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{{Short description|Expression that may be integrated over a region}}

In [[mathematics]], '''differential forms''' provide a unified approach to define [[integrand]]s over curves, surfaces, solids, and higher-dimensional [[manifold]]s.  The modern notion of differential forms was pioneered by [[Élie Cartan]].  It has many applications, especially in geometry, topology and physics.

For instance, the expression <math>f(x) \, dx</math> is an example of a [[1-form|{{math|1}}-form]], and can be [[integral|integrated]] over an interval <math>[a,b]</math> contained in the domain of <math>f</math>:
<math display="block">\int_a^b f(x)\,dx.</math>
Similarly, the expression <math display="block">f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz</math> is a '''{{math|2}}-form''' that can be integrated over a [[Surface (mathematics)|surface]] <math>S</math>:
<math display="block">\int_S \left(f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz\right).</math>
The symbol <math>\wedge</math> denotes the [[exterior product]], sometimes called the ''wedge product'', of two differential forms. Likewise, a '''{{math|3}}-form''' <math> f(x,y,z) \, dx \wedge dy \wedge dz</math> represents a [[volume element]] that can be integrated over a region of space. In general, a {{mvar|k}}-form is an object that may be integrated over a {{mvar|k}}-dimensional manifold, and is [[homogeneous polynomial|homogeneous]] of degree {{mvar|k}} in the coordinate differentials <math>dx, dy, \ldots.</math>
On an {{math|''n''}}-dimensional manifold, a top-dimensional form ({{math|''n''}}-form) is called a ''[[volume form]]''.

The differential forms form an [[alternating algebra]]. This implies that <math>dy \wedge dx = -dx \wedge dy</math> and <math>dx \wedge dx = 0.</math> This alternating property reflects the [[orientation (mathematics)|orientation]] of the domain of integration.  

The [[exterior derivative]] is an operation on differential forms  that, given a {{math|''k''}}-form <math>\varphi</math>, produces a {{math|(''k''+1)}}-form <math>d\varphi.</math> This operation extends the [[differential of a function]] (a function can be considered as a {{math|0}}-form, and its differential is <math>df(x) = f'(x) \, dx</math>). This allows expressing the [[fundamental theorem of calculus]], the [[divergence theorem]], [[Green's theorem]], and [[Stokes' theorem]] as special cases of a single general result, the [[generalized Stokes theorem]].

Differential {{math|1}}-forms are naturally dual to [[vector field]]s on a [[differentiable manifold]], and the pairing between vector fields and {{math|1}}-forms is extended to arbitrary differential forms by the [[interior product]].  The algebra of differential forms along with the exterior derivative defined on it is preserved by the [[pullback (differential geometry)|pullback]] under smooth functions between two manifolds.  This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided that the information is expressed in terms of differential forms.  As an example, the [[change of variables formula]] for integration becomes a simple statement that an integral is preserved under pullback.

== History ==
Differential forms are part of the field of differential geometry, influenced by linear algebra.  Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to [[Élie Cartan]] with reference to his 1899 paper.<ref>{{Citation |title=Sur certaines expressions différentielles et le problème de Pfaff | first=Élie  |last=Cartan |journal=Annales Scientifiques de l'École Normale Supérieure |url=http://www.numdam.org/item?id=ASENS_1899_3_16__239_0|year=1899 | volume=16 |pages=239–332| doi=10.24033/asens.467 |doi-access=free }}</ref> Some aspects of the [[exterior algebra]] of differential forms appears in [[Hermann Grassmann]]'s 1844 work, ''Die Lineale<!--sic--> Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics)''.

== 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.

== Intrinsic definitions ==
{{see also|Exterior algebra}}

Let {{math|''M''}} be a [[smooth manifold]]. A smooth differential form of degree {{math|''k''}} is a [[section (fiber bundle)|smooth section]] of the {{math|''k''}}th [[exterior algebra|exterior power]] of the [[cotangent bundle]] of {{math|''M''}}.  The set of all differential {{math|''k''}}-forms on a manifold {{math|''M''}} is a [[vector space]], often denoted <math>\Omega^k(M)</math>.

The definition of a differential form may be restated as follows.  At any point <math>p\in M</math>, a {{math|''k''}}-form <math>\beta</math> defines an element
<math display="block"> \beta_p \in {\textstyle\bigwedge}^k T_p^* M,</math>
where <math>T_pM</math> is the [[tangent space]] to {{math|''M''}} at {{math|''p''}} and <math>T^*_p(M)</math> is its [[dual space]].  This space is {{Clarify|date=December 2021|reason=Please specify the isomorphism. The linked MO answer does not give the isomorphism explicitly and is unclear for non-specialists. The definition of the isomorphism needs to be clear to be able to chose between two commonly used definitions of the wedge product.|text=[[natural isomorphism|naturally isomorphic]]<ref>{{Cite web|url=https://mathoverflow.net/q/68033|title = Linear algebra – "Natural" pairings between exterior powers of a vector space and its dual}}</ref>}} to the fiber at {{math|''p''}} of the dual bundle of the {{math|''k''}}th exterior power of the [[tangent bundle]] of {{math|''M''}}. That is, <math>\beta</math> is also a linear functional <math display="inline">\beta_p \colon {\textstyle\bigwedge}^k T_pM \to \mathbf{R}</math>, i.e. the dual of the {{math|''k''}}th exterior power is isomorphic to the {{math|''k''}}th exterior power of the dual:
<math display="block">{\textstyle\bigwedge}^k T^*_p M \cong \Big({\textstyle\bigwedge}^k T_p M\Big)^*</math>

By the universal property of exterior powers, this is equivalently an [[alternating form|alternating]] [[multilinear map]]:
<math display="block">\beta_p\colon \bigoplus_{n=1}^k T_p M \to \mathbf{R}.</math>
Consequently, a differential {{math|''k''}}-form may be evaluated against any {{math|''k''}}-tuple of tangent vectors to the same point {{math|''p''}} of {{math|''M''}}.  For example, a differential {{math|1}}-form {{math|''α''}} assigns to each point <math>p\in M</math> a [[linear functional]] {{math|''α''<sub>''p''</sub>}} on <math>T_pM</math>. In the presence of an [[inner product]] on <math>T_pM</math> (induced by a [[Riemannian metric]] on {{math|''M''}}), {{math|''α''<sub>''p''</sub>}} may be [[Riesz representation theorem|represented]] as the inner product with a [[tangent vector]] <math>X_p</math>. Differential {{math|1}}-forms are sometimes called [[covariance and contravariance of vectors|covariant vector fields]], covector fields, or "dual vector fields", particularly within physics.

