{"history":[],"article":{"id":"58f72b7a-dbb1-4ad9-8c23-c20542b6e6aa","title":"Differential form","slug":"differential-form","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **differential forms** provide a unified approach to define [integrands](https://bliptext.com/articles/integrand) over curves, surfaces, solids, and higher-dimensional [manifolds](https://bliptext.com/articles/manifold).\nThe modern notion of differential forms was pioneered by [Élie Cartan](https://bliptext.com/articles/lie-cartan).\nIt has many applications, especially in geometry, topology and physics.\nFor instance, the expression\nf(x) dx\nis an example of a [[1-form|\n1\n-form]], and can be [integrated](https://bliptext.com/articles/integral) over an interval\n[a, b]\ncontained in the domain of\nf\nSimilarly, the expression\nf(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz\nis a '''\n2\n-form''' that can be integrated over a [surface](https://bliptext.com/articles/surface-mathematics)\nS\nThe symbol\n∧\ndenotes the [exterior product](https://bliptext.com/articles/exterior-product), sometimes called the wedge product, of two differential forms.\nLikewise, a '''\n3\n-form'''\nf(x, y, z) dx ∧ dy ∧ dz\nrepresents a [volume element](https://bliptext.com/articles/volume-element) that can be integrated over a region of space.\nIn general, a k-form is an object that may be integrated over a k-dimensional manifold, and is [homogeneous](https://bliptext.com/articles/homogeneous-polynomial) of degree k in the coordinate differentials\nOn an\nn\n-dimensional manifold, the top-dimensional form (\nn\n-form) is called a [volume form](https://bliptext.com/articles/volume-form).\nThe differential forms form an [alternating algebra](https://bliptext.com/articles/alternating-algebra).\nThis implies that and This alternating property reflects the [orientation](https://bliptext.com/articles/orientation-mathematics) of the domain of integration.\nThe [exterior derivative](https://bliptext.com/articles/exterior-derivative) is an operation on differential forms that, given a\nk\n-form \\varphi, produces a\n(k+1)\n-form d\\varphi.\nThis operation extends the [differential of a function](https://bliptext.com/articles/differential-of-a-function) (a function can be considered as a\n0\n-form, and its differential is ). This allows expressing the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus), the [divergence theorem](https://bliptext.com/articles/divergence-theorem), [Green's theorem](https://bliptext.com/articles/green-s-theorem), and [Stokes' theorem](https://bliptext.com/articles/stokes-theorem) as special cases of a single general result, the [generalized Stokes theorem](https://bliptext.com/articles/generalized-stokes-theorem).\nDifferential\n1\n-forms are naturally dual to [vector fields](https://bliptext.com/articles/vector-field) on a [differentiable manifold](https://bliptext.com/articles/differentiable-manifold), and the pairing between [vector fields](https://bliptext.com/articles/vector-field) and\n1\n-forms is extended to arbitrary differential forms by the [interior product](https://bliptext.com/articles/interior-product).\nThe algebra of differential forms along with the exterior derivative defined on it is preserved by the [pullback](https://bliptext.com/articles/pullback-differential-geometry) under smooth functions between two manifolds.\nThis 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.\nAs an example, the [change of variables formula](https://bliptext.com/articles/change-of-variables-formula) for integration becomes a simple statement that an integral is preserved under pullback.\n\n# History\nDifferential forms are part of the field of differential geometry, influenced by linear algebra.\nAlthough the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to [Élie Cartan](https://bliptext.com/articles/lie-cartan) with reference to his 1899 paper.\nSome aspects of the [exterior algebra](https://bliptext.com/articles/exterior-algebra) of differential forms appears in [Hermann Grassmann](https://bliptext.com/articles/hermann-grassmann)'s 1844 work, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics).\n\n# Concept\nDifferential forms provide an approach to [multivariable calculus](https://bliptext.com/articles/multivariable-calculus) that is independent of [coordinates](https://bliptext.com/articles/coordinate).\n\n## Integration and orientation\nA differential k-form can be integrated over an oriented [manifold](https://bliptext.com/articles/manifold-mathematics) of dimension k. A differential\n1\n-form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density.\nA differential\n2\n-form can be thought of as measuring an infinitesimal oriented area, or 2-dimensional oriented density.\nAnd so on.\nIntegration of differential forms is well-defined only on [oriented](https://bliptext.com/articles/orientability) [manifolds](https://bliptext.com/articles/manifold-mathematics).\nAn example of a 1-dimensional manifold is an interval\n[a, b]\n, and intervals can be given an orientation: they are positively oriented if\na < b\n, and negatively oriented otherwise.\nIf\na < b\nthen the integral of the differential\n1\n-form\nf(x) dx\nover the interval\n[a, b]\n(with its natural positive orientation) is which is the negative of the integral of the same differential form over the same interval, when equipped with the opposite orientation.\nThat is:\nThis gives a geometrical context to the [conventions](https://bliptext.com/articles/integral) for one-dimensional integrals, that the sign changes when the orientation of the interval is reversed.\nA standard explanation of this in one-variable integration theory is that, when the limits of integration are in the opposite order (\nb < a\n), the increment\ndx\nis negative in the direction of integration.\nMore generally, an m-form is an oriented density that can be integrated over an m-dimensional oriented manifold.\n(For example, a\n1\n-form can be integrated over an oriented curve, a\n2\n-form can be integrated over an oriented surface, etc.) If M is an oriented m-dimensional manifold, and\nM\nis the same manifold with opposite orientation and ω is an m-form, then one has:\nThese conventions correspond to interpreting the integrand as a differential form, integrated over a [chain](https://bliptext.com/articles/chain-algebraic-topology).\nIn [measure theory](https://bliptext.com/articles/measure-theory), by contrast, one interprets the integrand as a function f with respect to a measure μ and integrates over a subset A, without any notion of orientation; one writes to indicate integration over a subset A. This is a minor distinction in one dimension, but becomes subtler on higher-dimensional manifolds; see [below](https://bliptext.com/articles/) for details.\nMaking the notion of an oriented density precise, and thus of a differential form, involves the [exterior algebra](https://bliptext.com/articles/exterior-algebra).\nThe differentials of a set of coordinates,\ndx1\n, ...,\ndx\ncan be used as a basis for all\n1\n-forms.\nEach of these represents a [covector](https://bliptext.com/articles/covector) at each point on the manifold that may be thought of as measuring a small displacement in the corresponding coordinate direction.\nA general\n1\n-form is a linear combination of these differentials at every point on the manifold:\nwhere the\nfk = fk(x1, ..., xn)\nare functions of all the coordinates.\nA differential\n1\n-form is integrated along an oriented curve as a line integral.\nThe expressions\ndx ∧ dx\n, where\ni < j\ncan be used as a basis at every point on the manifold for all\n2\n-forms.\nThis may be thought of as an infinitesimal oriented square parallel to the\nx\n–\nx\n-plane.\nA general\n2\n-form is a linear combination of these at every point on the manifold:, and it is integrated just like a surface integral.\nA fundamental operation defined on differential forms is the [exterior product](https://bliptext.com/articles/exterior-product) (the symbol is the [wedge](https://bliptext.com/articles/wedge-symbol)\n∧\n). This is similar to the [cross product](https://bliptext.com/articles/cross-product) from vector calculus, in that it is an alternating product. For instance,\nbecause the square whose first side is\ndx1\nand second side is\ndx2\nis to be regarded as having the opposite orientation as the square whose first side is\ndx2\nand whose second side is\ndx1\n. This is why we only need to sum over expressions\ndx ∧ dx\n, with\ni < j\na(dx ∧ dx) + b(dx ∧ dx) = (a − b) dx ∧ dx\n. 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](https://bliptext.com/articles/cross-product) in vector calculus allows one to compute the area vector of a parallelogram from vectors pointing up the two sides.\nAlternating also implies that\ndx ∧ dx = 0\n, 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.\nIn higher dimensions,\ndx ∧ ⋅⋅⋅ ∧ dx = 0\nif any two of the indices\ni1\n, ...,\nim\nare equal, in the same way that the \"volume\" enclosed by a [parallelotope](https://bliptext.com/articles/parallelepiped) whose edge vectors are [linearly dependent](https://bliptext.com/articles/linear-independence) is zero.\n\n## Multi-index notation\nA common notation for the wedge product of elementary k-forms is so called [multi-index notation](https://bliptext.com/articles/multi-index-notation): in an n-dimensional context, for, we define.\nAnother useful notation is obtained by defining the set of all strictly increasing multi-indices of length k, in a space of dimension n, denoted.\nThen locally (wherever the coordinates apply), spans the space of differential k-forms in a manifold M of dimension n, when viewed as a module over the ring\nC∞(M)\nof smooth functions on M. By calculating the size of combinatorially, the module of k-forms on an n-dimensional manifold, and in general space of k-covectors on an n-dimensional vector space, is n choose k:.\nThis also demonstrates that there are no nonzero differential forms of degree greater than the dimension of the underlying manifold.