In the

exterior product
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...

of two differential forms, the exterior derivative
On a differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-interse ...

of a single differential form, the

exterior algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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

mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

fields of differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...

and tensor calculus
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, differential forms are an approach to multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one Variable (mathematics), variable to calculus with function of several variables, functions of several variables: the Differential calculus, different ...

that is independent of coordinate
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

s. Differential forms provide a unified approach to define integrand
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s over curves, surfaces, solids, and higher-dimensional manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

s. The modern notion of differential forms was pioneered by Élie Cartan
Élie Joseph Cartan, ForMemRS
Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a ...

. It has many applications, especially in geometry, topology and physics.
For instance, the expression from one-variable calculus is an example of a '' -form'', and can be integrated
Integration may refer to:
Biology
*Modular integration, where different parts in a module have a tendency to vary together
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome ...

over an oriented interval in the domain of :
:$\backslash int\_a^b\; f(x)\backslash ,dx.$
Similarly, the expression is a -form that has a surface integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

over an oriented
is non-orientable
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical a ...

surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile.
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...

:
:$\backslash int\_S\; (f(x,y,z)\backslash ,dx\backslash wedge\; dy\; +\; g(x,y,z)\backslash ,dz\backslash wedge\; dx\; +\; h(x,y,z)\backslash ,dy\backslash wedge\; dz).$
The symbol denotes the exterior product
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...

, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form represents a volume elementIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

that can be integrated over an oriented region of space. In general, a -form is an object that may be integrated over a -dimensional oriented manifold, and is homogeneous of degree in the coordinate differentials.
On an -dimensional manifold, the top-dimensional form (-form) is called a ''volume formIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

''.
The algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

of differential forms is organized in a way that naturally reflects the orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building design ...

of the domain of integration. There is an operation on differential forms known as the exterior derivative
On a differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-interse ...

that, when given a -form as input, produces a -form as output. This operation extends the differential of a function
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...

, and is directly related to the divergence
In vector calculus
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

and the curl
Curl or CURL may refer to:
Science and technology
* Curl (mathematics)
In vector calculus
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimension ...

of a vector field in a manner that makes the fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating the gradient) with the concept of integral, integrating a function (calculating the area under t ...

, the divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the flux of a vector field through a closed surface (mathematics), surface to the divergence of the fie ...

, Green's theorem
In vector calculus, Green's theorem relates a line integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geome ...

, and Stokes' theorem
Stokes' theorem, also known as Kelvin–Stokes theorem
Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)"
:ja:培風館, Ba ...

special cases of the same general result, known in this context also as the generalized Stokes theorem
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integral, integration of differen ...

. In a deeper way, this theorem relates the topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of the domain of integration to the structure of the differential forms themselves; the precise connection is known as de Rham's theorem#Redirect DE {{Redirect category shell, 1=
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.
The general setting for the study of differential forms is on a differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...

. Differential -forms are naturally dual to vector field
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product ...

s on a manifold, and the pairing between vector fields and -forms is extended to arbitrary differential forms by the interior product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a Pushforward (disambiguation), pushforward.
Precomposition
Precomposition with a Function (mathematics), function probabl ...

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
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...

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 toÉlie Cartan
Élie Joseph Cartan, ForMemRS
Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a ...

with reference to his 1899 paper. Some aspects of the exterior algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of differential forms appears in Hermann Grassmann
Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath, known in his day as a linguistics, linguist and now also as a mathematics, mathematician. He was also a physics, physicist, gener ...

's 1844 work, ''Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics)''.
Concept

Differential forms provide an approach tomultivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one Variable (mathematics), variable to calculus with function of several variables, functions of several variables: the Differential calculus, different ...

that is independent of coordinate
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

s.
Integration and orientation

A differential -form can be integrated over an orientedmanifold
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of dimension . A differential -form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density. A differential -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 oriented
is non-orientable
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical a ...

manifolds
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...

. An example of a 1-dimensional manifold is an interval , and intervals can be given an orientation: they are positively oriented if , and negatively oriented otherwise. If then the integral of the differential -form over the interval (with its natural positive orientation) is
:$\backslash int\_a^b\; f(x)\; \backslash ,dx$
which is the negative of the integral of the same differential form over the same interval, when equipped with the opposite orientation. That is:
:$\backslash int\_b^a\; f(x)\backslash ,dx\; =\; -\backslash int\_a^b\; f(x)\backslash ,dx$
This gives a geometrical context to the 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 (), the increment is negative in the direction of integration.
More generally, an -form is an oriented density that can be integrated over an -dimensional oriented manifold. (For example, a -form can be integrated over an oriented curve, a -form can be integrated over an oriented surface, etc.) If is an oriented -dimensional manifold, and is the same manifold with opposite orientation and is an -form, then one has:
:$\backslash int\_M\; \backslash omega\; =\; -\; \backslash int\_\; \backslash omega\; \backslash ,.$
These conventions correspond to interpreting the integrand as a differential form, integrated over a chain
A chain is a wikt:series#Noun, serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression (physics), compression but line (g ...