The exterior algebra may be embedded in the tensor algebra by means of the alternation map. The alternation map is defined as a mapping  
<math display="block">\operatorname{Alt} \colon {\bigotimes}^k T^*M \to {\bigotimes}^k T^*M.</math>
For a tensor <math>\tau</math> at a point {{math|''p''}},
<math display="block">\operatorname{Alt}(\tau_p)(x_1, \dots, x_k) = \frac{1}{k!}\sum_{\sigma \in S_k} \sgn(\sigma) \tau_p(x_{\sigma(1)}, \dots, x_{\sigma(k)}),</math>
where {{math|''S''<sub>''k''</sub>}} is the [[symmetric group]] on {{math|''k''}} elements.  The alternation map is constant on the cosets of the ideal in the tensor algebra generated by the symmetric 2-forms, and therefore descends to an embedding
<math display="block">\operatorname{Alt} \colon {\textstyle\bigwedge}^k T^*M \to {\bigotimes}^k T^*M.</math>

This map exhibits <math>\beta</math> as a [[antisymmetric tensor|totally antisymmetric]] [[covariance and contravariance of vectors|covariant]] [[tensor field]] of rank {{math|''k''}}. The differential forms on {{math|''M''}} are in one-to-one correspondence with such tensor fields.

== Operations ==
As well as the addition and multiplication by scalar operations which arise from the vector space structure, there are several other standard operations defined on differential forms. The most important operations are the [[exterior product]] of two differential forms, the [[exterior derivative]] of a single differential form, the [[interior product]] of a differential form and a vector field, the [[Lie derivative]] of a differential form with respect to a vector field and the [[covariant derivative]] of a differential form with respect to a vector field on a manifold with a defined connection.

=== Exterior product ===
The exterior product of a {{math|''k''}}-form {{math|''α''}} and an {{math|''ℓ''}}-form {{math|''β''}}, denoted {{math|''α'' ∧ ''β''}}, is a ({{math|''k'' + ''ℓ''}})-form. At each point {{math|''p''}} of the manifold {{math|''M''}}, the forms {{math|''α''}} and {{math|''β''}} are elements of an exterior power of the cotangent space at {{math|''p''}}. When the exterior algebra is viewed as a quotient of the tensor algebra, the exterior product corresponds to the tensor product (modulo the equivalence relation defining the exterior algebra).

The antisymmetry inherent in the exterior algebra means that when {{math|''α'' ∧ ''β''}} is viewed as a multilinear functional, it is alternating. However, when the exterior algebra is embedded as a subspace of the tensor algebra by means of the alternation map, the tensor product {{math|''α'' ⊗ ''β''}} is not alternating. There is an explicit formula which describes the exterior product in this situation.  The exterior product is
<math display="block">\alpha \wedge \beta = \operatorname{Alt}(\alpha \otimes \beta).</math>
If the embedding of <math>{\textstyle\bigwedge}^n T^*M</math> into <math>{\bigotimes}^n T^*M</math> is done via the map <math>n!\operatorname{Alt}</math> instead of <math>\operatorname{Alt}</math>, the exterior product is
<math display="block">\alpha \wedge \beta = \frac{(k + \ell)!}{k!\ell!}\operatorname{Alt}(\alpha \otimes \beta).</math>
This description is useful for explicit computations. For example, if {{math|1=''k'' = ''ℓ'' = 1}}, then {{math|''α'' ∧ ''β''}} is the {{math|2}}-form whose value at a point {{math|''p''}} is the [[alternating bilinear form]] defined by
<math display="block"> (\alpha\wedge\beta)_p(v,w)=\alpha_p(v)\beta_p(w) - \alpha_p(w)\beta_p(v)</math>
for {{math|''v'', ''w'' ∈ T<sub>''p''</sub>''M''}}.

The exterior product is bilinear: If {{math|''α''}}, {{math|''β''}}, and {{math|''γ''}} are any differential forms, and if {{math|''f''}} is any smooth function, then
<math display="block">\alpha \wedge (\beta + \gamma) = \alpha \wedge \beta + \alpha \wedge \gamma,</math>
<math display="block">\alpha \wedge (f \cdot \beta) = f \cdot (\alpha \wedge \beta).</math>

It is ''skew commutative'' (also known as ''graded commutative''), meaning that it satisfies a variant of [[anticommutativity]] that depends on the degrees of the forms: if {{math|''α''}} is a {{math|''k''}}-form and {{math|''β''}} is an {{math|''ℓ''}}-form, then
<math display="block">\alpha \wedge \beta = (-1)^{k\ell} \beta \wedge \alpha .</math>
One also has the [[Differential graded algebra|graded Leibniz rule]]:<blockquote><math>d(\alpha\wedge\beta)=d\alpha\wedge\beta + (-1)^{k}\alpha\wedge d\beta.</math></blockquote>

=== Riemannian manifold ===
On a [[Riemannian manifold]], or more generally a [[pseudo-Riemannian manifold]], the metric defines a fibre-wise isomorphism of the tangent and cotangent bundles.  This makes it possible to convert vector fields to covector fields and vice versa.  It also enables the definition of additional operations such as the [[Hodge star operator]] <math>\star \colon \Omega^k(M)\ \stackrel{\sim}{\to}\ \Omega^{n-k}(M)</math> and the [[Hodge star#Codifferential|codifferential]] <math>\delta\colon \Omega^k(M)\rightarrow \Omega^{k-1}(M)</math>, which has degree {{math|−1}} and is [[Differential operator#Adjoint of an operator|adjoint]] to the exterior differential {{math|''d''}}.

==== Vector field structures ====
On a pseudo-Riemannian manifold, {{math|1}}-forms can be identified with vector fields; vector fields have additional distinct algebraic structures, which are listed here for context and to avoid confusion.

Firstly, each (co)tangent space generates a [[Clifford algebra]], where the product of a (co)vector with itself is given by the value of a quadratic form – in this case, the natural one induced by the [[metric tensor|metric]]. This algebra is ''distinct'' from the [[exterior algebra]] of differential forms, which can be viewed as a Clifford algebra where the quadratic form vanishes (since the exterior product of any vector with itself is zero). Clifford algebras are thus non-anticommutative ("quantum") deformations of the exterior algebra. They are studied in [[geometric algebra]].

Another alternative is to consider vector fields as derivations. The (noncommutative) algebra of [[differential operator]]s they generate is the [[Weyl algebra]] and is a noncommutative ("quantum") deformation of the ''symmetric'' algebra in the vector fields.