\n\n## The exterior derivative\nIn addition to the exterior product, there is also the [exterior derivative](https://bliptext.com/articles/exterior-derivative) operator\nd\n. The exterior derivative of a differential form is a generalization of the [differential of a function](https://bliptext.com/articles/differential-of-a-function), in the sense that the exterior derivative of\nf ∈ C∞(M) = Ω0(M)\nis exactly the differential of f. When generalized to higher forms, if\nω = f dx\nis a simple k-form, then its exterior derivative\ndω\nis a\n(k + 1)\n-form defined by taking the differential of the coefficient functions:\nwith extension to general k-forms through linearity: if, then its exterior derivative is\nIn\nR3\n, with the [Hodge star operator](https://bliptext.com/articles/hodge-star-operator), the exterior derivative corresponds to [gradient](https://bliptext.com/articles/gradient), [curl](https://bliptext.com/articles/curl-mathematics), and [divergence](https://bliptext.com/articles/divergence), although this correspondence, like the cross product, does not generalize to higher dimensions, and should be treated with some caution.\nThe exterior derivative itself applies in an arbitrary finite number of dimensions, and is a flexible and powerful tool with wide application in [differential geometry](https://bliptext.com/articles/differential-geometry), [differential topology](https://bliptext.com/articles/differential-topology), and many areas in physics.\nOf 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 [antiderivation](https://bliptext.com/articles/derivation-differential-algebra) of degree 1 on the [exterior algebra](https://bliptext.com/articles/exterior-algebra) of differential forms.\nThe benefit of this more general approach is that it allows for a natural coordinate-free approach to integrate on [manifolds](https://bliptext.com/articles/manifold).\nIt also allows for a natural generalization of the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus), called the (generalized) [Stokes' theorem](https://bliptext.com/articles/stokes-theorem), which is a central result in the theory of integration on manifolds.\n\n## Differential calculus\nLet U be an [open set](https://bliptext.com/articles/open-set) in\nRn\n. A differential\n0\n-form (\"zero-form\") is defined to be a [smooth function](https://bliptext.com/articles/smooth-function) f on U – the set of which is denoted\nC∞(U)\n. If\nv\nis any vector in\nRn\n, then\nf\nhas a [directional derivative](https://bliptext.com/articles/directional-derivative)\n∂v f\n, which is another function on U whose value at a point\np ∈ U\nis the rate of change (at p) of f in the\nv\ndirection:\n(This notion can be extended pointwise to the case that\nv\nis a [vector field](https://bliptext.com/articles/vector-field) on U by evaluating\nv\nat the point p in the definition.)\nIn particular, if\nv = ej\nis the jth [coordinate vector](https://bliptext.com/articles/coordinate-vector) then\n∂v f\nis the [partial derivative](https://bliptext.com/articles/partial-derivative) of f with respect to the jth coordinate vector, i.e.,\n∂f / ∂x\n, where\nx\n,\nx\n, ...,\nx\nare the coordinate vectors in U. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates\ny\n,\ny\n, ...,\ny\nare introduced, then\nThe first idea leading to differential forms is the observation that\n∂v f (p)\nis a [linear function](https://bliptext.com/articles/linear-function) of\nv\nfor any vectors\nv\n,\nw\nand any real number c. At each point p, this [linear map](https://bliptext.com/articles/linear-map) from\nRn\nto\nR\nis denoted\ndfp\nand called the [derivative](https://bliptext.com/articles/derivative) or [differential](https://bliptext.com/articles/differential-of-a-function) of f at p. Thus\ndfp(v) = ∂v f (p)\n. Extended over the whole set, the object\ndf\ncan be viewed as a function that takes a vector field on U, and returns a real-valued function whose value at each point is the derivative along the vector field of the function f. Note that at each p, the differential\ndfp\nis not a real number, but a linear functional on tangent vectors, and a prototypical example of a differential [[1-form|\n1\n-form]].\nSince any vector\nv\nis a [linear combination](https://bliptext.com/articles/linear-combination)\nΣ vej\nof its [components](https://bliptext.com/articles/euclidean-vector),\ndf\nis uniquely determined by\ndfp(ej)\nfor each\nj\nand each\np ∈ U\n, which are just the partial derivatives of f on U. Thus\ndf\nprovides a way of encoding the partial derivatives of f. It can be decoded by noticing that the coordinates\nx\n,\nx2\n, ...,\nx\nare themselves functions on U, and so define differential\n1\n-forms\ndx\n,\ndx\n, ...,\ndx\n. Let\nf = x\n. Since\n∂x / ∂x = δij\n, the [Kronecker delta function](https://bliptext.com/articles/kronecker-delta-function), it follows that\nThe meaning of this expression is given by evaluating both sides at an arbitrary point p: on the right hand side, the sum is defined \"[pointwise](https://bliptext.com/articles/pointwise)\", so that\nApplying both sides to\nej\n, the result on each side is the jth partial derivative of f at p. Since p and j were arbitrary, this proves the formula.\nMore generally, for any smooth functions\ngi\nand\nhi\non U, we define the differential\n1\n-form\nα = Σi gi dhi\npointwise by for each\np ∈ U\n. Any differential\n1\n-form arises this way, and by using it follows that any differential\n1\n-form α on U may be expressed in coordinates as for some smooth functions\nfi\non U.\nThe second idea leading to differential forms arises from the following question: given a differential\n1\n-form α on U, when does there exist a function f on U such that\nα = df\n? The above expansion reduces this question to the search for a function f whose partial derivatives\n∂f / ∂x\nare equal to n given functions\nfi\n. For\nn > 1\n, such a function does not always exist: any smooth function f satisfies\nso it will be impossible to find such an f unless\nfor all i and j.\nThe [skew-symmetry](https://bliptext.com/articles/skew-symmetry) of the left hand side in i and j suggests introducing an antisymmetric product\n∧\non differential\n1\n-forms, the [exterior product](https://bliptext.com/articles/exterior-product), so that these equations can be combined into a single condition\nwhere\n∧\nis defined so that:\nThis is an example of a differential\n2\n-form.\nThis\n2\n-form is called the [exterior derivative](https://bliptext.com/articles/exterior-derivative)\ndα\nof\nα = Σn j=1 fj dx\n. It is given by\nTo summarize:\ndα = 0\nis a necessary condition for the existence of a function f with\nα = df\n.\nDifferential\n0\n-forms,\n1\n-forms, and\n2\n-forms are special cases of differential forms.\nFor each k, there is a space of differential k-forms, which can be expressed in terms of the coordinates as for a collection of functions\nfi 1i2⋅⋅⋅ik\n. Antisymmetry, which was already present for\n2\n-forms, makes it possible to restrict the sum to those sets of indices for which\ni1 < i2 < ...\n< ik−1 < ik\n.\nDifferential forms can be multiplied together using the exterior product, and for any differential k-form α, there is a differential\n(k + 1)\n-form\ndα\ncalled the exterior derivative of α.\nDifferential forms, the exterior product and the exterior derivative are independent of a choice of coordinates.\nConsequently, they may be defined on any [smooth manifold](https://bliptext.com/articles/smooth-manifold) M. One way to do this is cover M with [coordinate charts](https://bliptext.com/articles/coordinate-chart) and define a differential k-form on M to be a family of differential k-forms on each chart which agree on the overlaps.\nHowever, there are more intrinsic definitions which make the independence of coordinates manifest.\n\n# Intrinsic definitions\nLet\nM\nbe a [smooth manifold](https://bliptext.com/articles/smooth-manifold).\nA smooth differential form of degree\nk\nis a [smooth section](https://bliptext.com/articles/section-fiber-bundle) of the\nk\nth [exterior power](https://bliptext.com/articles/exterior-algebra) of the [cotangent bundle](https://bliptext.com/articles/cotangent-bundle) of\nM\n. The set of all differential\nk\n-forms on a manifold\nM\nis a [vector space](https://bliptext.com/articles/vector-space), often denoted\nΩk(M)\n.\nThe definition of a differential form may be restated as follows.\nAt any point\np ∈ M\n, a\nk\n-form\nβ\ndefines an element\nwhere\nTpM\nis the [tangent space](https://bliptext.com/articles/tangent-space) to\nM\nat\np\nand\nTp*M\nis its [dual space](https://bliptext.com/articles/dual-space).\nThis space is to the fiber at\np\nof the dual bundle of the\nk\nth exterior power of the [tangent bundle](https://bliptext.com/articles/tangent-bundle) of\nM\n. That is,\nβ\nis also a linear functional, i.e. the dual of the\nk\nth exterior power is isomorphic to the\nk\nth exterior power of the dual:\nBy the universal property of exterior powers, this is equivalently an [alternating](https://bliptext.com/articles/alternating-form) [multilinear map](https://bliptext.com/articles/multilinear-map):\nConsequently, a differential\nk\n-form may be evaluated against any\nk\n-tuple of tangent vectors to the same point\np\nof\nM\n. For example, a differential\n1\n-form\nα\nassigns to each point\np ∈ M\na [linear functional](https://bliptext.com/articles/linear-functional)\nαp\non\nTpM\n. In the presence of an [inner product](https://bliptext.com/articles/inner-product) on\nTpM\n(induced by a [Riemannian metric](https://bliptext.com/articles/riemannian-metric) on\nM\n),\nαp\nmay be [represented](https://bliptext.com/articles/riesz-representation-theorem) as the inner product with a [tangent vector](https://bliptext.com/articles/tangent-vector)\nXp\n. Differential\n1\n-forms are sometimes called [covariant vector fields](https://bliptext.com/articles/covariance-and-contravariance-of-vectors), covector fields, or \"dual vector fields\", particularly within physics.\nThe exterior algebra may be embedded in the tensor algebra by means of the alternation map.\nThe alternation map is defined as a mapping\nFor a tensor \\tau at a point\np\n, where\nSk\nis the [symmetric group](https://bliptext.com/articles/symmetric-group) on\nk\nelements.\nThe 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\nThis map exhibits\nβ\nas a [totally antisymmetric](https://bliptext.com/articles/antisymmetric-tensor) [covariant](https://bliptext.com/articles/covariance-and-contravariance-of-vectors) [tensor field](https://bliptext.com/articles/tensor-field) of rank\nk\n. The differential forms on\nM\nare in one-to-one correspondence with such tensor fields.