. In measure theory
Measure is a fundamental concept of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...

, by contrast, one interprets the integrand as a function with respect to a measure and integrates over a subset , without any notion of orientation; one writes $\backslash int\_A\; f\backslash ,d\backslash mu\; =\; \backslash int\_\; f\backslash ,d\backslash mu$ to indicate integration over a subset . This is a minor distinction in one dimension, but becomes subtler on higher-dimensional manifolds; see below
Below may refer to:
*Earth
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*Soil
*Floor
*Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Fred Below (1926–1988), American blues drummer
*Fritz von Below (1853 ...

for details.
Making the notion of an oriented density precise, and thus of a differential form, involves the exterior algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. The differentials of a set of coordinates, , ..., can be used as a basis for all -forms. Each of these represents a covector
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

at each point on the manifold that may be thought of as measuring a small displacement in the corresponding coordinate direction. A general -form is a linear combination of these differentials at every point on the manifold:
:$f\_1\backslash ,dx^1+\backslash cdots+f\_n\backslash ,dx^n\; ,$
where the are functions of all the coordinates. A differential -form is integrated along an oriented curve as a line integral.
The expressions , where can be used as a basis at every point on the manifold for all -forms. This may be thought of as an infinitesimal oriented square parallel to the –-plane. A general -form is a linear combination of these at every point on the manifold: and it is integrated just like a surface integral.
A fundamental operation defined on differential forms is the exterior product
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...

(the symbol is the wedge
A wedge is a triangular
A triangle is a polygon
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branche ...

). This is similar to the cross product
In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

from vector calculus, in that it is an alternating product. For instance,
:$dx^1\backslash wedge\; dx^2=-dx^2\backslash wedge\; dx^1$
because the square whose first side is and second side is is to be regarded as having the opposite orientation as the square whose first side is and whose second side is . This is why we only need to sum over expressions , with ; for example: . 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 , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

in vector calculus allows one to compute the area vector of a parallelogram from vectors pointing up the two sides. Alternating also implies that , 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, if any two of the indices , ..., are equal, in the same way that the "volume" enclosed by a parallelotope whose edge vectors are linearly dependent
In the theory of vector spaces, a set of vectors is said to be if at least one of the vectors in the set can be defined as a linear combinationIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

is zero.
Multi-index notation

A common notation for the wedge product of elementary -forms is so calledmulti-index notation
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index
Index may refer to:
Arts, ...

: in an -dimensional context, for we define Another useful notation is obtained by defining the set of all strictly increasing multi-indices of length , in a space of dimension , denoted Then locally (wherever the coordinates apply), $\backslash \_$ spans the space of differential -forms in a manifold of dimension , when viewed as a module over the ring of smooth functions on . By calculating the size of $\backslash mathcal\_$ combinatorially, the module of -forms on a -dimensional manifold, and in general space of -covectors on an -dimensional vector space, is choose : 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 theexterior derivative
On a differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-interse ...

operator . The exterior derivative of a differential form is a generalization of the differential of a function
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...

, in the sense that the exterior derivative of is exactly the differential of . When generalized to higher forms, if is a simple -form, then its exterior derivative is a -form defined by taking the differential of the coefficient functions:
:$d\backslash omega\; =\; \backslash sum\_^n\; \backslash frac\; \backslash ,\; dx^i\; \backslash wedge\; dx^I.$
with extension to general -forms through linearity: if then its exterior derivative is
: $d\backslash tau\; =\; \backslash sum\_\backslash left(\backslash sum\_^n\; \backslash frac\; \backslash ,\; dx^j\backslash right)\backslash wedge\; dx^I\; \backslash in\; \backslash Omega^(M)$
In , with the Hodge star operator
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, the exterior derivative corresponds to gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

, curl
Curl or CURL may refer to:
Science and technology
* Curl (mathematics)
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the p ...

, and divergence
In vector calculus
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

, 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 geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...

, differential topology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, 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 antiderivationIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

of degree 1 on the exterior algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of differential forms. The benefit of this more general approach is that it allows for a natural coordinate-free approach to integration on manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

s. It also allows for a natural generalization of the fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating the gradient) with the concept of integral, integrating a function (calculating the area under t ...

, called the (generalized) Stokes' theorem
Stokes' theorem, also known as Kelvin–Stokes theorem
Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)"
:ja:培風館, Ba ...

, which is a central result in the theory of integration on manifolds.
Differential calculus

Let be anopen set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

in . A differential -form ("zero-form") is defined to be a smooth function
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mat ...

on – the set of which is denoted . If is any vector in , then has a directional derivative
In mathematics, the directional derivative of a multivariate differentiable function, differentiable (scalar) function along a given vector (mathematics), vector v at a given point x intuitively represents the instantaneous rate of change of the ...