=== Exterior differential complex ===
One important property of the exterior derivative is that {{math|1=''d''{{i sup|2}} = 0}}. This means that the exterior derivative defines a [[cochain complex]]:
<math display="block">0\ \to\ \Omega^0(M)\ \stackrel{d}{\to}\ \Omega^1(M)\ \stackrel{d}{\to}\ \Omega^2(M)\ \stackrel{d}{\to}\ \Omega^3(M)\ \to\ \cdots \ \to\ \Omega^n(M)\ \to \ 0.</math>

This complex is called the de Rham complex, and its [[cohomology]] is by definition the [[de Rham cohomology]] of {{math|''M''}}.  By the [[Poincaré lemma]], the de Rham complex is locally [[exact sequence|exact]] except at {{math|Ω<sup>0</sup>(''M'')}}.  The kernel at {{math|Ω<sup>0</sup>(''M'')}} is the space of [[locally constant function]]s on {{math|''M''}}.  Therefore, the complex is a resolution of the constant [[sheaf (mathematics)|sheaf]] {{math|{{underline|'''R'''}}}}, which in turn implies a form of de Rham's theorem: de Rham cohomology computes the [[sheaf cohomology]] of {{math|{{underline|'''R'''}}}}.

== Pullback ==
{{see also|Pullback (differential geometry)}}

Suppose that {{math|''f'' : ''M'' → ''N''}} is smooth. The differential of {{math|''f''}} is a smooth map {{math|''df'' : ''TM'' → ''TN''}} between the tangent bundles of {{math|''M''}} and {{math|''N''}}. This map is also denoted {{math|''f''<sub>∗</sub>}} and called the '''pushforward'''. For any point {{math|''p'' ∈ ''M''}} and any tangent vector {{math|''v'' &isin; ''T''<sub>''p''</sub>''M''}}, there is a well-defined pushforward vector {{math|''f''<sub>∗</sub>(''v'')}} in {{math|''T''<sub>''f''(''p'')</sub>''N''}}. However, the same is not true of a vector field.  If {{math|''f''}} is not injective, say because {{math|''q'' ∈ ''N''}} has two or more preimages, then the vector field may determine two or more distinct vectors in {{math|''T''<sub>''q''</sub>''N''}}. If {{math|''f''}} is not surjective, then there will be a point {{math|''q'' &isin; ''N''}} at which {{math|''f''<sub>∗</sub>}} does not determine any tangent vector at all. Since a vector field on {{math|''N''}} determines, by definition, a unique tangent vector at every point of {{math|''N''}}, the pushforward of a vector field does not always exist.

By contrast, it is always possible to pull back a differential form.  A differential form on {{math|''N''}} may be viewed as a linear functional on each tangent space.  Precomposing this functional with the differential {{math|''df'' : ''TM'' → ''TN''}} defines a linear functional on each tangent space of {{math|''M''}} and therefore a differential form on {{math|''M''}}.  The existence of pullbacks is one of the key features of the theory of differential forms.  It leads to the existence of pullback maps in other situations, such as pullback homomorphisms in de Rham cohomology.

Formally, let {{math|''f'' : ''M'' → ''N''}} be smooth, and let {{math|''ω''}} be a smooth {{math|''k''}}-form on {{math|''N''}}.  Then there is a differential form {{math|''f''{{i sup|∗}}''ω''}} on {{math|''M''}}, called the '''pullback''' of {{math|''ω''}}, which captures the behavior of {{math|''ω''}} as seen relative to {{math|''f''}}.  To define the pullback, fix a point {{math|''p''}} of {{math|''M''}} and tangent vectors {{math|''v''<sub>1</sub>}}, ..., {{math|''v''<sub>''k''</sub>}} to {{math|''M''}} at {{math|''p''}}.  The pullback of {{math|''ω''}} is defined by the formula
<math display="block">(f^*\omega)_p(v_1, \ldots, v_k) = \omega_{f(p)}(f_*v_1, \ldots, f_*v_k).</math>

There are several more abstract ways to view this definition.  If {{math|''ω''}} is a {{math|1}}-form on {{math|''N''}}, then it may be viewed as a section of the cotangent bundle {{math|''T''{{i sup|∗}}''N''}} of {{math|''N''}}.  Using {{i sup|∗}} to denote a dual map, the dual to the differential of {{math|''f''}} is {{math|(''df''){{i sup|∗}} : ''T''{{i sup|∗}}''N'' → ''T''{{i sup|∗}}''M''}}.  The pullback of {{math|''ω''}} may be defined to be the composite
<math display="block">M\ \stackrel{f}{\to}\ N\ \stackrel{\omega}{\to}\ T^*N\ \stackrel{(df)^*}{\longrightarrow}\ T^*M.</math>
This is a section of the cotangent bundle of {{math|''M''}} and hence a differential {{math|1}}-form on {{math|''M''}}.  In full generality, let <math display="inline">\bigwedge^k (df)^*</math> denote the {{math|''k''}}th exterior power of the dual map to the differential.  Then the pullback of a {{math|''k''}}-form {{math|''ω''}} is the composite
<math display="block">M\ \stackrel{f}{\to}\ N\ \stackrel{\omega}{\to}\ {\textstyle\bigwedge}^k T^*N\ \stackrel{{\bigwedge}^k (df)^*}{\longrightarrow}\ {\textstyle\bigwedge}^k T^*M.</math>

Another abstract way to view the pullback comes from viewing a {{math|''k''}}-form {{math|''ω''}} as a linear functional on tangent spaces.  From this point of view, {{math|''ω''}} is a morphism of [[vector bundle]]s
<math display="block">{\textstyle\bigwedge}^k TN\ \stackrel{\omega}{\to}\ N \times \mathbf{R},</math>
where {{math|''N'' × '''R'''}} is the trivial rank one bundle on {{math|''N''}}.  The composite map
<math display="block">{\textstyle\bigwedge}^k TM\ \stackrel{{\bigwedge}^k df}{\longrightarrow}\ {\textstyle\bigwedge}^k TN\ \stackrel{\omega}{\to}\ N \times \mathbf{R}</math>
defines a linear functional on each tangent space of {{math|''M''}}, and therefore it factors through the trivial bundle {{math|''M'' × '''R'''}}.  The vector bundle morphism <math display="inline">{\textstyle\bigwedge}^k TM \to M \times \mathbf{R}</math> defined in this way is {{math|''f''{{i sup|∗}}''ω''}}.

