given a differential {{math|1}}-form {{mvar|α}} on {{mvar|U}}, when does there exist a function {{mvar|f}} on {{mvar|U}} such that {{math|1=''α'' = ''df''}}? The above expansion reduces this question to the search for a function {{mvar|f}} whose partial derivatives {{math|∂''f'' / ∂''x''{{i sup|''i''}}}} are equal to {{mvar|n}} given functions {{math|''f''<sub>''i''</sub>}}. For {{math|''n'' > 1}}, such a function does not always exist: any smooth function {{mvar|f}} satisfies
<math display="block"> \frac{\partial^2 f}{\partial x^i \, \partial x^j} =  \frac{\partial^2 f}{\partial x^j \, \partial x^i} ,</math>
so it will be impossible to find such an {{mvar|f}} unless
<math display="block"> \frac{\partial f_j}{\partial x^i} - \frac{\partial f_i}{\partial x^j} = 0</math>
for all {{mvar|i}} and {{mvar|j}}.

The [[skew symmetry|skew-symmetry]] of the left hand side in {{mvar|i}} and {{mvar|j}} suggests introducing an antisymmetric product {{math|∧}} on differential {{math|1}}-forms, the [[exterior product]], so that these equations can be combined into a single condition
<math display="block"> \sum_{i,j=1}^n \frac{\partial f_j}{\partial x^i} \, dx^i \wedge dx^j = 0 ,</math>
where {{math|∧}} is defined so that:
<math display="block"> dx^i \wedge dx^j = - dx^j \wedge dx^i. </math>

This is an example of a differential {{math|2}}-form.  This {{math|2}}-form is called the [[exterior derivative]] {{math|''dα''}} of {{math|1=''α'' = ∑{{su|b=''j''=1|p=''n''}} ''f''<sub>''j''</sub> ''dx''{{i sup|''j''}}}}.  It is given by
<math display="block"> d\alpha = \sum_{j=1}^n df_j \wedge dx^j = \sum_{i,j=1}^n \frac{\partial f_j}{\partial x^i} \, dx^i \wedge dx^j .</math>

To summarize: {{math|1=''dα'' = 0}} is a necessary condition for the existence of a function {{mvar|f}} with {{math|1=''α'' = ''df''}}.

Differential {{math|0}}-forms, {{math|1}}-forms, and {{math|2}}-forms are special cases of differential forms. For each {{mvar|k}}, there is a space of differential {{mvar|k}}-forms, which can be expressed in terms of the coordinates as
<math display="block"> \sum_{i_1,i_2\ldots i_k=1}^n f_{i_1i_2\ldots i_k} \, dx^{i_1} \wedge dx^{i_2} \wedge\cdots \wedge dx^{i_k}</math>
for a collection of functions {{math|''f''<sub>''i''<sub>1</sub>''i''<sub>2</sub>⋅⋅⋅''i''<sub>''k''</sub></sub>}}. Antisymmetry, which was already present for {{math|2}}-forms, makes it possible to restrict the sum to those sets of indices for which {{math|''i''<sub>1</sub> < ''i''<sub>2</sub> < ... < ''i''<sub>''k''−1</sub> < ''i''<sub>''k''</sub>}}.

Differential forms can be multiplied together using the exterior product, and for any differential {{mvar|k}}-form {{mvar|α}}, there is a differential {{math|(''k'' + 1)}}-form {{math|''dα''}} called the exterior derivative of {{mvar|α}}.

Differential forms, the exterior product and the exterior derivative are independent of a choice of coordinates. Consequently, they may be defined on any [[smooth manifold]] {{mvar|M}}. One way to do this is cover {{mvar|M}} with [[coordinate chart]]s and define a differential {{mvar|k}}-form on {{mvar|M}} to be a family of differential {{mvar|k}}-forms on each chart which agree on the overlaps. However, there are more intrinsic definitions which make the independence of coordinates manifest.