, which is another function on whose value at a point is the rate of change (at ) of in the direction:
:$(\backslash partial\_v\; f)(p)\; =\; \backslash left.\; \backslash frac\; f(p+tv)\backslash \_\; .$
(This notion can be extended point-wise to the case that is a vector field
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product ...

on by evaluating at the point in the definition.)
In particular, if is the th coordinate vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis. Coordinates are always specified relative to an ordered basis. Bases and their a ...

then is the partial derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of with respect to the th coordinate function, i.e., , where , , ..., are the coordinate functions on . By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates , , ..., are introduced, then
:$\backslash frac\; =\; \backslash sum\_^n\backslash frac\backslash frac\; .$
The first idea leading to differential forms is the observation that is a linear function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of :
:$\backslash begin\; (\backslash partial\_\; f)(p)\; \&=\; (\backslash partial\_v\; f)(p)\; +\; (\backslash partial\_w\; f)(p)\; \backslash \backslash \; (\backslash partial\_\; f)(p)\; \&=\; c\; (\backslash partial\_v\; f)(p)\; \backslash end$
for any vectors , and any real number . At each point ''p'', this linear map
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

from to is denoted and called the derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

or differential of at . Thus . Extended over the whole set, the object can be viewed as a function that takes a vector field on , and returns a real-valued function whose value at each point is the derivative along the vector field of the function . Note that at each , the differential is not a real number, but a linear functional on tangent vectors, and a prototypical example of a differential -form.
Since any vector is a linear combination
In mathematics, a linear combination is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be ...

of its components
Component may refer to:
In engineering, science, and technology Generic systems
*System
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A system, surrounde ...

, is uniquely determined by for each and each , which are just the partial derivatives of on . Thus provides a way of encoding the partial derivatives of . It can be decoded by noticing that the coordinates , , ..., are themselves functions on , and so define differential -forms , , ..., . Let . Since , the Kronecker delta function
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a Function (mathematics), function of two Variable (mathematics), variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
...

, it follows that
The meaning of this expression is given by evaluating both sides at an arbitrary point : on the right hand side, the sum is defined "pointwiseIn mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined o ...

", so that
:$df\_p\; =\; \backslash sum\_^n\; \backslash frac(p)\; (dx^i)\_p\; .$
Applying both sides to , the result on each side is the th partial derivative of at . Since and were arbitrary, this proves the formula .
More generally, for any smooth functions and on , we define the differential -form pointwise by
:$\backslash alpha\_p\; =\; \backslash sum\_i\; g\_i(p)\; (dh\_i)\_p$
for each . Any differential -form arises this way, and by using it follows that any differential -form on may be expressed in coordinates as
:$\backslash alpha\; =\; \backslash sum\_^n\; f\_i\backslash ,\; dx^i$
for some smooth functions on .
The second idea leading to differential forms arises from the following question: given a differential -form on , when does there exist a function on such that ? The above expansion reduces this question to the search for a function whose partial derivatives are equal to given functions . For , such a function does not always exist: any smooth function satisfies
:$\backslash frac\; =\; \backslash frac\; ,$
so it will be impossible to find such an unless
:$\backslash frac\; -\; \backslash frac\; =\; 0$
for all and .
The skew-symmetry of the left hand side in and suggests introducing an antisymmetric product on differential -forms, the exterior product
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...

, so that these equations can be combined into a single condition
:$\backslash sum\_^n\; \backslash frac\; \backslash ,\; dx^i\; \backslash wedge\; dx^j\; =\; 0\; ,$
where is defined so that:
: $dx^i\; \backslash wedge\; dx^j\; =\; -\; dx^j\; \backslash wedge\; dx^i.$
This is an example of a differential -form. This -form is called the exterior derivative
On a differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-interse ...

of . It is given by
:$d\backslash alpha\; =\; \backslash sum\_^n\; df\_j\; \backslash wedge\; dx^j\; =\; \backslash sum\_^n\; \backslash frac\; \backslash ,\; dx^i\; \backslash wedge\; dx^j\; .$
To summarize: is a necessary condition for the existence of a function with .
Differential -forms, -forms, and -forms are special cases of differential forms. For each , there is a space of differential -forms, which can be expressed in terms of the coordinates as
:$\backslash sum\_^n\; f\_\; \backslash ,\; dx^\; \backslash wedge\; dx^\; \backslash wedge\backslash cdots\; \backslash wedge\; dx^$
for a collection of functions . Antisymmetry, which was already present for -forms, makes it possible to restrict the sum to those sets of indices for which .
Differential forms can be multiplied together using the exterior product, and for any differential -form , there is a differential -form called the exterior derivative of .
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
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

. One way to do this is cover with coordinate chartIn topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological spa ...

s and define a differential -form on to be a family of differential -forms on each chart which agree on the overlaps. However, there are more intrinsic definitions which make the independence of coordinates manifest.
Intrinsic definitions