Pullback respects all of the basic operations on forms.  If {{math|''ω''}} and {{math|''η''}} are forms and {{math|''c''}} is a real number, then
<math display="block">\begin{align}
             f^*(c\omega) &= c(f^*\omega), \\
       f^*(\omega + \eta) &= f^*\omega + f^*\eta, \\
  f^*(\omega \wedge \eta) &= f^*\omega \wedge f^*\eta, \\
             f^*(d\omega) &= d(f^*\omega).
\end{align}</math>

The pullback of a form can also be written in coordinates. Assume that {{math|''x''<sup>1</sup>}}, ..., {{math|''x''<sup>''m''</sup>}} are coordinates on {{math|''M''}}, that {{math|''y''<sup>1</sup>}}, ..., {{math|''y''<sup>''n''</sup>}} are coordinates on {{math|''N''}}, and that these coordinate systems are related by the formulas {{math|1=''y''<sup>''i''</sup> = ''f''<sub>''i''</sub>(''x''<sup>1</sup>, ..., ''x''<sup>''m''</sup>)}} for all {{math|''i''}}. Locally on {{math|''N''}}, {{math|''ω''}} can be written as
<math display="block">\omega = \sum_{i_1 < \cdots < i_k} \omega_{i_1\cdots i_k} \, dy^{i_1} \wedge \cdots \wedge dy^{i_k},</math>
where, for each choice of {{math|''i''<sub>1</sub>}}, ..., {{math|''i''<sub>''k''</sub>}}, {{math|''ω''<sub>''i''<sub>1</sub>⋅⋅⋅''i''<sub>''k''</sub></sub>}} is a real-valued function of {{math|''y''<sup>1</sup>}}, ..., {{math|''y''<sup>''n''</sup>}}. Using the linearity of pullback and its compatibility with exterior product, the pullback of {{math|''ω''}} has the formula
<math display="block">f^*\omega = \sum_{i_1 < \cdots < i_k} (\omega_{i_1\cdots i_k}\circ f) \, df_{i_1} \wedge \cdots \wedge df_{i_k}.</math>

Each exterior derivative {{math|''df''<sub>''i''</sub>}} can be expanded in terms of {{math|''dx''<sup>1</sup>}}, ..., {{math|''dx''<sup>''m''</sup>}}. The resulting {{math|''k''}}-form can be written using [[Jacobian matrix and determinant|Jacobian]] matrices:
<math display="block"> f^*\omega = \sum_{i_1 < \cdots < i_k} \sum_{j_1 < \cdots < j_k} (\omega_{i_1\cdots i_k}\circ f)\frac{\partial(f_{i_1}, \ldots, f_{i_k})}{\partial(x^{j_1}, \ldots, x^{j_k})} \, dx^{j_1} \wedge \cdots \wedge dx^{j_k}.</math>

Here, <math display="inline>\frac{\partial(f_{i_1}, \ldots, f_{i_k})}{\partial(x^{j_1}, \ldots, x^{j_k})}</math> denotes the determinant of the matrix whose entries are <math display="inline">\frac{\partial f_{i_m}}{\partial x^{j_n}}</math>, <math>1\leq m,n\leq k</math>.

== Integration ==
A differential {{math|''k''}}-form can be integrated over an oriented {{math|''k''}}-dimensional manifold.  When the {{math|''k''}}-form is defined on an {{math|''n''}}-dimensional manifold with {{math|''n'' &gt; ''k''}}, then the {{math|''k''}}-form can be integrated over oriented {{math|''k''}}-dimensional submanifolds.  If {{math|1=''k'' = 0}}, integration over oriented 0-dimensional submanifolds is just the summation of the integrand evaluated at points, according to the orientation of those points. Other values of {{math|1=''k'' = 1, 2, 3, ...}} correspond to line integrals, surface integrals, volume integrals, and so on.  There are several equivalent ways to formally define the integral of a differential form, all of which depend on reducing to the case of Euclidean space.

=== Integration on Euclidean space ===
Let {{math|''U''}} be an open subset of {{math|'''R'''<sup>''n''</sup>}}.  Give {{math|'''R'''<sup>''n''</sup>}} its standard orientation and {{math|''U''}} the restriction of that orientation.  Every smooth {{math|''n''}}-form {{math|''ω''}} on {{math|''U''}} has the form
<math display="block">\omega = f(x)\,dx^1 \wedge \cdots \wedge dx^n</math>
for some smooth function {{math|''f'' : '''R'''<sup>''n''</sup> → '''R'''}}.  Such a function has an integral in the usual Riemann or Lebesgue sense.  This allows us to define the integral of {{math|''ω''}} to be the integral of {{math|''f''}}:
<math display="block">\int_U \omega\ \stackrel{\text{def}}{=} \int_U f(x)\,dx^1 \cdots dx^n.</math>
Fixing an orientation is necessary for this to be well-defined.  The skew-symmetry of differential forms means that the integral of, say, {{math|''dx''<sup>1</sup> ∧ ''dx''<sup>2</sup>}} must be the negative of the integral of {{math|''dx''<sup>2</sup> ∧ ''dx''<sup>1</sup>}}.  Riemann and Lebesgue integrals cannot see this dependence on the ordering of the coordinates, so they leave the sign of the integral undetermined.  The orientation resolves this ambiguity.

=== Integration over chains ===
Let {{math|''M''}} be an {{math|''n''}}-manifold and {{math|''ω''}} an {{math|''n''}}-form on {{math|''M''}}.  First, assume that there is a parametrization of {{math|''M''}} by an open subset of Euclidean space.  That is, assume that there exists a diffeomorphism
<math display="block">\varphi \colon D \to M</math>
where {{math|''D'' ⊆ '''R'''<sup>''n''</sup>}}.  Give {{math|''M''}} the orientation induced by {{math|''φ''}}.  Then {{Harv|Rudin|1976}} defines the integral of {{math|''ω''}} over {{math|''M''}} to be the integral of {{math|''φ''<sup>∗</sup>''ω''}} over {{math|''D''}}.  In coordinates, this has the following expression.  Fix an embedding of {{math|''M''}} in {{math|'''R'''<sup>''I''</sup>}} with coordinates {{math|''x''<sup>1</sup>, ..., ''x''<sup>''I''</sup>}}.  Then
<math display="block">\omega = \sum_{i_1 < \cdots < i_n} a_{i_1,\ldots,i_n}({\mathbf x})\,dx^{i_1} \wedge \cdots \wedge dx^{i_n}.</math>
Suppose that {{math|''φ''}} is defined by
<math display="block">\varphi({\mathbf u}) = (x^1({\mathbf u}),\ldots,x^I({\mathbf u})).</math>
Then the integral may be written in coordinates as
<math display="block">\int_M \omega = \int_D \sum_{i_1 < \cdots < i_n} a_{i_1,\ldots,i_n}(\varphi({\mathbf u})) \frac{\partial(x^{i_1},\ldots,x^{i_n})}{\partial(u^{1},\dots,u^{n})}\,du^1 \cdots du^n,</math>
where
<math display="block">\frac{\partial(x^{i_1},\ldots,x^{i_n})}{\partial(u^{1},\ldots,u^{n})}</math>
is the determinant of the [[Jacobian matrix and determinant|Jacobian]].  The Jacobian exists because {{math|''φ''}} is differentiable.