\n\n# Operations\nAs 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.\nThe most important operations are the [exterior product](https://bliptext.com/articles/exterior-product) of two differential forms, the [exterior derivative](https://bliptext.com/articles/exterior-derivative) of a single differential form, the [interior product](https://bliptext.com/articles/interior-product) of a differential form and a vector field, the [Lie derivative](https://bliptext.com/articles/lie-derivative) of a differential form with respect to a vector field and the [covariant derivative](https://bliptext.com/articles/covariant-derivative) of a differential form with respect to a vector field on a manifold with a defined connection.\n\n## Exterior product\nThe exterior product of a\nk\n-form\nα\nand an\nℓ\n-form\nβ\n, denoted\nα ∧ β\n, is a (\nk + ℓ\n)-form. At each point\np\nof the manifold\nM\n, the forms\nα\nand\nβ\nare elements of an exterior power of the cotangent space at\np\n. 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).\nThe antisymmetry inherent in the exterior algebra means that when\nα ∧ β\nis viewed as a multilinear functional, it is alternating.\nHowever, when the exterior algebra is embedded as a subspace of the tensor algebra by means of the alternation map, the tensor product\nα ⊗ β\nis not alternating.\nThere is an explicit formula which describes the exterior product in this situation.\nThe exterior product is If the embedding of into is done via the map instead of, the exterior product is This description is useful for explicit computations.\nFor example, if\nk = ℓ = 1\n, then\nα ∧ β\nis the\n2\n-form whose value at a point\np\nis the [alternating bilinear form](https://bliptext.com/articles/alternating-bilinear-form) defined by for\nv, w ∈ TpM\n.\nThe exterior product is bilinear: If\nα\n,\nβ\n, and\nγ\nare any differential forms, and if\nf\nis any smooth function, then\nIt is skew commutative (also known as graded commutative), meaning that it satisfies a variant of [anticommutativity](https://bliptext.com/articles/anticommutativity) that depends on the degrees of the forms: if\nα\nis a\nk\n-form and\nβ\nis an\nℓ\n-form, then\nOne also has the [graded Leibniz rule](https://bliptext.com/articles/differential-graded-algebra):\"\"\n\n## Riemannian manifold\nOn a [Riemannian manifold](https://bliptext.com/articles/riemannian-manifold), or more generally a pseudo-[Riemannian manifold](https://bliptext.com/articles/riemannian-manifold), the metric defines a fibre-wise isomorphism of the tangent and cotangent bundles.\nThis makes it possible to convert vector fields to covector fields and vice versa.\nIt also enables the definition of additional operations such as the [Hodge star operator](https://bliptext.com/articles/hodge-star-operator) and the [codifferential](https://bliptext.com/articles/hodge-star), which has degree\n−1\nand is [adjoint](https://bliptext.com/articles/differential-operator) to the exterior differential\nd\n.\n\n### Vector field structures\nOn a pseudo-Riemannian manifold,\n1\n-forms can be identified with vector fields; vector fields have additional distinct algebraic structures, which are listed here for context and to avoid confusion.\nFirstly, each (co)tangent space generates a [Clifford algebra](https://bliptext.com/articles/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](https://bliptext.com/articles/metric-tensor).\nThis algebra is distinct from the [exterior algebra](https://bliptext.com/articles/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).\nClifford algebras are thus non-anticommutative (\"quantum\") deformations of the exterior algebra.\nThey are studied in [geometric algebra](https://bliptext.com/articles/geometric-algebra).\nAnother alternative is to consider vector fields as derivations.\nThe (noncommutative) algebra of [differential operators](https://bliptext.com/articles/differential-operator) they generate is the [Weyl algebra](https://bliptext.com/articles/weyl-algebra) and is a noncommutative (\"quantum\") deformation of the symmetric algebra in the vector fields.\n\n## Exterior differential complex\nOne important property of the exterior derivative is that\nd = 0\n. This means that the exterior derivative defines a [cochain complex](https://bliptext.com/articles/cochain-complex):\nThis complex is called the de Rham complex, and its [cohomology](https://bliptext.com/articles/cohomology) is by definition the de Rham [cohomology](https://bliptext.com/articles/cohomology) of\nM\n. By the [Poincaré lemma](https://bliptext.com/articles/poincar-lemma), the de Rham complex is locally [exact](https://bliptext.com/articles/exact-sequence) except at\nΩ0(M)\n. The kernel at\nΩ0(M)\nis the space of [locally constant functions](https://bliptext.com/articles/locally-constant-function) on\nM\n. Therefore, the complex is a resolution of the constant [sheaf](https://bliptext.com/articles/sheaf-mathematics)\nR\n, which in turn implies a form of de Rham's theorem: de Rham cohomology computes the [sheaf cohomology](https://bliptext.com/articles/sheaf-cohomology) of\nR\n.\n\n# Pullback\nSuppose that\nf : M → N\nis smooth.\nThe differential of\nf\nis a smooth map\ndf : TM → TN\nbetween the tangent bundles of\nM\nand\nN\n. This map is also denoted\nf∗\nand called the **pushforward**.\nFor any point\np ∈ M\nand any tangent vector\nv ∈ TpM\n, there is a well-defined pushforward vector\nf∗(v)\nin\nTf(p)N\n. However, the same is not true of a vector field.\nIf\nf\nis not injective, say because\nq ∈ N\nhas two or more preimages, then the vector field may determine two or more distinct vectors in\nTqN\n. If\nf\nis not surjective, then there will be a point\nq ∈ N\nat which\nf∗\ndoes not determine any tangent vector at all.\nSince a vector field on\nN\ndetermines, by definition, a unique tangent vector at every point of\nN\n, the pushforward of a vector field does not always exist.\nBy contrast, it is always possible to pull back a differential form.\nA differential form on\nN\nmay be viewed as a linear functional on each tangent space.\nPrecomposing this functional with the differential\ndf : TM → TN\ndefines a linear functional on each tangent space of\nM\nand therefore a differential form on\nM\n. The existence of pullbacks is one of the key features of the theory of differential forms.\nIt leads to the existence of pullback maps in other situations, such as pullback homomorphisms in de Rham cohomology.\nFormally, let\nf : M → N\nbe smooth, and let\nω\nbe a smooth\nk\n-form on\nN\n. Then there is a differential form\nfω\non\nM\n, called the **pullback** of\nω\n, which captures the behavior of\nω\nas seen relative to\nf\n. To define the pullback, fix a point\np\nof\nM\nand tangent vectors\nv1\n, ...,\nvk\nto\nM\nat\np\n. The pullback of\nω\nis defined by the formula\nThere are several more abstract ways to view this definition.\nIf\nω\nis a\n1\n-form on\nN\n, then it may be viewed as a section of the cotangent bundle\nTN\nof\nN\n. Using to denote a dual map, the dual to the differential of\nf\nis\n(df) : T'N → T'M\n. The pullback of\nω\nmay be defined to be the composite\nThis is a section of the cotangent bundle of\nM\nand hence a differential\n1\n-form on\nM\n. In full generality, let denote the\nk\nth exterior power of the dual map to the differential.\nThen the pullback of a\nk\n-form\nω\nis the composite\nAnother abstract way to view the pullback comes from viewing a\nk\n-form\nω\nas a linear functional on tangent spaces.\nFrom this point of view,\nω\nis a morphism of [vector bundles](https://bliptext.com/articles/vector-bundle)\nwhere\nN × R\nis the trivial rank one bundle on\nN\n. The composite map\ndefines a linear functional on each tangent space of\nM\n, and therefore it factors through the trivial bundle\nM × R\n. The vector bundle morphism defined in this way is\nfω\n.\nPullback respects all of the basic operations on forms.\nIf\nω\nand\nη\nare forms and\nc\nis a real number, then\nThe pullback of a form can also be written in coordinates.\nAssume that\nx1\n, ...,\nxm\nare coordinates on\nM\n, that\ny1\n, ...,\nyn\nare coordinates on\nN\n, and that these coordinate systems are related by the formulas\nyi = fi(x1, ..., xm)\nfor all\ni\n. Locally on\nN\n,\nω\ncan be written as where, for each choice of\ni1\n, ...,\nik\n,\nωi 1⋅⋅⋅ik\nis a real-valued function of\ny1\n, ...,\nyn\n. Using the linearity of pullback and its compatibility with exterior product, the pullback of\nω\nhas the formula\nEach exterior derivative\ndfi\ncan be expanded in terms of\ndx1\n, ...,\ndxm\n. The resulting\nk\n-form can be written using [Jacobian](https://bliptext.com/articles/jacobian-matrix-and-determinant) matrices:\nHere, denotes the determinant of the matrix whose entries are,.\n\n# Integration\nA differential\nk\n-form can be integrated over an oriented\nk\n-dimensional manifold.\nWhen the\nk\n-form is defined on an\nn\n-dimensional manifold with\nn > k\n, then the\nk\n-form can be integrated over oriented\nk\n-dimensional submanifolds.\nIf\nk = 0\n, integration over oriented 0-dimensional submanifolds is just the summation of the integrand evaluated at points, according to the orientation of those points.\nOther values of\nk = 1, 2, 3, ...\ncorrespond to line integrals, surface integrals, volume integrals, and so on.\nThere 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.\n\n## Integration on Euclidean space\nLet\nU\nbe an open subset of\nRn\n. Give\nRn\nits standard orientation and\nU\nthe restriction of that orientation.\nEvery smooth\nn\n-form\nω\non\nU\nhas the form\nfor some smooth function\nf : Rn → R\n. Such a function has an integral in the usual Riemann or Lebesgue sense.\nThis allows us to define the integral of\nω\nto be the integral of\nf\nFixing an orientation is necessary for this to be well-defined.\nThe skew-symmetry of differential forms means that the integral of, say,\ndx1 ∧ dx2\nmust be the negative of the integral of\ndx2 ∧ dx1\n. Riemann and Lebesgue integrals cannot see this dependence on the ordering of the coordinates, so they leave the sign of the integral undetermined.\nThe orientation resolves this ambiguity.\n\n## Integration over chains\nLet\nM\nbe an\nn\n-manifold and\nω\nan\nn\n-form on\nM\n. First, assume that there is a parametrization of\nM\nby an open subset of Euclidean space.