Let be asmooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

. A smooth differential form of degree is a smooth section of the th exterior power
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...

of the cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...

of . The set of all differential -forms on a manifold is a vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

, often denoted .
The definition of a differential form may be restated as follows. At any point , a -form defines an element
:$\backslash beta\_p\; \backslash in\; ^k\; T\_p^*\; M,$
where is the tangent space
In , the tangent space of a generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can ...

to at and is its dual space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

. This space is to the fiber at of the dual bundle of the th exterior power of the tangent bundle
Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).
In differen ...

of . That is, is also a linear functional $\backslash beta\_p\; \backslash colon\; ^k\; T\_pM\; \backslash to\; \backslash mathbf$, i.e. the dual of the th exterior power is isomorphic to the th exterior power of the dual:
: $^k\; T^*\_p\; M\; \backslash cong\; \backslash Big(^k\; T\_p\; M\backslash Big)^*$
By the universal property of exterior powers, this is equivalently an alternating multilinear map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
:f\colon V_1 \times \cdots \times V_n \to W\text
where V_1,\ldots,V_n and W ar ...

:
:$\backslash beta\_p\backslash colon\; \backslash bigoplus\_^k\; T\_p\; M\; \backslash to\; \backslash mathbf.$
Consequently, a differential -form may be evaluated against any -tuple of tangent vectors to the same point of . For example, a differential -form assigns to each point a linear functional
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

on . In the presence of an inner product
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

on (induced by a Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g'p'' on the tangent space ''T'p'M'' at each poin ...

on ), may be represented as the inner product with a tangent vector
:''For a more general — but much more technical — treatment of tangent vectors, see tangent space.''
In mathematics, a tangent vector is a Vector (geometry), vector that is tangent to a curve or Surface (mathematics), surface at a given point. T ...

. Differential -forms are sometimes called 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
: $\backslash operatorname\; \backslash colon\; ^k\; T^*M\; \backslash to\; ^k\; T^*M.$
For a tensor $\backslash tau$ at a point ,
:$\backslash operatorname(\backslash tau\_p)(x\_1,\; \backslash dots,\; x\_k)\; =\; \backslash frac\backslash sum\_\; \backslash sgn(\backslash sigma)\; \backslash tau\_p(x\_,\; \backslash dots,\; x\_),$
where is the symmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

on 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
: $\backslash operatorname\; \backslash colon\; ^k\; T^*M\; \backslash to\; ^k\; T^*M.$
This map exhibits as a totally antisymmetric covariant tensor field
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of rank . The differential forms on 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 theinterior product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of a differential form and a vector field, the Lie derivative
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral ...

of a differential form with respect to a vector field and the covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection (mathematics), connection on a manifold ...

of a differential form with respect to a vector field on a manifold with a defined connection.
Exterior product

The exterior product of a -form and an -form , denoted , is a ()-form. At each point of the manifold , the forms and are elements of an exterior power of the cotangent space at . 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 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 is not alternating. There is an explicit formula which describes the exterior product in this situation. The exterior product is :$\backslash alpha\; \backslash wedge\; \backslash beta\; =\; \backslash operatorname(\backslash alpha\; \backslash otimes\; \backslash beta).$ If the embedding of $^n\; T^*M$ into $^n\; T^*M$ is done via the map $n!\backslash operatorname$ instead of $\backslash operatorname$, the exterior product is :$\backslash alpha\; \backslash wedge\; \backslash beta\; =\; \backslash frac\backslash operatorname(\backslash alpha\; \backslash otimes\; \backslash beta).$ This description is useful for explicit computations. For example, if , then is the -form whose value at a point is the alternating bilinear form defined by :$(\backslash alpha\backslash wedge\backslash beta)\_p(v,w)=\backslash alpha\_p(v)\backslash beta\_p(w)\; -\; \backslash alpha\_p(w)\backslash beta\_p(v)$ for . The exterior product is bilinear: If , , and are any differential forms, and if is any smooth function, then :$\backslash alpha\; \backslash wedge\; (\backslash beta\; +\; \backslash gamma)\; =\; \backslash alpha\; \backslash wedge\; \backslash beta\; +\; \backslash alpha\; \backslash wedge\; \backslash gamma,$ :$\backslash alpha\; \backslash wedge\; (f\; \backslash cdot\; \backslash beta)\; =\; f\; \backslash cdot\; (\backslash alpha\; \backslash wedge\; \backslash beta).$ It is ''skew commutative'' (also known as ''graded commutative''), meaning that it satisfies a variant ofanticommutativity
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

that depends on the degrees of the forms: if is a -form and is an -form, then
:$\backslash alpha\; \backslash wedge\; \backslash beta\; =\; (-1)^\; \backslash beta\; \backslash wedge\; \backslash alpha\; .$
Riemannian manifold

On aRiemannian manifold
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integr ...

, or more generally a pseudo-Riemannian manifold
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differenti ...