In general, an {{math|''n''}}-manifold cannot be parametrized by an open subset of {{math|'''R'''<sup>''n''</sup>}}.  But such a parametrization is always possible locally, so it is possible to define integrals over arbitrary manifolds by defining them as sums of integrals over collections of local parametrizations.  Moreover, it is also possible to define parametrizations of {{math|''k''}}-dimensional subsets for {{math|''k'' &lt; ''n''}}, and this makes it possible to define integrals of {{math|''k''}}-forms.  To make this precise, it is convenient to fix a standard domain {{math|''D''}} in {{math|'''R'''<sup>''k''</sup>}}, usually a cube or a simplex.  A {{math|''k''}}-'''chain''' is a formal sum of smooth embeddings {{math|''D'' → ''M''}}.  That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity.  Each smooth embedding determines a {{math|''k''}}-dimensional submanifold of {{math|''M''}}.  If the chain is
<math display="block">c = \sum_{i=1}^r m_i \varphi_i,</math>
then the integral of a {{math|''k''}}-form {{math|''ω''}} over {{math|''c''}} is defined to be the sum of the integrals over the terms of {{math|''c''}}:
<math display="block">\int_c \omega = \sum_{i=1}^r m_i \int_D \varphi_i^*\omega.</math>

This approach to defining integration does not assign a direct meaning to integration over the whole manifold {{math|''M''}}.  However, it is still possible to assign such a meaning indirectly because every smooth manifold may be smoothly [[Triangulation (topology)|triangulated]] in an essentially unique way, and the integral over {{math|''M''}} may be defined to be the integral over the chain determined by a triangulation.

=== Integration using partitions of unity ===
There is another approach, expounded in {{Harv|Dieudonné|1972}}, which does directly assign a meaning to integration over {{math|''M''}}, but this approach requires fixing an orientation of {{math|''M''}}.  The integral of an {{math|''n''}}-form {{math|''ω''}} on an {{math|''n''}}-dimensional manifold is defined by working in charts.  Suppose first that {{math|''ω''}} is supported on a single positively oriented chart.  On this chart, it may be pulled back to an {{math|''n''}}-form on an open subset of {{math|'''R'''<sup>''n''</sup>}}.  Here, the form has a well-defined Riemann or Lebesgue integral as before.  The change of variables formula and the assumption that the chart is positively oriented together ensure that the integral of {{math|''ω''}} is independent of the chosen chart.  In the general case, use a partition of unity to write {{math|''ω''}} as a sum of {{math|''n''}}-forms, each of which is supported in a single positively oriented chart, and define the integral of {{math|''ω''}} to be the sum of the integrals of each term in the partition of unity.

It is also possible to integrate {{math|''k''}}-forms on oriented {{math|''k''}}-dimensional submanifolds using this more intrinsic approach.  The form is pulled back to the submanifold, where the integral is defined using charts as before.  For example, given a path {{math|''γ''(''t'') : [0, 1] → '''R'''<sup>2</sup>}}, integrating a {{math|1}}-form on the path is simply pulling back the form to a form {{math|''f''(''t''){{thin space}}''dt''}} on {{math|[0, 1]}}, and this integral is the integral of the function {{math|''f''(''t'')}} on the interval.

=== Integration along fibers ===
{{main|Integration along fibers}}

[[Fubini's theorem]] states that the integral over a set that is a product may be computed as an iterated integral over the two factors in the product.  This suggests that the integral of a differential form over a product ought to be computable as an iterated integral as well.  The geometric flexibility of differential forms ensures that this is possible not just for products, but in more general situations as well.  Under some hypotheses, it is possible to integrate along the fibers of a smooth map, and the analog of Fubini's theorem is the case where this map is the projection from a product to one of its factors.

Because integrating a differential form over a submanifold requires fixing an orientation, a prerequisite to integration along fibers is the existence of a well-defined orientation on those fibers.  Let {{math|''M''}} and {{math|''N''}} be two orientable manifolds of pure dimensions {{math|''m''}} and {{math|''n''}}, respectively.  Suppose that {{math|''f'' : ''M'' → ''N''}} is a surjective submersion.  This implies that each fiber {{math|''f''{{i sup|&minus;1}}(''y'')}} is {{math|(''m'' &minus; ''n'')}}-dimensional and that, around each point of {{math|''M''}}, there is a chart on which {{math|''f''}} looks like the projection from a product onto one of its factors.  Fix {{math|''x'' ∈ ''M''}} and set {{math|1=''y'' = ''f''(''x'')}}.  Suppose that
<math display="block">\begin{align}
\omega_x &\in {\textstyle\bigwedge}^m T_x^*M, \\[2pt]
\eta_y &\in {\textstyle\bigwedge}^n T_y^*N,
\end{align}</math>
and that {{math|''η''<sub>''y''</sub>}} does not vanish.  Following {{Harv|Dieudonné|1972}}, there is a unique
<math display="block">\sigma_x \in {\textstyle\bigwedge}^{m-n} T_x^*(f^{-1}(y))</math>
which may be thought of as the fibral part of {{math|''ω''<sub>''x''</sub>}} with respect to {{math|''η''<sub>''y''</sub>}}.  More precisely, define {{math|''j'' : ''f''{{i sup|&minus;1}}(''y'') → ''M''}} to be the inclusion.  Then {{math|''σ''<sub>''x''</sub>}} is defined by the property that
<math display="block">\omega_x = (f^*\eta_y)_x \wedge \sigma'_x \in {\textstyle\bigwedge}^m T_x^*M,</math>
where
<math display="block">\sigma'_x \in {\textstyle\bigwedge}^{m-n} T_x^*M</math>
is any {{math|(''m'' &minus; ''n'')}}-covector for which
<math display="block">\sigma_x = j^*\sigma'_x.</math>
The form {{math|''σ''<sub>''x''</sub>}} may also be notated {{math|''ω''<sub>''x''</sub> / ''η''<sub>''y''</sub>}}.