\nThat is, assume that there exists a diffeomorphism\nwhere\nD ⊆ Rn\n. Give\nM\nthe orientation induced by\nφ\n. Then defines the integral of\nω\nover\nM\nto be the integral of\nφ∗ω\nover\nD\n. In coordinates, this has the following expression.\nFix an embedding of\nM\nin\nRI\nwith coordinates\nx1, ..., xI\n. Then\nSuppose that\nφ\nis defined by Then the integral may be written in coordinates as where\nis the determinant of the [Jacobian](https://bliptext.com/articles/jacobian-matrix-and-determinant).\nThe Jacobian exists because\nφ\nis differentiable.\nIn general, an\nn\n-manifold cannot be parametrized by an open subset of\nRn\n. 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.\nMoreover, it is also possible to define parametrizations of\nk\n-dimensional subsets for\nk < n\n, and this makes it possible to define integrals of\nk\n-forms.\nTo make this precise, it is convenient to fix a standard domain\nD\nin\nRk\n, usually a cube or a simplex.\nA\nk\n-**chain** is a formal sum of smooth embeddings\nD → M\n. That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity.\nEach smooth embedding determines a\nk\n-dimensional submanifold of\nM\n. If the chain is then the integral of a\nk\n-form\nω\nover\nc\nis defined to be the sum of the integrals over the terms of\nc\nThis approach to defining integration does not assign a direct meaning to integration over the whole manifold\nM\n. However, it is still possible to assign such a meaning indirectly because every smooth manifold may be smoothly [triangulated](https://bliptext.com/articles/triangulation-topology) in an essentially unique way, and the integral over\nM\nmay be defined to be the integral over the chain determined by a triangulation.\n\n## Integration using partitions of unity\nThere is another approach, expounded in, which does directly assign a meaning to integration over\nM\n, but this approach requires fixing an orientation of\nM\n. The integral of an\nn\n-form\nω\non an\nn\n-dimensional manifold is defined by working in charts.\nSuppose first that\nω\nis supported on a single positively oriented chart.\nOn this chart, it may be pulled back to an\nn\n-form on an open subset of\nRn\n. Here, the form has a well-defined Riemann or Lebesgue integral as before.\nThe change of variables formula and the assumption that the chart is positively oriented together ensure that the integral of\nω\nis independent of the chosen chart.\nIn the general case, use a partition of unity to write\nω\nas a sum of\nn\n-forms, each of which is supported in a single positively oriented chart, and define the integral of\nω\nto be the sum of the integrals of each term in the partition of unity.\nIt is also possible to integrate\nk\n-forms on oriented\nk\n-dimensional submanifolds using this more intrinsic approach.\nThe form is pulled back to the submanifold, where the integral is defined using charts as before.\nFor example, given a path\nγ(t) : [0, 1] → R2\n, integrating a\n1\n-form on the path is simply pulling back the form to a form\nf(t) dt\non\n[0, 1]\n, and this integral is the integral of the function\nf(t)\non the interval.\n\n## Integration along fibers\n[Fubini's theorem](https://bliptext.com/articles/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.\nThis suggests that the integral of a differential form over a product ought to be computable as an iterated integral as well.\nThe geometric flexibility of differential forms ensures that this is possible not just for products, but in more general situations as well.\nUnder 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.\nBecause 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.\nLet\nM\nand\nN\nbe two orientable manifolds of pure dimensions\nm\nand\nn\n, respectively.\nSuppose that\nf : M → N\nis a surjective submersion.\nThis implies that each fiber\nf(y)\nis\n(m − n)\n-dimensional and that, around each point of\nM\n, there is a chart on which\nf\nlooks like the projection from a product onto one of its factors.\nFix\nx ∈ M\nand set\ny = f(x)\n. Suppose that\nand that\nηy\ndoes not vanish.\nFollowing, there is a unique\nwhich may be thought of as the fibral part of\nωx\nwith respect to\nηy\n. More precisely, define\nj : f(y) → M\nto be the inclusion.\nThen\nσx\nis defined by the property that\nwhere\nis any\n(m − n)\n-covector for which\nThe form\nσx\nmay also be notated\nωx / ηy\n.\nMoreover, for fixed\ny\n,\nσx\nvaries smoothly with respect to\nx\n. That is, suppose that\nis a smooth section of the projection map; we say that\nω\nis a smooth differential\nm\n-form on\nM\nalong\nf(y)\n. Then there is a smooth differential\n(m − n)\n-form\nσ\non\nf(y)\nsuch that, at each\nx ∈ f(y)\n, This form is denoted\nω / ηy\n. The same construction works if\nω\nis an\nm\n-form in a neighborhood of the fiber, and the same notation is used.\nA consequence is that each fiber\nf(y)\nis orientable.\nIn particular, a choice of orientation forms on\nM\nand\nN\ndefines an orientation of every fiber of\nf\n.\nThe analog of Fubini's theorem is as follows.\nAs before,\nM\nand\nN\nare two orientable manifolds of pure dimensions\nm\nand\nn\n, and\nf : M → N\nis a surjective submersion.\nFix orientations of\nM\nand\nN\n, and give each fiber of\nf\nthe induced orientation.\nLet\nω\nbe an\nm\n-form on\nM\n, and let\nη\nbe an\nn\n-form on\nN\nthat is almost everywhere positive with respect to the orientation of\nN\n. Then, for almost every\ny ∈ N\n, the form\nω / ηy\nis a well-defined integrable\nm − n\nform on\nf(y)\n. Moreover, there is an integrable\nn\n-form on\nN\ndefined by Denote this form by Then proves the generalized Fubini formula\nIt is also possible to integrate forms of other degrees along the fibers of a submersion.\nAssume the same hypotheses as before, and let\nα\nbe a compactly supported\n(m − n + k)\n-form on\nM\n. Then there is a\nk\n-form\nγ\non\nN\nwhich is the result of integrating\nα\nalong the fibers of\nf\n. The form\nα\nis defined by specifying, at each\ny ∈ N\n, how\nγ\npairs with each\nk\n-vector\nv\nat\ny\n, and the value of that pairing is an integral over\nf(y)\nthat depends only on\nα\n,\nv\n, and the orientations of\nM\nand\nN\n. More precisely, at each\ny ∈ N\n, there is an isomorphism\ndefined by the interior product\nfor any choice of volume form\nζ\nin the orientation of\nN\n. If\nx ∈ f(y)\n, then a\nk\n-vector\nv\nat\ny\ndetermines an\n(n − k)\n-covector at\nx\nby pullback:\nEach of these covectors has an exterior product against\nα\n, so there is an\n(m − n)\n-form\nβv\non\nM\nalong\nf(y)\ndefined by This form depends on the orientation of\nN\nbut not the choice of\nζ\n. Then the\nk\n-form\nγ\nis uniquely defined by the property\nand\nγ\nis smooth.\nThis form also denoted\nα♭\nand called the '''integral of\nα\nalong the fibers of\nf\n'''. Integration along fibers is important for the construction of Gysin maps in de Rham cohomology.\nIntegration along fibers satisfies the **projection formula**.\nIf\nλ\nis any\nℓ\n-form on\nN\n, then\n\n## Stokes's theorem\nThe fundamental relationship between the exterior derivative and integration is given by the [Stokes' theorem](https://bliptext.com/articles/stokes-theorem): If\nω\nis an (\nn − 1\n)-form with compact support on\nM\nand\n∂M\ndenotes the [boundary](https://bliptext.com/articles/manifold) of\nM\nwith its induced [orientation](https://bliptext.com/articles/orientation-mathematics), then\nA key consequence of this is that \"the integral of a closed form over homologous chains is equal\": If\nω\nis a closed\nk\n-form and\nM\nand\nN\nare\nk\n-chains that are homologous (such that\nM − N\nis the boundary of a\n(k + 1)\n-chain\nW\n), then, since the difference is the integral.\nFor example, if\nω = df\nis the derivative of a potential function on the plane or\nRn\n, then the integral of\nω\nover a path from\na\nto\nb\ndoes not depend on the choice of path (the integral is\nf(b) − f(a)\n), since different paths with given endpoints are [homotopic](https://bliptext.com/articles/homotopic), hence homologous (a weaker condition). This case is called the [gradient theorem](https://bliptext.com/articles/gradient-theorem), and generalizes the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus). This path independence is very useful in [contour integration](https://bliptext.com/articles/contour-integration).\nThis theorem also underlies the duality between [de Rham cohomology](https://bliptext.com/articles/de-rham-co[homology](https://bliptext.com/articles/homology-mathematics)) and the [homology](https://bliptext.com/articles/homology-mathematics) of chains.\n\n## Relation with measures\nOn 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.\nThe simplest example is attempting to integrate the\n1\n-form\ndx\nover the interval\n[0, 1]\n. Assuming the usual distance (and thus measure) on the real line, this integral is either\n1\nor\n−1\n, depending on orientation:, while.\nBy contrast, the integral of the measure\non the interval is unambiguously\n1\n(i.e. the integral of the constant function\n1\nwith respect to this measure is\n1\n). Similarly, under a change of coordinates a differential\nn\n-form changes by the [Jacobian determinant](https://bliptext.com/articles/jacobian-determinant)\nJ\n, while a measure changes by the absolute value of the Jacobian determinant,\n, which further reflects the issue of orientation.\nFor example, under the map\nx ↦ −x\non the line, the differential form\ndx\npulls back to\n−dx\n, pulls back to\nIn the presence of the additional data of an orientation, it is possible to integrate\nn\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](https://bliptext.com/articles/fundamental-class) of the manifold,\n[M]\n. Formally, in the presence of an orientation, one may identify\nn\n-forms with [densities on a manifold](https://bliptext.com/articles/densities-on-a-manifold); densities in turn define a measure, and thus can be integrated.\nOn an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate\nn\n-forms over compact subsets, with the two choices differing by a sign.\nOn a non-orientable manifold,\nn\n-forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere (there are no [volume forms](https://bliptext.com/articles/volume-form) on non-orientable manifolds), but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate\nn\n-forms.\nOne can instead identify densities with top-dimensional [pseudoforms](https://bliptext.com/articles/volume-form).