, 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

$\backslash star\; \backslash colon\; \backslash Omega^k(M)\backslash \; \backslash stackrel\backslash \; \backslash Omega^(M)$ and the codifferential $\backslash delta\backslash colon\; \backslash Omega^k(M)\backslash rightarrow\; \backslash Omega^(M)$, which has degree and is adjoint to the exterior differential .
Vector field structures

On a pseudo-Riemannian manifold, -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 aClifford algebra
In mathematics, a Clifford algebra is an algebra over a field, algebra generated by a vector space with a quadratic form, and is a Unital algebra, unital associative algebra. As algebra over a field, ''K''-algebras, they generalize the real nu ...

, 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
METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...

. This algebra is ''distinct'' from the geometric algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.
Another alternative is to consider vector fields as derivations. The (noncommutative) algebra of differential operator
300px, A harmonic function defined on an annulus. Harmonic functions are exactly those functions which lie in the kernel of the Laplace operator, an important differential operator.
In mathematics, a differential operator is an Operator (mathe ...

s they generate is the Weyl algebra
In abstract algebra, the Weyl algebra is the ring of differential operator
300px, A harmonic function defined on an annulus. Harmonic functions are exactly those functions which lie in the kernel of the Laplace operator, an important differe ...

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 . This means that the exterior derivative defines acochain complex
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

:
:$0\backslash \; \backslash to\backslash \; \backslash Omega^0(M)\backslash \; \backslash stackrel\backslash \; \backslash Omega^1(M)\backslash \; \backslash stackrel\backslash \; \backslash Omega^2(M)\backslash \; \backslash stackrel\backslash \; \backslash Omega^3(M)\backslash \; \backslash to\backslash \; \backslash cdots\; \backslash \; \backslash to\backslash \; \backslash Omega^n(M)\backslash \; \backslash to\; \backslash \; 0.$
This complex is called the de Rham complex, and its cohomology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

is by definition the de Rham cohomology of . By the Poincaré lemma, the de Rham complex is locally exact except at . The kernel at is the space of locally constant function
In mathematics, a function (mathematics), function ''f'' from a topological space ''A'' to a Set (mathematics), set ''B'' is called locally constant if for every ''a'' in ''A'' there exists a neighborhood (topology), neighborhood ''U'' of ''a'' suc ...

s on . Therefore, the complex is a resolution of the constant sheaf (mathematics), sheaf , which in turn implies a form of de Rham's theorem: de Rham cohomology computes the sheaf cohomology of .
Pullback

Suppose that is smooth. The differential of is a smooth map between the tangent bundles of and . This map is also denoted and called the pushforward. For any point and any tangent vector , there is a well-defined pushforward vector in . However, the same is not true of a vector field. If is not injective, say because has two or more preimages, then the vector field may determine two or more distinct vectors in . If is not surjective, then there will be a point at which does not determine any tangent vector at all. Since a vector field on determines, by definition, a unique tangent vector at every point of , 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 may be viewed as a linear functional on each tangent space. Precomposing this functional with the differential defines a linear functional on each tangent space of and therefore a differential form on . 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 be smooth, and let be a smooth -form on . Then there is a differential form on , called the pullback of , which captures the behavior of as seen relative to . To define the pullback, fix a point of and tangent vectors , ..., to at . The pullback of is defined by the formula :$(f^*\backslash omega)\_p(v\_1,\; \backslash ldots,\; v\_k)\; =\; \backslash omega\_(f\_*v\_1,\; \backslash ldots,\; f\_*v\_k).$ There are several more abstract ways to view this definition. If is a -form on , then it may be viewed as a section of the cotangent bundle of . Using to denote a dual map, the dual to the differential of is . The pullback of may be defined to be the composite :$M\backslash \; \backslash stackrel\backslash \; N\backslash \; \backslash stackrel\backslash \; T^*N\backslash \; \backslash stackrel\backslash \; T^*M.$ This is a section of the cotangent bundle of and hence a differential -form on . In full generality, let $\backslash bigwedge^k\; (df)^*$ denote the th exterior power of the dual map to the differential. Then the pullback of a -form is the composite :$M\backslash \; \backslash stackrel\backslash \; N\backslash \; \backslash stackrel\backslash \; ^k\; T^*N\backslash \; \backslash stackrel\backslash \; ^k\; T^*M.$ Another abstract way to view the pullback comes from viewing a -form as a linear functional on tangent spaces. From this point of view, is a morphism of vector bundles :$^k\; TN\backslash \; \backslash stackrel\backslash \; N\; \backslash times\; \backslash mathbf,$ where is the trivial rank one bundle on . The composite map :$^k\; TM\backslash \; \backslash stackrel\backslash \; ^k\; TN\backslash \; \backslash stackrel\backslash \; N\; \backslash times\; \backslash mathbf$ defines a linear functional on each tangent space of , and therefore it factors through the trivial bundle . The vector bundle morphism $^k\; TM\; \backslash to\; M\; \backslash times\; \backslash mathbf$ defined in this way is . Pullback respects all of the basic operations on forms. If and are forms and is a real number, then :$\backslash begin\; f^*(c\backslash omega)\; \&=\; c(f^*\backslash omega),\; \backslash \backslash \; f^*(\backslash omega\; +\; \backslash eta)\; \&=\; f^*\backslash omega\; +\; f^*\backslash eta,\; \backslash \backslash \; f^*(\backslash omega\; \backslash wedge\; \backslash eta)\; \&=\; f^*\backslash omega\; \backslash wedge\; f^*\backslash eta,\; \backslash \backslash \; f^*(d\backslash omega)\; \&=\; d(f^*\backslash omega).\; \backslash end$ The pullback of a form can also be written in coordinates. Assume that , ..., are coordinates on , that , ..., are coordinates on , and that these coordinate systems are related by the formulas for all . Locally on , can be written as :$\backslash omega\; =\; \backslash sum\_\; \backslash omega\_\; \backslash ,\; dy^\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dy^,$ where, for each choice of , ..., , is a real-valued function of , ..., . Using the linearity of pullback and its compatibility with exterior product, the pullback of has the formula :$f^*\backslash omega\; =\; \backslash sum\_\; (\backslash omega\_\backslash circ\; f)\; \backslash ,\; df\_\; \backslash wedge\; \backslash cdots\; \backslash wedge\; df\_.$ Each exterior derivative can be expanded in terms of , ..., . The resulting -form can be written using Jacobian matrix and determinant, Jacobian matrices: :$f^*\backslash omega\; =\; \backslash sum\_\; \backslash sum\_\; (\backslash omega\_\backslash circ\; f)\backslash frac\; \backslash ,\; dx^\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^.$ Here, $\backslash frac$ denotes the determinant of the matrix whose entries are $\backslash frac$, $1\backslash leq\; m,n\backslash leq\; k$.Integration