Moreover, for fixed {{math|''y''}}, {{math|''σ''<sub>''x''</sub>}} varies smoothly with respect to {{math|''x''}}.  That is, suppose that
<math display="block">\omega \colon f^{-1}(y) \to T^*M</math>
is a smooth section of the projection map; we say that {{math|''ω''}} is a smooth differential {{math|''m''}}-form on {{math|''M''}} along {{math|''f''{{i sup|&minus;1}}(''y'')}}.  Then there is a smooth differential {{math|(''m'' &minus; ''n'')}}-form {{math|''σ''}} on {{math|''f''{{i sup|&minus;1}}(''y'')}} such that, at each {{math|''x'' ∈ ''f''{{i sup|&minus;1}}(''y'')}},
<math display="block">\sigma_x = \omega_x / \eta_y.</math>
This form is denoted {{math|''ω'' / ''η''<sub>''y''</sub>}}.  The same construction works if {{math|''ω''}} is an {{math|''m''}}-form in a neighborhood of the fiber, and the same notation is used.  A consequence is that each fiber {{math|''f''{{i sup|&minus;1}}(''y'')}} is orientable.  In particular, a choice of orientation forms on {{math|''M''}} and {{math|''N''}} defines an orientation of every fiber of {{math|''f''}}.

The analog of Fubini's theorem is as follows.  As before, {{math|''M''}} and {{math|''N''}} are two orientable manifolds of pure dimensions {{math|''m''}} and {{math|''n''}}, and {{math|''f'' : ''M'' → ''N''}} is a surjective submersion.  Fix orientations of {{math|''M''}} and {{math|''N''}}, and give each fiber of {{math|''f''}} the induced orientation.  Let {{math|''ω''}} be an {{math|''m''}}-form on {{math|''M''}}, and let {{math|''η''}} be an {{math|''n''}}-form on {{math|''N''}} that is almost everywhere positive with respect to the orientation of {{math|''N''}}.  Then, for almost every {{math|''y'' ∈ ''N''}}, the form {{math|''ω'' / ''η''<sub>''y''</sub>}} is a well-defined integrable {{math|''m'' &minus; ''n''}} form on {{math|''f''{{i sup|&minus;1}}(''y'')}}.  Moreover, there is an integrable {{math|''n''}}-form on {{math|''N''}} defined by
<math display="block">y \mapsto \bigg(\int_{f^{-1}(y)} \omega / \eta_y\bigg)\,\eta_y.</math>
Denote this form by
<math display="block">\bigg(\int_{f^{-1}(y)} \omega / \eta\bigg)\,\eta.</math>
Then {{Harv|Dieudonné|1972}} proves the generalized Fubini formula
<math display="block">\int_M \omega = \int_N \bigg(\int_{f^{-1}(y)} \omega / \eta\bigg)\,\eta.</math>

It is also possible to integrate forms of other degrees along the fibers of a submersion.  Assume the same hypotheses as before, and let {{math|''α''}} be a compactly supported {{math|(''m'' &minus; ''n'' + ''k'')}}-form on {{math|''M''}}.  Then there is a {{math|''k''}}-form {{math|''γ''}} on {{math|''N''}} which is the result of integrating {{math|''α''}} along the fibers of {{math|''f''}}.  The form {{math|''α''}} is defined by specifying, at each {{math|''y'' ∈ ''N''}}, how {{math|''γ''}} pairs with each {{math|''k''}}-vector {{math|'''v'''}} at {{math|''y''}}, and the value of that pairing is an integral over {{math|''f''{{i sup|&minus;1}}(''y'')}} that depends only on {{math|''α''}}, {{math|'''v'''}}, and the orientations of {{math|''M''}} and {{math|''N''}}.  More precisely, at each {{math|''y'' ∈ ''N''}}, there is an isomorphism
<math display="block">{\textstyle\bigwedge}^k T_yN \to {\textstyle\bigwedge}^{n-k} T_y^*N</math>
defined by the interior product
<math display="block">\mathbf{v} \mapsto \mathbf{v}\,\lrcorner\,\zeta_y,</math>
for any choice of volume form {{math|''ζ''}} in the orientation of {{math|''N''}}. If {{math|''x'' ∈ ''f''{{i sup|&minus;1}}(''y'')}}, then a {{math|''k''}}-vector {{math|'''v'''}} at {{math|''y''}} determines an {{math|(''n'' &minus; ''k'')}}-covector at {{math|''x''}} by pullback:
<math display="block">f^*(\mathbf{v}\,\lrcorner\,\zeta_y) \in {\textstyle\bigwedge}^{n-k} T_x^*M.</math>
Each of these covectors has an exterior product against {{math|''α''}}, so there is an {{math|(''m'' &minus; ''n'')}}-form {{math|''β''<sub>'''v'''</sub>}} on {{math|''M''}} along {{math|''f''{{i sup|&minus;1}}(''y'')}} defined by
<math display="block">(\beta_{\mathbf{v}})_x = \left(\alpha_x \wedge f^*(\mathbf{v}\,\lrcorner\,\zeta_y)\right) \big/ \zeta_y \in {\textstyle\bigwedge}^{m-n} T_x^*M.</math>
This form depends on the orientation of {{math|''N''}} but not the choice of {{math|''ζ''}}.  Then the {{math|''k''}}-form {{math|''γ''}} is uniquely defined by the property
<math display="block">\langle\gamma_y, \mathbf{v}\rangle = \int_{f^{-1}(y)} \beta_{\mathbf{v}},</math>
and {{math|''γ''}} is smooth {{Harv|Dieudonné|1972}}.  This form also denoted {{math|''α''<sup>♭</sup>}} and called the '''integral of {{math|''α''}} along the fibers of {{math|''f''}}'''.  Integration along fibers is important for the construction of Gysin maps in de Rham cohomology.

Integration along fibers satisfies the '''projection formula''' {{Harv|Dieudonné|1972}}.  If {{math|''λ''}} is any {{math|''ℓ''}}-form on {{math|''N''}}, then
<math display="block">\alpha^\flat \wedge \lambda = (\alpha \wedge f^*\lambda)^\flat.</math>

===Stokes's theorem===
{{main|Stokes's theorem}}
The fundamental relationship between the exterior derivative and integration is given by the [[Stokes' theorem]]: If {{math|''ω''}} is an ({{math|''n'' − 1}})-form with compact support on {{math|''M''}} and {{math|''∂M''}} denotes the [[manifold#Manifold with boundary|boundary]] of {{math|''M''}} with its induced [[Orientation (mathematics)|orientation]], then
<math display="block">\int_M d\omega = \int_{\partial M} \omega.</math>

A key consequence of this is that "the integral of a closed form over homologous chains is equal": If {{math|''ω''}} is a closed {{math|''k''}}-form and {{math|''M''}} and {{math|''N''}} are {{math|''k''}}-chains that are homologous (such that {{math|''M'' − ''N''}} is the boundary of a {{math|(''k'' + 1)}}-chain {{math|''W''}}), then <math>\textstyle{\int_M \omega = \int_N \omega}</math>, since the difference is the integral <math>\textstyle\int_W d\omega = \int_W 0 = 0</math>.