\nEven in the presence of an orientation, there is in general no meaningful way to integrate\nk\n-forms over subsets for\nk < n\nbecause there is no consistent way to use the ambient orientation to orient\nk\n-dimensional subsets.\nGeometrically, a\nk\n-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.\nCompare the [Gram determinant](https://bliptext.com/articles/gram-determinant) of a set of\nk\nvectors in an\nn\n-dimensional space, which, unlike the determinant of\nn\nvectors, is always positive, corresponding to a squared number.\nAn orientation of a\nk\n-submanifold is therefore extra data not derivable from the ambient manifold.\nOn a Riemannian manifold, one may define a\nk\n-dimensional [Hausdorff measure](https://bliptext.com/articles/hausdorff-measure) for any\nk\n(integer or real), which may be integrated over\nk\n-dimensional subsets of the manifold.\nA function times this Hausdorff measure can then be integrated over\nk\n-dimensional subsets, providing a measure-theoretic analog to integration of\nk\n-forms.\nThe\nn\n-dimensional Hausdorff measure yields a density, as above.\n\n## Currents\nThe differential form analog of a [distribution](https://bliptext.com/articles/distribution-mathematics) or generalized function is called a **[current](https://bliptext.com/articles/current-mathematics)**.\nThe space of\nk\n-currents on\nM\nis the dual space to an appropriate space of differential\nk\n-forms.\nCurrents play the role of generalized domains of integration, similar to but even more flexible than chains.\n\n# Applications in physics\nDifferential forms arise in some important physical contexts.\nFor example, in Maxwell's theory of [electromagnetism](https://bliptext.com/articles/electromagnetism), the **Faraday 2-form**, or [electromagnetic field strength](https://bliptext.com/articles/electromagnetic-field-strength), is where the\nfab\nare formed from the electromagnetic fields \\vec E and \\vec B; e.g.,\nf12 = Ez/c\n,\nf23 = −Bz\n, or equivalent definitions.\nThis form is a special case of the [curvature form](https://bliptext.com/articles/curvature-form) on the\nU(1)\n[principal bundle](https://bliptext.com/articles/principal-bundle) on which both electromagnetism and general [gauge theories](https://bliptext.com/articles/gauge-theories) may be described.\nThe [connection form](https://bliptext.com/articles/connection-form) for the principal bundle is the vector potential, typically denoted by\nA\n, when represented in some gauge.\nOne then has\nThe '''current\n3\n-form''' is where\nja\nare the four components of the current density.\n(Here it is a matter of convention to write\nFab\ninstead of\nfab\n, i.e. to use capital letters, and to write\nJa\ninstead of\nja\n. However, the vector rsp.\ntensor components and the above-mentioned forms have different physical dimensions.\nMoreover, by decision of an international commission of the [International Union of Pure and Applied Physics](https://bliptext.com/articles/international-union-of-pure-and-applied-physics), the magnetic polarization vector has been called \\vec J for several decades, and by some publishers\nJ\nUsing the above-mentioned definitions, [Maxwell's equations](https://bliptext.com/articles/maxwell-s-equations) can be written very compactly in [geometrized units](https://bliptext.com/articles/geometrized-units) as where \\star denotes the [Hodge star](https://bliptext.com/articles/hodge-star) operator.\nSimilar considerations describe the geometry of gauge theories in general.\nThe\n2\n-form, which is [dual](https://bliptext.com/articles/duality-mathematics) to the Faraday form, is also called **Maxwell 2-form**.\nElectromagnetism is an example of a\nU(1)\n[gauge theory](https://bliptext.com/articles/gauge-theory).\nHere the [Lie group](https://bliptext.com/articles/lie-group) is\nU(1)\n, the one-dimensional [unitary group](https://bliptext.com/articles/unitary-group), which is in particular [abelian](https://bliptext.com/articles/abelian-group).\nThere are gauge theories, such as [Yang–Mills theory](https://bliptext.com/articles/yang-mills-theory), in which the Lie group is not abelian.\nIn that case, one gets relations which are similar to those described here.\nThe analog of the field\nF\nin such theories is the curvature form of the connection, which is represented in a gauge by a [Lie algebra](https://bliptext.com/articles/lie-algebra)-valued one-form\nA\n. The Yang–Mills field\nF\nis then defined by\nIn the abelian case, such as electromagnetism,\nA ∧ A = 0\n, but this does not hold in general.\nLikewise the field equations are modified by additional terms involving exterior products of\nA\nand\nF\n, owing to the [structure equations](https://bliptext.com/articles/maurer-cartan-form) of the gauge group.\n\n# Applications in geometric measure theory\nNumerous minimality results for complex analytic manifolds are based on the [Wirtinger inequality for 2-forms](https://bliptext.com/articles/wirtinger-inequality-2-forms).\nA succinct proof may be found in [Herbert Federer](https://bliptext.com/articles/herbert-federer)'s classic text Geometric Measure Theory.\nThe Wirtinger inequality is also a key ingredient in [Gromov's inequality for complex projective space](https://bliptext.com/articles/gromov-s-inequality-for-complex-projective-space) in [systolic geometry](https://bliptext.com/articles/systolic-geometry).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:45:14.538Z","updated_at":"2025-01-25T18:45:14.538Z","search_vector":"'differenti':1 'form':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **differential forms** provide a unified approach to define [integrands](https://bliptext.com/articles/integrand) over curves, surfaces, solids, and higher-dimensional [manifolds](https://bliptext.com/articles/manifold).\nThe modern notion of differential forms was pioneered by [Élie Cartan](https://bliptext.com/articles/lie-cartan).\nIt has many applications, especially in geometry, topology and physics.\nFor instance, the expression\nf(x) dx\nis an example of a [[1-form|\n1\n-form]], and can be [integrated](https://bliptext.com/articles/integral) over an interval\n[a, b]\ncontained in the domain of\nf\nSimilarly, the expression\nf(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz\nis a '''\n2\n-form''' that can be integrated over a [surface](https://bliptext.com/articles/surface-mathematics)\nS\nThe symbol\n∧\ndenotes the [exterior product](https://bliptext.com/articles/exterior-product), sometimes called the wedge product, of two differential forms.\nLikewise, a '''\n3\n-form'''\nf(x, y, z) dx ∧ dy ∧ dz\nrepresents a [volume element](https://bliptext.com/articles/volume-element) that can be integrated over a region of space.\nIn general, a k-form is an object that may be integrated over a k-dimensional manifold, and is [homogeneous](https://bliptext.com/articles/homogeneous-polynomial) of degree k in the coordinate differentials\nOn an\nn\n-dimensional manifold, the top-dimensional form (\nn\n-form) is called a [volume form](https://bliptext.com/articles/volume-form).\nThe differential forms form an [alternating algebra](https://bliptext.com/articles/alternating-algebra).\nThis implies that and This alternating property reflects the [orientation](https://bliptext.com/articles/orientation-mathematics) of the domain of integration.\nThe [exterior derivative](https://bliptext.com/articles/exterior-derivative) is an operation on differential forms that, given a\nk\n-form \\varphi, produces a\n(k+1)\n-form d\\varphi.\nThis operation extends the [differential of a function](https://bliptext.com/articles/differential-of-a-function) (a function can be considered as a\n0\n-form, and its differential is ). This allows expressing the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus), the [divergence theorem](https://bliptext.com/articles/divergence-theorem), [Green's theorem](https://bliptext.com/articles/green-s-theorem), and [Stokes' theorem](https://bliptext.com/articles/stokes-theorem) as special cases of a single general result, the [generalized Stokes theorem](https://bliptext.com/articles/generalized-stokes-theorem).\nDifferential\n1\n-forms are naturally dual to [vector fields](https://bliptext.com/articles/vector-field) on a [differentiable manifold](https://bliptext.com/articles/differentiable-manifold), and the pairing between [vector fields](https://bliptext.com/articles/vector-field) and\n1\n-forms is extended to arbitrary differential forms by the [interior product](https://bliptext.com/articles/interior-product).\nThe algebra of differential forms along with the exterior derivative defined on it is preserved by the [pullback](https://bliptext.com/articles/pullback-differential-geometry) under smooth functions between two manifolds.\nThis 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.\nAs an example, the [change of variables formula](https://bliptext.com/articles/change-of-variables-formula) for integration becomes a simple statement that an integral is preserved under pullback.\n\n# History\nDifferential forms are part of the field of differential geometry, influenced by linear algebra.\nAlthough the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to [Élie Cartan](https://bliptext.com/articles/lie-cartan) with reference to his 1899 paper.\nSome aspects of the [exterior algebra](https://bliptext.com/articles/exterior-algebra) of differential forms appears in [Hermann Grassmann](https://bliptext.com/articles/hermann-grassmann)'s 1844 work, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics).\n\n# Concept\nDifferential forms provide an approach to [multivariable calculus](https://bliptext.com/articles/multivariable-calculus) that is independent of [coordinates](https://bliptext.com/articles/coordinate).\n\n## Integration and orientation\nA differential k-form can be integrated over an oriented [manifold](https://bliptext.com/articles/manifold-mathematics) of dimension k. A differential\n1\n-form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density.\nA differential\n2\n-form can be thought of as measuring an infinitesimal oriented area, or 2-dimensional oriented density.\nAnd so on.\nIntegration of differential forms is well-defined only on [oriented](https://bliptext.com/articles/orientability) [manifolds](https://bliptext.com/articles/manifold-mathematics).