A differential -form can be integrated over an oriented -dimensional manifold. When the -form is defined on an -dimensional manifold with , then the -form can be integrated over oriented -dimensional submanifolds. If , integration over oriented 0-dimensional submanifolds is just the summation of the integrand evaluated at points, with according to the orientation of those points. Other values of 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 be an open subset of . Give its standard orientation and the restriction of that orientation. Every smooth -form on has the form :$\backslash omega\; =\; f(x)\backslash ,dx^1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^n$ for some smooth function . Such a function has an integral in the usual Riemann or Lebesgue sense. This allows us to define the integral of to be the integral of : :$\backslash int\_U\; \backslash omega\backslash \; \backslash stackrel\; \backslash int\_U\; f(x)\backslash ,dx^1\; \backslash cdots\; dx^n.$ Fixing an orientation is necessary for this to be well-defined. The skew-symmetry of differential forms means that the integral of, say, must be the negative of the integral of . 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 be an -manifold and an -form on . First, assume that there is a parametrization of by an open subset of Euclidean space. That is, assume that there exists a diffeomorphism :$\backslash varphi\; \backslash colon\; D\; \backslash to\; M$ where . Give the orientation induced by . Then defines the integral of over to be the integral of over . In coordinates, this has the following expression. Fix a chart on with coordinates . Then :$\backslash omega\; =\; \backslash sum\_\; a\_()\backslash ,dx^\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^.$ Suppose that is defined by :$\backslash varphi()\; =\; (x^1(),\backslash ldots,x^n()).$ Then the integral may be written in coordinates as :$\backslash int\_M\; \backslash omega\; =\; \backslash int\_D\; \backslash sum\_\; a\_(\backslash varphi())\; \backslash frac\backslash ,du^1\; \backslash cdots\; du^n,$ where :$\backslash frac$ is the determinant of the Jacobian matrix and determinant, Jacobian. The Jacobian exists because is differentiable. In general, an -manifold cannot be parametrized by an open subset of . 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 -dimensional subsets for , and this makes it possible to define integrals of -forms. To make this precise, it is convenient to fix a standard domain in , usually a cube or a simplex. A -chain is a formal sum of smooth embeddings . That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity. Each smooth embedding determines a -dimensional submanifold of . If the chain is :$c\; =\; \backslash sum\_^r\; m\_i\; \backslash varphi\_i,$ then the integral of a -form over is defined to be the sum of the integrals over the terms of : :$\backslash int\_c\; \backslash omega\; =\; \backslash sum\_^r\; m\_i\; \backslash int\_D\; \backslash varphi\_i^*\backslash omega.$ This approach to defining integration does not assign a direct meaning to integration over the whole manifold . 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 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 , which does directly assign a meaning to integration over , but this approach requires fixing an orientation of . The integral of an -form on an -dimensional manifold is defined by working in charts. Suppose first that is supported on a single positively oriented chart. On this chart, it may be pulled back to an -form on an open subset of . 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 is independent of the chosen chart. In the general case, use a partition of unity to write as a sum of -forms, each of which is supported in a single positively oriented chart, and define the integral of to be the sum of the integrals of each term in the partition of unity. It is also possible to integrate -forms on oriented -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 , integrating a -form on the path is simply pulling back the form to a form on , and this integral is the integral of the function on the interval.