For example, if {{math|1=''ω'' = ''df''}} is the derivative of a potential function on the plane or {{math|'''R'''<sup>''n''</sup>}}, then the integral of {{math|''ω''}} over a path from {{math|''a''}} to {{math|''b''}} does not depend on the choice of path (the integral is {{math|''f''(''b'') − ''f''(''a'')}}), since different paths with given endpoints are [[homotopic]], hence homologous (a weaker condition). This case is called the [[gradient theorem]], and generalizes the [[fundamental theorem of calculus]]. This path independence is very useful in [[contour integration]].

This theorem also underlies the duality between [[de Rham cohomology]] and the [[homology (mathematics)|homology]] of chains.

=== Relation with measures ===
{{details|Density on a manifold}}

On a ''general'' differentiable manifold (without additional structure), differential forms ''cannot'' be integrated over subsets of the manifold; this distinction is key to the distinction between differential forms, which are integrated over chains or oriented submanifolds, and measures, which are integrated over subsets. The simplest example is attempting to integrate the {{math|1}}-form {{math|''dx''}} over the interval {{math|[0, 1]}}. Assuming the usual distance (and thus measure) on the real line, this integral is either {{math|1}} or {{math|&minus;1}}, depending on ''orientation'': {{nowrap|<math display="inline"> \int_0^1 dx = 1</math>,}} while {{nowrap|<math display="inline"> \int_1^0 dx = - \int_0^1 dx = -1</math>.}} By contrast, the integral of the ''measure'' {{math|{{abs|''dx''}}}} on the interval is unambiguously {{math|1}} (i.e. the integral of the constant function {{math|1}} with respect to this measure is {{math|1}}). Similarly, under a change of coordinates a differential {{math|''n''}}-form changes by the [[Jacobian determinant]] {{math|''J''}}, while a measure changes by the ''absolute value'' of the Jacobian determinant, {{math|{{abs|''J''}}}}, which further reflects the issue of orientation. For example, under the map {{math|''x'' ↦ −''x''}} on the line, the differential form {{math|''dx''}} pulls back to {{math|−''dx''}}; orientation has reversed; while the [[Lebesgue measure]], which here we denote {{math|{{abs|''dx''}}}}, pulls back to {{math|{{abs|''dx''}}}}; it does not change.

In the presence of the additional data of an ''orientation'', it is possible to integrate {{math|''n''}}-forms (top-dimensional forms) over the entire manifold or over compact subsets; integration over the entire manifold corresponds to integrating the form over the [[fundamental class]] of the manifold, {{math|[''M'']}}. Formally, in the presence of an orientation, one may identify {{math|''n''}}-forms with [[densities on a manifold]]; densities in turn define a measure, and thus can be integrated {{Harv |Folland |1999 |loc = Section 11.4, pp.&nbsp;361&ndash;362}}.

On an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate {{math|''n''}}-forms over compact subsets, with the two choices differing by a sign. On a non-orientable manifold, {{math|''n''}}-forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere (there are no [[volume form]]s on non-orientable manifolds), but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate {{math|''n''}}-forms. One can instead identify densities with top-dimensional [[Volume form#Relation to measures|pseudoform]]s.

Even in the presence of an orientation, there is in general no meaningful way to integrate {{math|''k''}}-forms over subsets for {{math|''k'' < ''n''}} because there is no consistent way to use the ambient orientation to orient {{math|''k''}}-dimensional subsets.  Geometrically, a {{math|''k''}}-dimensional subset can be turned around in place, yielding the same subset with the opposite orientation; for example, the horizontal axis in a plane can be rotated by 180 degrees.  Compare the [[Gram determinant]] of a set of {{math|''k''}} vectors in an {{math|''n''}}-dimensional space, which, unlike the determinant of {{math|''n''}} vectors, is always positive, corresponding to a squared number.  An orientation of a {{math|''k''}}-submanifold is therefore extra data not derivable from the ambient manifold.

On a Riemannian manifold, one may define a {{math|''k''}}-dimensional [[Hausdorff measure]] for any {{math|''k''}} (integer or real), which may be integrated over {{math|''k''}}-dimensional subsets of the manifold. A function times this Hausdorff measure can then be integrated over {{math|''k''}}-dimensional subsets, providing a measure-theoretic analog to integration of {{math|''k''}}-forms. The {{math|''n''}}-dimensional Hausdorff measure yields a density, as above.

=== Currents ===
The differential form analog of a [[distribution (mathematics)|distribution]] or generalized function is called a '''[[Current (mathematics)|current]]'''.  The space of {{math|''k''}}-currents on {{math|''M''}} is the dual space to an appropriate space of differential {{math|''k''}}-forms.  Currents play the role of generalized domains of integration, similar to but even more flexible than chains.

== Applications in physics ==
<!-- This section is linked from [[Maxwell's equations]] -->

Differential forms arise in some important physical contexts. For example, in Maxwell's theory of [[electromagnetism]], the '''Faraday 2-form''', or [[electromagnetic field strength]], is
<math display="block">\mathbf{F} = \frac 1 2 f_{ab}\, dx^a \wedge dx^b\,,</math>
where the {{math|''f''<sub>''ab''</sub>}} are formed from the electromagnetic fields <math>\vec E</math> and <math>\vec B</math>; e.g., {{math|1=''f''<sub>12</sub> = ''E''<sub>''z''</sub>/''c''}}, {{math|1=''f''<sub>23</sub> = −''B''<sub>''z''</sub>}}, or equivalent definitions.

This form is a special case of the [[curvature form]] on the {{math|[[U(1)]]}} [[principal bundle]] on which both electromagnetism and general [[gauge theories]] may be described.  The [[connection form]] for the principal bundle is the vector potential, typically denoted by {{math|'''A'''}}, when represented in some gauge.  One then has
<math display="block">\mathbf{F} = d\mathbf{A}.</math>

The '''current {{math|3}}-form''' is
<math display="block"> \mathbf{J} = \frac 1 6 j^a\, \varepsilon_{abcd}\, dx^b \wedge dx^c \wedge dx^d\,,</math>
where {{math|''j''<sup>''a''</sup>}} are the four components of the current density. (Here it is a matter of convention to write {{math|''F''<sub>''ab''</sub>}} instead of {{math|''f''<sub>''ab''</sub>}}, i.e. to use capital letters, and to write {{math|''J''<sup>''a''</sup>}} instead of {{math|''j''<sup>''a''</sup>}}. However, the vector rsp. tensor components and the above-mentioned forms have different physical dimensions. Moreover, by decision of an international commission of the [[International Union of Pure and Applied Physics]], the magnetic polarization vector has been called <math>\vec J</math> for several decades, and by some publishers {{math|'''J'''}}; i.e., the same name is used for different quantities.)

