Differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds.In general, a k-form is an object that may be integrated over a k-dimensional manifold, and is homogeneous of degree k in the coordinate differentialsThe algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds.[1] Some aspects of the exterior algebra of differential forms appears in Hermann Grassmann's 1844 work, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics).That is: This gives a geometrical context to the conventions for one-dimensional integrals, that the sign changes when the orientation of the interval is reversed.Making the notion of an oriented density precise, and thus of a differential form, involves the exterior algebra.Each of these represents a covector at each point on the manifold that may be thought of as measuring a small displacement in the corresponding coordinate direction.A fundamental operation defined on differential forms is the exterior product (the symbol is the wedge ∧).In higher dimensions, dxi1 ∧ ⋅⋅⋅ ∧ dxim = 0 if any two of the indices i1, ..., im are equal, in the same way that the "volume" enclosed by a parallelotope whose edge vectors are linearly dependent is zero.[2] Another useful notation is obtained by defining the set of all strictly increasing multi-indices of length k, in a space of dimension n, denotedspans the space of differential k-forms in a manifold M of dimension n, when viewed as a module over the ring C∞(M) of smooth functions on M. By calculating the size ofThis also demonstrates that there are no nonzero differential forms of degree greater than the dimension of the underlying manifold.By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates y1, y2, ..., yn are introduced, then The first idea leading to differential forms is the observation that ∂v f (p) is a linear function of v: for any vectors v, w and any real number c. At each point p, this linear map from Rn to R is denoted dfp and called the derivative or differential of f at p. Thus dfp(v) = ∂v f (p).Since any vector v is a linear combination Σ vjej of its components, df is uniquely determined by dfp(ej) for each j and each p ∈ U, which are just the partial derivatives of f on U.Thus df provides a way of encoding the partial derivatives of f. It can be decoded by noticing that the coordinates x1, x2, ..., xn are themselves functions on U, and so define differential 1-forms dx1, dx2, ..., dxn.The antisymmetry inherent in the exterior algebra means that when α ∧ β is viewed as a multilinear functional, it is alternating.For example, if k = ℓ = 1, then α ∧ β is the 2-form whose value at a point p is the alternating bilinear form defined by for v, w ∈ TpM.The exterior product is bilinear: If α, β, and γ are any differential forms, and if f is any smooth function, then It is skew commutative (also known as graded commutative), meaning that it satisfies a variant of anticommutativity that depends on the degrees of the forms: if α is a k-form and β is an ℓ-form, thenFirstly, each (co)tangent space generates a Clifford algebra, where the product of a (co)vector with itself is given by the value of a quadratic form – in this case, the natural one induced by the metric.The kernel at Ω0(M) is the space of locally constant functions on M. Therefore, the complex is a resolution of the constant sheaf R, which in turn implies a form of de Rham's theorem: de Rham cohomology computes the sheaf cohomology of R. Suppose that f : M → N is smooth.The pullback of ω may be defined to be the composite This is a section of the cotangent bundle of M and hence a differential 1-form on M. In full generality, letThere 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.Let M be an n-manifold and ω an n-form on M. First, assume that there is a parametrization of M by an open subset of Euclidean space.To make this precise, it is convenient to fix a standard domain D in Rk, usually a cube or a simplex.In the general case, use a partition of unity to write ω as a sum of n-forms, each of which is supported in a single positively oriented chart, and define the integral of ω to be the sum of the integrals of each term in the partition of unity.If λ is any ℓ-form on N, then The fundamental relationship between the exterior derivative and integration is given by the Stokes' theorem: If ω is an (n − 1)-form with compact support on M and ∂M denotes the boundary of M with its induced orientation, then A key consequence of this is that "the integral of a closed form over homologous chains is equal": If ω is a closed k-form and M and N are k-chains that are homologous (such that M − N is the boundary of a (k + 1)-chain W), thenThe connection form for the principal bundle is the vector potential, typically denoted by A, when represented in some gauge.The analog of the field F in such theories is the curvature form of the connection, which is represented in a gauge by a Lie algebra-valued one-form A.Numerous minimality results for complex analytic manifolds are based on the Wirtinger inequality for 2-forms.A succinct proof may be found in Herbert Federer's classic text Geometric Measure Theory.