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 and be two orientable manifolds of pure dimensions and , respectively. Suppose that is a surjective submersion. This implies that each fiber is -dimensional and that, around each point of , there is a chart on which looks like the projection from a product onto one of its factors. Fix and set . Suppose that :$\backslash begin\; \backslash omega\_x\; \&\backslash in\; ^m\; T\_x^*M,\; \backslash \backslash \; \backslash eta\_y\; \&\backslash in\; ^n\; T\_y^*N,\; \backslash end$ and that does not vanish. Following , there is a unique :$\backslash sigma\_x\; \backslash in\; ^\; T\_x^*(f^(y))$ which may be thought of as the fibral part of with respect to . More precisely, define to be the inclusion. Then is defined by the property that :$\backslash omega\_x\; =\; (f^*\backslash eta\_y)\_x\; \backslash wedge\; \backslash sigma\text{'}\_x\; \backslash in\; ^m\; T\_x^*M,$ where :$\backslash sigma\text{'}\_x\; \backslash in\; ^\; T\_x^*M$ is any -covector for which :$\backslash sigma\_x\; =\; j^*\backslash sigma\text{'}\_x.$ The form may also be notated . Moreover, for fixed , varies smoothly with respect to . That is, suppose that :$\backslash omega\; \backslash colon\; f^(y)\; \backslash to\; T^*M$ is a smooth section of the projection map; we say that is a smooth differential -form on along . Then there is a smooth differential -form on such that, at each , :$\backslash sigma\_x\; =\; \backslash omega\_x\; /\; \backslash eta\_y.$ This form is denoted . The same construction works if is an -form in a neighborhood of the fiber, and the same notation is used. A consequence is that each fiber is orientable. In particular, a choice of orientation forms on and defines an orientation of every fiber of . The analog of Fubini's theorem is as follows. As before, and are two orientable manifolds of pure dimensions and , and is a surjective submersion. Fix orientations of and , and give each fiber of the induced orientation. Let be an -form on , and let be an -form on that is almost everywhere positive with respect to the orientation of . Then, for almost every , the form is a well-defined integrable form on . Moreover, there is an integrable -form on defined by :$y\; \backslash mapsto\; \backslash bigg(\backslash int\_\; \backslash theta\; /\; \backslash zeta\_y\backslash bigg)\backslash ,\backslash zeta\_y.$ Denote this form by :$\backslash bigg(\backslash int\_\; \backslash theta\; /\; \backslash zeta\backslash bigg)\backslash ,\backslash zeta.$ Then proves the generalized Fubini formula :$\backslash int\_M\; \backslash theta\; =\; \backslash int\_N\; \backslash bigg(\backslash int\_\; \backslash theta\; /\; \backslash zeta\backslash bigg)\backslash ,\backslash zeta.$ It is also possible to integrate forms of other degrees along the fibers of a submersion. Assume the same hypotheses as before, and let be a compactly supported -form on . Then there is a -form on which is the result of integrating along the fibers of . The form is defined by specifying, at each , how pairs against each -vector at , and the value of that pairing is an integral over that depends only on , , and the orientations of and . More precisely, at each , there is an isomorphism :$^k\; T\_yN\; \backslash to\; ^\; T\_y^*N$ defined by the interior product :$\backslash mathbf\; \backslash mapsto\; \backslash mathbf\backslash ,\backslash lrcorner\backslash ,\backslash zeta\_y.$ If , then a -vector at determines an -covector at by pullback: :$f^*(\backslash mathbf\backslash ,\backslash lrcorner\backslash ,\backslash zeta\_y)\; \backslash in\; ^\; T\_x^*M.$ Each of these covectors has an exterior product against , so there is an -form on along defined by :$(\backslash beta\_)\_x\; =\; \backslash left(\backslash alpha\_x\; \backslash wedge\; f^*(\backslash mathbf\backslash ,\backslash lrcorner\backslash ,\backslash zeta\_y)\backslash right)\; \backslash big/\; \backslash zeta\_y\; \backslash in\; ^\; T\_x^*M.$ This form depends on the orientation of but not the choice of . Then the -form is uniquely defined by the property :$\backslash langle\backslash gamma\_y,\; \backslash mathbf\backslash rangle\; =\; \backslash int\_\; \backslash beta\_(x),$ and is smooth . This form also denoted and called the integral of along the fibers of . Integration along fibers is important for the construction of Gysin maps in de Rham cohomology. Integration along fibers satisfies the projection formula . If is any -form on , then :$\backslash alpha^\backslash flat\; \backslash wedge\; \backslash lambda\; =\; (\backslash alpha\; \backslash wedge\; f^*\backslash lambda)^\backslash flat.$Stokes's theorem