Using the above-mentioned definitions, [[Maxwell's equations]] can be written very compactly in [[geometrized units]] as
<math display="block">\begin{align}
        d {\mathbf{F}} &= \mathbf{0} \\
  d {\star \mathbf{F}} &= \mathbf{J},
\end{align}</math>
where <math>\star</math> denotes the [[Hodge star]] operator.  Similar considerations describe the geometry of gauge theories in general.

The {{math|2}}-form <math>{\star} \mathbf{F}</math>, which is [[duality (mathematics)|dual]] to the Faraday form, is also called '''Maxwell 2-form'''.

Electromagnetism is an example of a {{math|[[U(1)]]}} [[gauge theory]].  Here the [[Lie group]] is {{math|U(1)}}, the one-dimensional [[unitary group]], which is in particular [[abelian group|abelian]].  There are gauge theories, such as [[Yang–Mills theory]], in which the Lie group is not abelian.  In that case, one gets relations which are similar to those described here. The analog of the field {{math|'''F'''}} in such theories is the curvature form of the connection, which is represented in a gauge by a [[Lie algebra]]-valued one-form {{math|'''A'''}}.  The Yang–Mills field {{math|'''F'''}} is then defined by
<math display="block">\mathbf{F} = d\mathbf{A} + \mathbf{A}\wedge\mathbf{A}.</math>

In the abelian case, such as electromagnetism, {{math|1='''A''' ∧ '''A''' = 0}}, but this does not hold in general.  Likewise the field equations are modified by additional terms involving exterior products of {{math|'''A'''}} and {{math|'''F'''}}, owing to the [[Maurer–Cartan form|structure equations]] of the gauge group.

== Applications in geometric measure theory ==
Numerous minimality results for complex analytic manifolds are based on the [[Wirtinger inequality (2-forms)|Wirtinger inequality for 2-forms]].  A succinct proof may be found in [[Herbert Federer]]'s classic text ''Geometric Measure Theory''.  The Wirtinger inequality is also a key ingredient in [[Gromov's inequality for complex projective space]] in [[systolic geometry]].

== See also ==
* [[Closed and exact differential forms]]
* [[Complex differential form]]
* [[Vector-valued differential form]]
* [[Equivariant differential form]]
* [[Calculus on Manifolds (book)|''Calculus on Manifolds'']]
* [[Multilinear form]]
* [[Polynomial differential form]]
* [[Presymplectic form]]

== Notes ==
{{reflist}}

== References ==
{{refbegin}}
* {{citation | first=David|last= Bachman | title=A Geometric Approach to Differential Forms | publisher=Birkhäuser | year = 2006 | isbn =978-0-8176-4499-4 }}
* {{citation | first=David|last= Bachman | title=A Geometric Approach to Differential Forms   | year = 2003 | 	arxiv=math/0306194v1 |bibcode=2003math......6194B}}
* {{citation | first=Henri|last= Cartan | author-link=Henri Cartan |title=Differential Forms | publisher=Dover  | year = 2006 | 	isbn= 0-486-45010-4 }}—Translation of ''Formes différentielles'' (1967)
* {{citation | author-link=Jean Dieudonné|first=Jean|last=Dieudonné| title=Treatise on Analysis | volume=3 | location=New York-London| publisher=Academic Press, Inc. | year=1972 | mr=0350769}}
* {{citation | first=Harold M. |last= Edwards | author-link = Harold M. Edwards | title=Advanced Calculus; A Differential Forms Approach |series= Modern Birkhäuser Classics | location=Boston, Basel, Berlin | publisher= Birkhäuser | year=1994 |doi=10.1007/978-0-8176-8412-9 | isbn = 978-0-8176-8411-2}}
* {{Citation | first = Gerald B. | last = Folland | author-link = Gerald Folland | title = Real Analysis: Modern Techniques and Their Applications | edition = Second | isbn = 978-0-471-31716-6 | year = 1999 | publisher = John Wiley & Sons | postscript =, }} provides a brief discussion of integration on manifolds from the point of view of measure theory in the last section.
* {{citation | first=Harley|last= Flanders | author-link=Harley Flanders|title=Differential forms with applications to the physical sciences | location=Mineola, New York | publisher=Dover Publications | year=1989 | orig-year=1964|isbn = 0-486-66169-5}}
* {{citation|first=Wendell H.|last= Fleming|author-link=Wendell Fleming|year=1965|title=Functions of Several Variables|publisher= Addison-Wesley|chapter=Chapter 6: Exterior algebra and differential calculus|pages=205&ndash;238 | postscript =. }} This textbook in [[multivariate calculus]] introduces the exterior algebra of differential forms at the college calculus level.
* {{citation | first=Shigeyuki | last=Morita | title=Geometry of Differential Forms | publisher=AMS | year=2001 | isbn=0-8218-1045-6 | url-access=registration | url=https://archive.org/details/geometryofdiffer00mori }}
* {{citation | author-link=Walter Rudin|first=Walter|last=Rudin | title=Principles of Mathematical Analysis | location=New York | publisher=McGraw-Hill | year=1976 | isbn = 0-07-054235-X}}
* {{citation |last=Spivak |first=Michael| title=Calculus on Manifolds |url=https://archive.org/details/SpivakM.CalculusOnManifolds_201703| year = 1965 | postscript =, | author-link=Michael Spivak| location=Menlo Park, California | publisher=W. A. Benjamin | isbn = 0-8053-9021-9 }} standard introductory text.
* {{citation | first=Loring W.|last= Tu | author-link=Loring W. Tu| title=An Introduction to Manifolds | publisher=Springer | year = 2008 | isbn = 978-0-387-48098-5 | doi=10.1007/978-1-4419-7400-6 |series= Universitext }}
* {{citation | author-link=Vladimir A. Zorich|first=Vladimir A.|last= Zorich | title=Mathematical Analysis II | publisher=Springer | year = 2004 | isbn = 3-540-40633-6 }}
{{refend}}

== External links ==
* {{mathworld|Differentialk-Form}}
* {{citation|url=http://pi.math.cornell.edu/~sjamaar/manifolds/manifold.pdf|title=Manifolds and differential forms lecture notes|first=Reyer|last=Sjamaar|year=2006}}, a course taught at [[Cornell University]].
* {{citation|arxiv=math/0306194 |title=A Geometric Approach to Differential Forms|first=David|last=Bachman|year=2003|bibcode=2003math......6194B}}, an undergraduate text.
* [[Tristan Needham|Needham, Tristan]]. ''[https://www.vdgf.space/table-of-contents Visual differential geometry and forms: a mathematical drama in five acts]''. Princeton University Press, 2021.

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[[Category:Differential forms| ]]
[[Category:Differential geometry]]