\nAn example of a 1-dimensional manifold is an interval\n[a, b]\n, and intervals can be given an orientation: they are positively oriented if\na 1\n, such a function does not always exist: any smooth function f satisfies\nso it will be impossible to find such an f unless\nfor all i and j.\nThe [skew-symmetry](https://bliptext.com/articles/skew-symmetry) of the left hand side in i and j suggests introducing an antisymmetric product\n∧\non differential\n1\n-forms, the [exterior product](https://bliptext.com/articles/exterior-product), so that these equations can be combined into a single condition\nwhere\n∧\nis defined so that:\nThis is an example of a differential\n2\n-form.\nThis\n2\n-form is called the [exterior derivative](https://bliptext.com/articles/exterior-derivative)\ndα\nof\nα = Σn j=1 fj dx\n. It is given by\nTo summarize:\ndα = 0\nis a necessary condition for the existence of a function f with\nα = df\n.\nDifferential\n0\n-forms,\n1\n-forms, and\n2\n-forms are special cases of differential forms.\nFor each k, there is a space of differential k-forms, which can be expressed in terms of the coordinates as for a collection of functions\nfi 1i2⋅⋅⋅ik\n. Antisymmetry, which was already present for\n2\n-forms, makes it possible to restrict the sum to those sets of indices for which\ni1 < i2 < ...\n< ik−1 < ik\n.\nDifferential forms can be multiplied together using the exterior product, and for any differential k-form α, there is a differential\n(k + 1)\n-form\ndα\ncalled the exterior derivative of α.\nDifferential forms, the exterior product and the exterior derivative are independent of a choice of coordinates.\nConsequently, they may be defined on any [smooth manifold](https://bliptext.com/articles/smooth-manifold) M. One way to do this is cover M with [coordinate charts](https://bliptext.com/articles/coordinate-chart) and define a differential k-form on M to be a family of differential k-forms on each chart which agree on the overlaps.\nHowever, there are more intrinsic definitions which make the independence of coordinates manifest.\n\n# Intrinsic definitions\nLet\nM\nbe a [smooth manifold](https://bliptext.com/articles/smooth-manifold).\nA smooth differential form of degree\nk\nis a [smooth section](https://bliptext.com/articles/section-fiber-bundle) of the\nk\nth [exterior power](https://bliptext.com/articles/exterior-algebra) of the [cotangent bundle](https://bliptext.com/articles/cotangent-bundle) of\nM\n. The set of all differential\nk\n-forms on a manifold\nM\nis a [vector space](https://bliptext.com/articles/vector-space), often denoted\nΩk(M)\n.\nThe definition of a differential form may be restated as follows.\nAt any point\np ∈ M\n, a\nk\n-form\nβ\ndefines an element\nwhere\nTpM\nis the [tangent space](https://bliptext.com/articles/tangent-space) to\nM\nat\np\nand\nTp*M\nis its [dual space](https://bliptext.com/articles/dual-space).\nThis space is to the fiber at\np\nof the dual bundle of the\nk\nth exterior power of the [tangent bundle](https://bliptext.com/articles/tangent-bundle) of\nM\n. That is,\nβ\nis also a linear functional, i.e. the dual of the\nk\nth exterior power is isomorphic to the\nk\nth exterior power of the dual:\nBy the universal property of exterior powers, this is equivalently an [alternating](https://bliptext.com/articles/alternating-form) [multilinear map](https://bliptext.com/articles/multilinear-map):\nConsequently, a differential\nk\n-form may be evaluated against any\nk\n-tuple of tangent vectors to the same point\np\nof\nM\n. For example, a differential\n1\n-form\nα\nassigns to each point\np ∈ M\na [linear functional](https://bliptext.com/articles/linear-functional)\nαp\non\nTpM\n. In the presence of an [inner product](https://bliptext.com/articles/inner-product) on\nTpM\n(induced by a [Riemannian metric](https://bliptext.com/articles/riemannian-metric) on\nM\n),\nαp\nmay be [represented](https://bliptext.com/articles/riesz-representation-theorem) as the inner product with a [tangent vector](https://bliptext.com/articles/tangent-vector)\nXp\n. Differential\n1\n-forms are sometimes called [covariant vector fields](https://bliptext.com/articles/covariance-and-contravariance-of-vectors), covector fields, or \"dual vector fields\", particularly within physics.\nThe exterior algebra may be embedded in the tensor algebra by means of the alternation map.\nThe alternation map is defined as a mapping\nFor a tensor \\tau at a point\np\n, where\nSk\nis the [symmetric group](https://bliptext.com/articles/symmetric-group) on\nk\nelements.\nThe 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\nThis map exhibits\nβ\nas a [totally antisymmetric](https://bliptext.com/articles/antisymmetric-tensor) [covariant](https://bliptext.com/articles/covariance-and-contravariance-of-vectors) [tensor field](https://bliptext.com/articles/tensor-field) of rank\nk\n. The differential forms on\nM\nare in one-to-one correspondence with such tensor fields.\n\n# Operations\nAs 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.\nThe most important operations are the [exterior product](https://bliptext.com/articles/exterior-product) of two differential forms, the [exterior derivative](https://bliptext.com/articles/exterior-derivative) of a single differential form, the [interior product](https://bliptext.com/articles/interior-product) of a differential form and a vector field, the [Lie derivative](https://bliptext.com/articles/lie-derivative) of a differential form with respect to a vector field and the [covariant derivative](https://bliptext.com/articles/covariant-derivative) of a differential form with respect to a vector field on a manifold with a defined connection.\n\n## Exterior product\nThe exterior product of a\nk\n-form\nα\nand an\nℓ\n-form\nβ\n, denoted\nα ∧ β\n, is a (\nk + ℓ\n)-form. At each point\np\nof the manifold\nM\n, the forms\nα\nand\nβ\nare elements of an exterior power of the cotangent space at\np\n. 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).\nThe antisymmetry inherent in the exterior algebra means that when\nα ∧ β\nis viewed as a multilinear functional, it is alternating.\nHowever, when the exterior algebra is embedded as a subspace of the tensor algebra by means of the alternation map, the tensor product\nα ⊗ β\nis not alternating.\nThere is an explicit formula which describes the exterior product in this situation.\nThe exterior product is If the embedding of into is done via the map instead of, the exterior product is This description is useful for explicit computations.\nFor example, if\nk = ℓ = 1\n, then\nα ∧ β\nis the\n2\n-form whose value at a point\np\nis the [alternating bilinear form](https://bliptext.com/articles/alternating-bilinear-form) defined by for\nv, w ∈ TpM\n.\nThe exterior product is bilinear: If\nα\n,\nβ\n, and\nγ\nare any differential forms, and if\nf\nis any smooth function, then\nIt is skew commutative (also known as graded commutative), meaning that it satisfies a variant of [anticommutativity](https://bliptext.com/articles/anticommutativity) that depends on the degrees of the forms: if\nα\nis a\nk\n-form and\nβ\nis an\nℓ\n-form, then\nOne also has the [graded Leibniz rule](https://bliptext.com/articles/differential-graded-algebra):\"\"\n\n## Riemannian manifold\nOn a [Riemannian manifold](https://bliptext.com/articles/riemannian-manifold), or more generally a pseudo-[Riemannian manifold](https://bliptext.com/articles/riemannian-manifold), the metric defines a fibre-wise isomorphism of the tangent and cotangent bundles.\nThis makes it possible to convert vector fields to covector fields and vice versa.\nIt also enables the definition of additional operations such as the [Hodge star operator](https://bliptext.com/articles/hodge-star-operator) and the [codifferential](https://bliptext.com/articles/hodge-star), which has degree\n−1\nand is [adjoint](https://bliptext.com/articles/differential-operator) to the exterior differential\nd\n.\n\n### Vector field structures\nOn a pseudo-Riemannian manifold,\n1\n-forms can be identified with vector fields; vector fields have additional distinct algebraic structures, which are listed here for context and to avoid confusion.\nFirstly, each (co)tangent space generates a [Clifford algebra](https://bliptext.com/articles/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](https://bliptext.com/articles/metric-tensor).\nThis algebra is distinct from the [exterior algebra](https://bliptext.com/articles/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).\nClifford algebras are thus non-anticommutative (\"quantum\") deformations of the exterior algebra.\nThey are studied in [geometric algebra](https://bliptext.com/articles/geometric-algebra).\nAnother alternative is to consider vector fields as derivations.\nThe (noncommutative) algebra of [differential operators](https://bliptext.com/articles/differential-operator) they generate is the [Weyl algebra](https://bliptext.com/articles/weyl-algebra) and is a noncommutative (\"quantum\") deformation of the symmetric algebra in the vector fields.\n\n## Exterior differential complex\nOne important property of the exterior derivative is that\nd = 0\n. This means that the exterior derivative defines a [cochain complex](https://bliptext.com/articles/cochain-complex):\nThis complex is called the de Rham complex, and its [cohomology](https://bliptext.com/articles/cohomology) is by definition the de Rham [cohomology](https://bliptext.com/articles/cohomology) of\nM\n. By the [Poincaré lemma](https://bliptext.com/articles/poincar-lemma), the de Rham complex is locally [exact](https://bliptext.com/articles/exact-sequence) except at\nΩ0(M)\n. The kernel at\nΩ0(M)\nis the space of [locally constant functions](https://bliptext.com/articles/locally-constant-function) on\nM\n. Therefore, the complex is a resolution of the constant [sheaf](https://bliptext.com/articles/sheaf-mathematics)\nR\n, which in turn implies a form of de Rham's theorem: de Rham cohomology computes the [sheaf cohomology](https://bliptext.com/articles/sheaf-cohomology) of\nR\n.\n\n# Pullback\nSuppose that\nf : M → N\nis smooth.\nThe differential of\nf\nis a smooth map\ndf : TM → TN\nbetween the tangent bundles of\nM\nand\nN\n. This map is also denoted\nf∗\nand called the **pushforward**.\nFor any point\np ∈ M\nand any tangent vector\nv ∈ TpM\n, there is a well-defined pushforward vector\nf∗(v)\nin\nTf(p)N\n. However, the same is not true of a vector field.