Differential form

differential forms2-formtwo-formdifferential 1-formform1-formdifferential 2-formdifferential one-formdifferentialdifferential ''k''-forms
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.wikipedia
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Exterior derivative

exterior calculusexterior differentiationdifferentials
on differential forms known as the exterior derivative that, when given a The most important operations are the exterior product of two differential forms, the exterior derivative of a single differential form, the interior product of a differential form and a vector field, the Lie derivative of a differential form with respect to a vector field and the covariant derivative of a differential form with respect to a vector field on a manifold with a defined connection.
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.

Differential of a function

total differentialdifferentialdifferentials
This operation extends the differential of a function, and is directly related to the divergence and the curl of a vector field in a manner that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the generalized Stokes' theorem.
The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or analytical significance if the differential is regarded as a linear approximation to the increment of a function.

Stokes' theorem

Stokes theoremStokes's theoremKelvin–Stokes theorem
This operation extends the differential of a function, and is directly related to the divergence and the curl of a vector field in a manner that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the generalized Stokes' theorem.
In vector calculus, and more generally differential geometry, Stokes' theorem (sometimes spelled Stokes's theorem, and also called the generalized Stokes theorem or the Stokes–Cartan theorem ) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.

Differential geometry

differentialdifferential geometerdifferential geometry and topology
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.
An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space, i.e., a nondegenerate 2-form ω, called the symplectic form.

Integral

integrationintegral calculusdefinite integral
Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds.
Some common interpretations of dx include: an integrator function in Riemann-Stieltjes integration (indicated by dα(x) in general), a measure in Lebesgue theory (indicated by dμ in general), or a differential form in exterior calculus (indicated by in general).

Multivariable calculus

multivariate calculusmultivariablemultivariate
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.
In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem, which applies to the integration of differential forms over manifolds.

Integration by substitution

change of variablessubstitutionchange of variables formula
As an example, the change of variables formula for integration becomes a simple statement that an integral is preserved under pullback.
(This equation may be put on a rigorous foundation by interpreting it as a statement about differential forms.) One may view the method of integration by substitution as a partial justification of Leibniz's notation for integrals and derivatives.

Interior product

interior multiplicationCartan's magic formulaCartan's formula
-forms is extended to arbitrary differential forms by the interior product. The most important operations are the exterior product of two differential forms, the exterior derivative of a single differential form, the interior product of a differential form and a vector field, the Lie derivative of a differential form with respect to a vector field and the covariant derivative of a differential form with respect to a vector field on a manifold with a defined connection.
In mathematics, the interior product ( interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold.

Orientability

orientableorientednon-orientable
-form that has a surface integral over an oriented surface
Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms.

Cross product

vector cross productvector productcross-product
). This is similar to the cross product from vector calculus, in that it is an alternating product.
The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector, or in arbitrary dimensions the exterior product of vectors can be used with a bivector or two-form result.

Volume element

area elementdifferential volume elementelement
represents a volume element that can be integrated over an oriented region of space.
On an orientable differentiable manifold, a volume element typically arises from a volume form: a top degree differential form.

Surface integral

surface elementsurface integralsarea element
-form that has a surface integral over an oriented surface
:be a differential 2-form defined on the surface S, and let

Exterior algebra

exterior productexterior powerwedge product
denotes the exterior product, sometimes called the wedge product, of two differential forms. The most important operations are the exterior product of two differential forms, the exterior derivative of a single differential form, the interior product of a differential form and a vector field, the Lie derivative of a differential form with respect to a vector field and the covariant derivative of a differential form with respect to a vector field on a manifold with a defined connection. 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).
It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry.

Gradient

gradientsgradient vectorvector gradient
It is worth noting that in, after equating vectors with their dual covectors, the exterior derivative corresponds to gradient, curl, and divergence, although this correspondence, like the cross product, does not generalize to higher dimensions, and should be treated with some caution.
If the coordinates are orthogonal we can easily express the gradient (and the differential) in terms of the normalized bases, which we refer to as and, using the scale factors (also known as Lamé coefficients) :

Lie derivative

Lie bracketLie commutatorcommuting vector fields
The most important operations are the exterior product of two differential forms, the exterior derivative of a single differential form, the interior product of a differential form and a vector field, the Lie derivative of a differential form with respect to a vector field and the covariant derivative of a differential form with respect to a vector field on a manifold with a defined connection.
The Lie derivative commutes with contraction and the exterior derivative on differential forms.

Pullback (differential geometry)

pullbackpull backpullbacks
The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds.
Now suppose that α is a section of T * N (a 1-form on N), and precompose α with φ to obtain a pullback section of φ * T * N.

Élie Cartan

CartanÉlie Joseph CartanE. Cartan
The modern notion of differential forms was pioneered by Élie Cartan.

Hermann Grassmann

GrassmannHermann Günther GrassmannGrassman
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).
With the rise of differential geometry the exterior algebra was applied to differential forms.

Green's theorem

Cauchy–GreenGreen–Stokes formula
This operation extends the differential of a function, and is directly related to the divergence and the curl of a vector field in a manner that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the generalized Stokes' theorem.
Green's theorem is also a straightforward result of the general Stokes' theorem using differential forms and exterior derivatives:

Antisymmetric tensor

antisymmetricantisymmetrizationcompletely antisymmetric tensor
as a totally antisymmetric covariant tensor field of rank
A completely antisymmetric covariant tensor of order p may be referred to as a p-form, and a completely antisymmetric contravariant tensor may be referred to as a p-vector.

Manifold

manifoldsboundarymanifold with boundary
Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds.

Fundamental theorem of calculus

First Fundamental Theorem Of Calculusfundamental theorem of real calculusfundamental theorem of the calculus
This operation extends the differential of a function, and is directly related to the divergence and the curl of a vector field in a manner that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the generalized Stokes' theorem.
One of the most powerful generalizations in this direction is Stokes' theorem (sometimes known as the fundamental theorem of multivariable calculus): Let M be an oriented piecewise smooth manifold of dimension n and let \omega be a smooth compactly supported (n–1)-form on M.

De Rham cohomology

de Rham complexde Rham's theoremde Rham
In a deeper way, this theorem relates the topology of the domain of integration to the structure of the differential forms themselves; the precise connection is known as de Rham's theorem.
It is a cohomology theory based on the existence of differential forms with prescribed properties.

Geometric algebra

geometric productgeometric algebra formulationgrade projection
They are studied in geometric algebra.
Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis, differential geometry, e.g. by using the Clifford algebra instead of differential forms.

Vector field

vector fieldsvectorgradient flow
-forms are naturally dual to vector fields on a manifold, and the pairing between vector fields and
Replacing vectors by p-vectors (pth exterior power of vectors) yields p-vector fields; taking the dual space and exterior powers yields differential k-forms, and combining these yields general tensor fields.