The fundamental relationship between the exterior derivative and integration is given by theStokes' theorem
Stokes' theorem, also known as Kelvin–Stokes theorem
Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)"
:ja:培風館, Ba ...

: If is an ()-form with compact support on and denotes the manifold, boundary of with its induced Orientation (mathematics), orientation, then
:$\backslash int\_M\; d\backslash omega\; =\; \backslash int\_\; \backslash omega.$
A key consequence of this is that "the integral of a closed form over homologous chains is equal": If is a closed -form and and are -chains that are homologous (such that is the boundary of a -chain ), then $\backslash textstyle$, since the difference is the integral $\backslash textstyle\backslash int\_W\; d\backslash omega\; =\; \backslash int\_W\; 0\; =\; 0$.
For example, if is the derivative of a potential function on the plane or , then the integral of over a path from to does not depend on the choice of path (the integral is ), 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
The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating the gradient) with the concept of integral, integrating a function (calculating the area under t ...

. 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

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 -form over the interval . Assuming the usual distance (and thus measure) on the real line, this integral is either or , depending on ''orientation:'' $\backslash textstyle$, while $\backslash textstyle$. By contrast, the integral of the ''measure'' on the interval is unambiguously (i.e. the integral of the constant function with respect to this measure is ). Similarly, under a change of coordinates a differential -form changes by the Jacobian determinant , while a measure changes by the ''absolute value'' of the Jacobian determinant, , which further reflects the issue of orientation. For example, under the map on the line, the differential form pulls back to ; orientation has reversed; while the Lebesgue measure, which here we denote , pulls back to ; it does not change. In the presence of the additional data of an ''orientation'', it is possible to integrate -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, . Formally, in the presence of an orientation, one may identify -forms with densities on a manifold; densities in turn define a measure, and thus can be integrated . On an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate -forms over compact subsets, with the two choices differing by a sign. On non-orientable manifold, -forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere (there are novolume formIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s on non-orientable manifolds), but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate -forms. One can instead identify densities with top-dimensional Volume form#Relation to measures, pseudoforms.
Even in the presence of an orientation, there is in general no meaningful way to integrate -forms over subsets for because there is no consistent way to use the ambient orientation to orient -dimensional subsets. Geometrically, a -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 vectors in an -dimensional space, which, unlike the determinant of vectors, is always positive, corresponding to a squared number. An orientation of a -submanifold is therefore extra data not derivable from the ambient manifold.
On a Riemannian manifold, one may define a -dimensional Hausdorff measure for any (integer or real), which may be integrated over -dimensional subsets of the manifold. A function times this Hausdorff measure can then be integrated over -dimensional subsets, providing a measure-theoretic analog to integration of -forms. The -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 -currents on is the dual space to an appropriate space of differential -forms. Currents play the role of generalized domains of integration, similar to but even more flexible than chains.Applications in physics

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 :$\backslash textbf\; =\; \backslash frac\; 1\; 2\; f\_\backslash ,\; dx^a\; \backslash wedge\; dx^b\backslash ,,$ where the are formed from the electromagnetic fields $\backslash vec\; E$ and $\backslash vec\; B$; e.g., , , or equivalent definitions. This form is a special case of the curvature form on the 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 , when represented in some gauge. One then has :$\backslash textbf\; =\; d\backslash textbf.$ The current -form is : $\backslash textbf\; =\; \backslash frac\; 1\; 6\; j^a\backslash ,\; \backslash varepsilon\_\backslash ,\; dx^b\; \backslash wedge\; dx^c\; \backslash wedge\; dx^d\backslash ,,$ where are the four components of the current density. (Here it is a matter of convention to write instead of , i.e. to use capital letters, and to write instead of . 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 $\backslash vec\; J$ for several decades, and by some publishers ; 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 :$\backslash begin\; d\; \&=\; \backslash textbf\; \backslash \backslash \; d\; \&=\; \backslash textbf,\; \backslash end$ where $\backslash star$ denotes the Hodge star operator. Similar considerations describe the geometry of gauge theories in general. The -form $\backslash mathbf$, which is duality (mathematics), dual to the Faraday form, is also called Maxwell 2-form. Electromagnetism is an example of a gauge theory. Here the Lie group is , 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 in such theories is the curvature form of the connection, which is represented in a gauge by a Lie algebra-valued one-form . The Yang–Mills field is then defined by :$\backslash mathbf\; =\; d\backslash mathbf\; +\; \backslash mathbf\backslash wedge\backslash mathbf.$ In the abelian case, such as electromagnetism, , but this does not hold in general. Likewise the field equations are modified by additional terms involving exterior products of and , 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 formNotes

References

* * * —Translation of ''Formes différentielles'' (1967) * * * * * * * * * *External links

* * , a course taught at Cornell University. * , an undergraduate text. * {{Authority control Differential forms, Differential geometry