\nIf\nf\nis not injective, say because\nq ∈ N\nhas two or more preimages, then the vector field may determine two or more distinct vectors in\nTqN\n. If\nf\nis not surjective, then there will be a point\nq ∈ N\nat which\nf∗\ndoes not determine any tangent vector at all.\nSince a vector field on\nN\ndetermines, by definition, a unique tangent vector at every point of\nN\n, the pushforward of a vector field does not always exist.\nBy contrast, it is always possible to pull back a differential form.\nA differential form on\nN\nmay be viewed as a linear functional on each tangent space.\nPrecomposing this functional with the differential\ndf : TM → TN\ndefines a linear functional on each tangent space of\nM\nand therefore a differential form on\nM\n. The existence of pullbacks is one of the key features of the theory of differential forms.\nIt leads to the existence of pullback maps in other situations, such as pullback homomorphisms in de Rham cohomology.\nFormally, let\nf : M → N\nbe smooth, and let\nω\nbe a smooth\nk\n-form on\nN\n. Then there is a differential form\nfω\non\nM\n, called the **pullback** of\nω\n, which captures the behavior of\nω\nas seen relative to\nf\n. To define the pullback, fix a point\np\nof\nM\nand tangent vectors\nv1\n, ...,\nvk\nto\nM\nat\np\n. The pullback of\nω\nis defined by the formula\nThere are several more abstract ways to view this definition.\nIf\nω\nis a\n1\n-form on\nN\n, then it may be viewed as a section of the cotangent bundle\nTN\nof\nN\n. Using to denote a dual map, the dual to the differential of\nf\nis\n(df) : T'N → T'M\n. The pullback of\nω\nmay be defined to be the composite\nThis is a section of the cotangent bundle of\nM\nand hence a differential\n1\n-form on\nM\n. In full generality, let denote the\nk\nth exterior power of the dual map to the differential.\nThen the pullback of a\nk\n-form\nω\nis the composite\nAnother abstract way to view the pullback comes from viewing a\nk\n-form\nω\nas a linear functional on tangent spaces.\nFrom this point of view,\nω\nis a morphism of [vector bundles](https://bliptext.com/articles/vector-bundle)\nwhere\nN × R\nis the trivial rank one bundle on\nN\n. The composite map\ndefines a linear functional on each tangent space of\nM\n, and therefore it factors through the trivial bundle\nM × R\n. The vector bundle morphism defined in this way is\nfω\n.\nPullback respects all of the basic operations on forms.\nIf\nω\nand\nη\nare forms and\nc\nis a real number, then\nThe pullback of a form can also be written in coordinates.\nAssume that\nx1\n, ...,\nxm\nare coordinates on\nM\n, that\ny1\n, ...,\nyn\nare coordinates on\nN\n, and that these coordinate systems are related by the formulas\nyi = fi(x1, ..., xm)\nfor all\ni\n. Locally on\nN\n,\nω\ncan be written as where, for each choice of\ni1\n, ...,\nik\n,\nωi 1⋅⋅⋅ik\nis a real-valued function of\ny1\n, ...,\nyn\n. Using the linearity of pullback and its compatibility with exterior product, the pullback of\nω\nhas the formula\nEach exterior derivative\ndfi\ncan be expanded in terms of\ndx1\n, ...,\ndxm\n. The resulting\nk\n-form can be written using [Jacobian](https://bliptext.com/articles/jacobian-matrix-and-determinant) matrices:\nHere, denotes the determinant of the matrix whose entries are,.\n\n# Integration\nA differential\nk\n-form can be integrated over an oriented\nk\n-dimensional manifold.\nWhen the\nk\n-form is defined on an\nn\n-dimensional manifold with\nn > k\n, then the\nk\n-form can be integrated over oriented\nk\n-dimensional submanifolds.\nIf\nk = 0\n, integration over oriented 0-dimensional submanifolds is just the summation of the integrand evaluated at points, according to the orientation of those points.\nOther values of\nk = 1, 2, 3, ...\ncorrespond to line integrals, surface integrals, volume integrals, and so on.\nThere 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.\n\n## Integration on Euclidean space\nLet\nU\nbe an open subset of\nRn\n. Give\nRn\nits standard orientation and\nU\nthe restriction of that orientation.\nEvery smooth\nn\n-form\nω\non\nU\nhas the form\nfor some smooth function\nf : Rn → R\n. Such a function has an integral in the usual Riemann or Lebesgue sense.\nThis allows us to define the integral of\nω\nto be the integral of\nf\nFixing an orientation is necessary for this to be well-defined.\nThe skew-symmetry of differential forms means that the integral of, say,\ndx1 ∧ dx2\nmust be the negative of the integral of\ndx2 ∧ dx1\n. Riemann and Lebesgue integrals cannot see this dependence on the ordering of the coordinates, so they leave the sign of the integral undetermined.\nThe orientation resolves this ambiguity.\n\n## Integration over chains\nLet\nM\nbe an\nn\n-manifold and\nω\nan\nn\n-form on\nM\n. First, assume that there is a parametrization of\nM\nby an open subset of Euclidean space.\nThat is, assume that there exists a diffeomorphism\nwhere\nD ⊆ Rn\n. Give\nM\nthe orientation induced by\nφ\n. Then defines the integral of\nω\nover\nM\nto be the integral of\nφ∗ω\nover\nD\n. In coordinates, this has the following expression.\nFix an embedding of\nM\nin\nRI\nwith coordinates\nx1, ..., xI\n. Then\nSuppose that\nφ\nis defined by Then the integral may be written in coordinates as where\nis the determinant of the [Jacobian](https://bliptext.com/articles/jacobian-matrix-and-determinant).\nThe Jacobian exists because\nφ\nis differentiable.\nIn general, an\nn\n-manifold cannot be parametrized by an open subset of\nRn\n. 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.\nMoreover, it is also possible to define parametrizations of\nk\n-dimensional subsets for\nk < n\n, and this makes it possible to define integrals of\nk\n-forms.\nTo make this precise, it is convenient to fix a standard domain\nD\nin\nRk\n, usually a cube or a simplex.\nA\nk\n-**chain** is a formal sum of smooth embeddings\nD → M\n. That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity.\nEach smooth embedding determines a\nk\n-dimensional submanifold of\nM\n. If the chain is then the integral of a\nk\n-form\nω\nover\nc\nis defined to be the sum of the integrals over the terms of\nc\nThis approach to defining integration does not assign a direct meaning to integration over the whole manifold\nM\n. However, it is still possible to assign such a meaning indirectly because every smooth manifold may be smoothly [triangulated](https://bliptext.com/articles/triangulation-topology) in an essentially unique way, and the integral over\nM\nmay be defined to be the integral over the chain determined by a triangulation.\n\n## Integration using partitions of unity\nThere is another approach, expounded in, which does directly assign a meaning to integration over\nM\n, but this approach requires fixing an orientation of\nM\n. The integral of an\nn\n-form\nω\non an\nn\n-dimensional manifold is defined by working in charts.\nSuppose first that\nω\nis supported on a single positively oriented chart.\nOn this chart, it may be pulled back to an\nn\n-form on an open subset of\nRn\n. Here, the form has a well-defined Riemann or Lebesgue integral as before.\nThe change of variables formula and the assumption that the chart is positively oriented together ensure that the integral of\nω\nis independent of the chosen chart.\nIn the general case, use a partition of unity to write\nω\nas a sum of\nn\n-forms, each of which is supported in a single positively oriented chart, and define the integral of\nω\nto be the sum of the integrals of each term in the partition of unity.\nIt is also possible to integrate\nk\n-forms on oriented\nk\n-dimensional submanifolds using this more intrinsic approach.\nThe form is pulled back to the submanifold, where the integral is defined using charts as before.\nFor example, given a path\nγ(t) : [0, 1] → R2\n, integrating a\n1\n-form on the path is simply pulling back the form to a form\nf(t) dt\non\n[0, 1]\n, and this integral is the integral of the function\nf(t)\non the interval.\n\n## Integration along fibers\n[Fubini's theorem](https://bliptext.com/articles/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.\nThis suggests that the integral of a differential form over a product ought to be computable as an iterated integral as well.\nThe geometric flexibility of differential forms ensures that this is possible not just for products, but in more general situations as well.\nUnder 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.\nBecause 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.\nLet\nM\nand\nN\nbe two orientable manifolds of pure dimensions\nm\nand\nn\n, respectively.\nSuppose that\nf : M → N\nis a surjective submersion.\nThis implies that each fiber\nf(y)\nis\n(m − n)\n-dimensional and that, around each point of\nM\n, there is a chart on which\nf\nlooks like the projection from a product onto one of its factors.\nFix\nx ∈ M\nand set\ny = f(x)\n. Suppose that\nand that\nηy\ndoes not vanish.\nFollowing, there is a unique\nwhich may be thought of as the fibral part of\nωx\nwith respect to\nηy\n. More precisely, define\nj : f(y) → M\nto be the inclusion.\nThen\nσx\nis defined by the property that\nwhere\nis any\n(m − n)\n-covector for which\nThe form\nσx\nmay also be notated\nωx / ηy\n.\nMoreover, for fixed\ny\n,\nσx\nvaries smoothly with respect to\nx\n. That is, suppose that\nis a smooth section of the projection map; we say that\nω\nis a smooth differential\nm\n-form on\nM\nalong\nf(y)\n. Then there is a smooth differential\n(m − n)\n-form\nσ\non\nf(y)\nsuch that, at each\nx ∈ f(y)\n, This form is denoted\nω / ηy\n. The same construction works if\nω\nis an\nm\n-form in a neighborhood of the fiber, and the same notation is used.\nA consequence is that each fiber\nf(y)\nis orientable.\nIn particular, a choice of orientation forms on\nM\nand\nN\ndefines an orientation of every fiber of\nf\n.\nThe analog of Fubini's theorem is as follows.\nAs before,\nM\nand\nN\nare two orientable manifolds of pure dimensions\nm\nand\nn\n, and\nf : M → N\nis a surjective submersion.\nFix orientations of\nM\nand\nN\n, and give each fiber of\nf\nthe induced orientation.\nLet\nω\nbe an\nm\n-form on\nM\n, and let\nη\nbe an\nn\n-form on\nN\nthat is almost everywhere positive with respect to the orientation of\nN\n. Then, for almost every\ny ∈ N\n, the form\nω / ηy\nis a well-defined integrable\nm − n\nform on\nf(y)\n. Moreover, there is an integrable\nn\n-form on\nN\ndefined by Denote this form by Then proves the generalized Fubini formula\nIt is also possible to integrate forms of other degrees along the fibers of a submersion.\nAssume the same hypotheses as before, and let\nα\nbe a compactly supported\n(m − n + k)\n-form on\nM\n. Then there is a\nk\n-form\nγ\non\nN\nwhich is the result of integrating\nα\nalong the fibers of\nf\n. The form\nα\nis defined by specifying, at each\ny ∈ N\n, how\nγ\npairs with each\nk\n-vector\nv\nat\ny\n, and the value of that pairing is an integral over\nf(y)\nthat depends only on\nα\n,\nv\n, and the orientations of\nM\nand\nN\n. More precisely, at each\ny ∈ N\n, there is an isomorphism\ndefined by the interior product\nfor any choice of volume form\nζ\nin the orientation of\nN\n. If\nx ∈ f(y)\n, then a\nk\n-vector\nv\nat\ny\ndetermines an\n(n − k)\n-covector at\nx\nby pullback:\nEach of these covectors has an exterior product against\nα\n, so there is an\n(m − n)\n-form\nβv\non\nM\nalong\nf(y)\ndefined by This form depends on the orientation of\nN\nbut not the choice of\nζ\n. Then the\nk\n-form\nγ\nis uniquely defined by the property\nand\nγ\nis smooth.\nThis form also denoted\nα♭\nand called the '''integral of\nα\nalong the fibers of\nf\n'''. Integration along fibers is important for the construction of Gysin maps in de Rham cohomology.\nIntegration along fibers satisfies the **projection formula**.\nIf\nλ\nis any\nℓ\n-form on\nN\n, then\n\n## Stokes's theorem\nThe fundamental relationship between the exterior derivative and integration is given by the [Stokes' theorem](https://bliptext.com/articles/stokes-theorem): If\nω\nis an (\nn − 1\n)-form with compact support on\nM\nand\n∂M\ndenotes the [boundary](https://bliptext.com/articles/manifold) of\nM\nwith its induced [orientation](https://bliptext.com/articles/orientation-mathematics), then\nA key consequence of this is that \"the integral of a closed form over homologous chains is equal\": If\nω\nis a closed\nk\n-form and\nM\nand\nN\nare\nk\n-chains that are homologous (such that\nM − N\nis the boundary of a\n(k + 1)\n-chain\nW\n), then, since the difference is the integral.\nFor example, if\nω = df\nis the derivative of a potential function on the plane or\nRn\n, then the integral of\nω\nover a path from\na\nto\nb\ndoes not depend on the choice of path (the integral is\nf(b) − f(a)\n), since different paths with given endpoints are [homotopic](https://bliptext.com/articles/homotopic), hence homologous (a weaker condition). This case is called the [gradient theorem](https://bliptext.com/articles/gradient-theorem), and generalizes the [fundamental theorem of calculus](https://bliptext.com/articles/fundamental-theorem-of-calculus). This path independence is very useful in [contour integration](https://bliptext.com/articles/contour-integration).\nThis theorem also underlies the duality between [de Rham co[homology](https://bliptext.com/articles/homology-mathematics)](https://bliptext.com/articles/de-rham-co[homology](https://bliptext.com/articles/homology-mathematics)) and the [homology](https://bliptext.com/articles/homology-mathematics) of chains.\n\n## Relation with measures\nOn 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.\nThe simplest example is attempting to integrate the\n1\n-form\ndx\nover the interval\n[0, 1]\n. Assuming the usual distance (and thus measure) on the real line, this integral is either\n1\nor\n−1\n, depending on orientation:, while.\nBy contrast, the integral of the measure\non the interval is unambiguously\n1\n(i.e. the integral of the constant function\n1\nwith respect to this measure is\n1\n). Similarly, under a change of coordinates a differential\nn\n-form changes by the [Jacobian determinant](https://bliptext.com/articles/jacobian-determinant)\nJ\n, while a measure changes by the absolute value of the Jacobian determinant,\n, which further reflects the issue of orientation.\nFor example, under the map\nx ↦ −x\non the line, the differential form\ndx\npulls back to\n−dx\n, pulls back to\nIn the presence of the additional data of an orientation, it is possible to integrate\nn\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](https://bliptext.com/articles/fundamental-class) of the manifold,\n[M]\n. Formally, in the presence of an orientation, one may identify\nn\n-forms with [densities on a manifold](https://bliptext.com/articles/densities-on-a-manifold); densities in turn define a measure, and thus can be integrated.\nOn an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate\nn\n-forms over compact subsets, with the two choices differing by a sign.\nOn a non-orientable manifold,\nn\n-forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere (there are no [volume forms](https://bliptext.com/articles/volume-form) on non-orientable manifolds), but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate\nn\n-forms.\nOne can instead identify densities with top-dimensional [pseudoforms](https://bliptext.com/articles/volume-form).\nEven in the presence of an orientation, there is in general no meaningful way to integrate\nk\n-forms over subsets for\nk < n\nbecause there is no consistent way to use the ambient orientation to orient\nk\n-dimensional subsets.\nGeometrically, a\nk\n-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.\nCompare the [Gram determinant](https://bliptext.com/articles/gram-determinant) of a set of\nk\nvectors in an\nn\n-dimensional space, which, unlike the determinant of\nn\nvectors, is always positive, corresponding to a squared number.\nAn orientation of a\nk\n-submanifold is therefore extra data not derivable from the ambient manifold.\nOn a Riemannian manifold, one may define a\nk\n-dimensional [Hausdorff measure](https://bliptext.com/articles/hausdorff-measure) for any\nk\n(integer or real), which may be integrated over\nk\n-dimensional subsets of the manifold.\nA function times this Hausdorff measure can then be integrated over\nk\n-dimensional subsets, providing a measure-theoretic analog to integration of\nk\n-forms.\nThe\nn\n-dimensional Hausdorff measure yields a density, as above.\n\n## Currents\nThe differential form analog of a [distribution](https://bliptext.com/articles/distribution-mathematics) or generalized function is called a **[current](https://bliptext.com/articles/current-mathematics)**.\nThe space of\nk\n-currents on\nM\nis the dual space to an appropriate space of differential\nk\n-forms.\nCurrents play the role of generalized domains of integration, similar to but even more flexible than chains.\n\n# Applications in physics\nDifferential forms arise in some important physical contexts.\nFor example, in Maxwell's theory of [electromagnetism](https://bliptext.com/articles/electromagnetism), the **Faraday 2-form**, or [electromagnetic field strength](https://bliptext.com/articles/electromagnetic-field-strength), is where the\nfab\nare formed from the electromagnetic fields \\vec E and \\vec B; e.g.,\nf12 = Ez/c\n,\nf23 = −Bz\n, or equivalent definitions.\nThis form is a special case of the [curvature form](https://bliptext.com/articles/curvature-form) on the\nU(1)\n[principal bundle](https://bliptext.com/articles/principal-bundle) on which both electromagnetism and general [gauge theories](https://bliptext.com/articles/gauge-theories) may be described.\nThe [connection form](https://bliptext.com/articles/connection-form) for the principal bundle is the vector potential, typically denoted by\nA\n, when represented in some gauge.\nOne then has\nThe '''current\n3\n-form''' is where\nja\nare the four components of the current density.\n(Here it is a matter of convention to write\nFab\ninstead of\nfab\n, i.e. to use capital letters, and to write\nJa\ninstead of\nja\n. However, the vector rsp.\ntensor components and the above-mentioned forms have different physical dimensions.\nMoreover, by decision of an international commission of the [International Union of Pure and Applied Physics](https://bliptext.com/articles/international-union-of-pure-and-applied-physics), the magnetic polarization vector has been called \\vec J for several decades, and by some publishers\nJ\nUsing the above-mentioned definitions, [Maxwell's equations](https://bliptext.com/articles/maxwell-s-equations) can be written very compactly in [geometrized units](https://bliptext.com/articles/geometrized-units) as where \\star denotes the [Hodge star](https://bliptext.com/articles/hodge-star) operator.\nSimilar considerations describe the geometry of gauge theories in general.\nThe\n2\n-form, which is [dual](https://bliptext.com/articles/duality-mathematics) to the Faraday form, is also called **Maxwell 2-form**.\nElectromagnetism is an example of a\nU(1)\n[gauge theory](https://bliptext.com/articles/gauge-theory).\nHere the [Lie group](https://bliptext.com/articles/lie-group) is\nU(1)\n, the one-dimensional [unitary group](https://bliptext.com/articles/unitary-group), which is in particular [abelian](https://bliptext.com/articles/abelian-group).\nThere are gauge theories, such as [Yang–Mills theory](https://bliptext.com/articles/yang-mills-theory), in which the Lie group is not abelian.\nIn that case, one gets relations which are similar to those described here.\nThe analog of the field\nF\nin such theories is the curvature form of the connection, which is represented in a gauge by a [Lie algebra](https://bliptext.com/articles/lie-algebra)-valued one-form\nA\n. The Yang–Mills field\nF\nis then defined by\nIn the abelian case, such as electromagnetism,\nA ∧ A = 0\n, but this does not hold in general.\nLikewise the field equations are modified by additional terms involving exterior products of\nA\nand\nF\n, owing to the [structure equations](https://bliptext.com/articles/maurer-cartan-form) of the gauge group.\n\n# Applications in geometric measure theory\nNumerous minimality results for complex analytic manifolds are based on the [Wirtinger inequality for 2-forms](https://bliptext.com/articles/wirtinger-inequality-2-forms).\nA succinct proof may be found in [Herbert Federer](https://bliptext.com/articles/herbert-federer)'s classic text Geometric Measure Theory.\nThe Wirtinger inequality is also a key ingredient in [Gromov's inequality for complex projective space](https://bliptext.com/articles/gromov-s-inequality-for-complex-projective-space) in [systolic geometry](https://bliptext.com/articles/systolic-geometry).","revisionCount":0},"hasMore":false,"nextCursor":null}