differential form
1 Notation and Preliminaries.
Let be an -dimensional differential manifold. Let denotethe manifold’s tangent bundle, the algebra of smoothfunctions, and the Lie algebra of smooth vector fields. Thedirectional derivative
makes into a module. Using local coordinates, the directional derivative operation
can be expressed as
2 Definitions.
Differential forms.
Let be a module. An -linear mapping is said to be tensorial if it is a-homomorphism, in other words, if it satisfies
for allfor all vector fields and functions .More generally, a multilinear map is called tensorial if it satisfies
for all vector fields and all functions .
We now define a differential 1-form to be a tensorial linear mappingfrom to . More generally, for wedefine a differential -form to be a tensorial multilinear,antisymmetric, mapping from (times) to . Using slightly fancier language
, the above amountsto saying that a -form is a section
of the cotangent bundle , while a differential -form as a section of.
Henceforth, we let denote the -module ofdifferential -forms. In particular, a differential -form is thesame thing as a function. Since the tangent spaces of are-dimensional vector spaces
, we also have for .We let
denote the vector space of all differential forms. There is a naturaloperator, called the exterior product, that endows withthe structure of a graded algebra. We describe this operation below.
Exterior and Interior Product.
Let be a vector field and adifferential form. We define , the interior productof and , to be the differential form given by
Theinterior product of a vector field with a -form is defined to bezero.
Let and be differentialforms. We define the exterior, or wedge product to be theunique differential form such that
for all vector fields . Equivalently, we could have defined
where the sum is takenover all permutations of such that and , and where according to whether is an evenor odd permutation
.
Exterior derivative.
The exterior derivative is afirst-order differential operator , that can be defined as the unique linear mappingsatisfying
3 Local coordinates.
Let be a system of local coordinates on , andlet denote the corresponding frame ofcoordinate vector fields. In other words,
where the right hand side is theusual Kronecker delta symbol. By the definition of theexterior derivative,
In other words, the 1-forms form the dual coframe.
Locally, the freely generate , meaningthat every vector field has the form
where the coordinate components are uniquely determined as
Similarly, locally the freely generate . Thismeans thateveryone-form takes the form
where
More generally, locally is a freely generated by thedifferential -forms
Thus, a differential form is given by
(1) | ||||
where
Consequently, for vector fields , we have
In terms of local coordinates and the skew-symmetrization indexnotation, the interior and exterior product, and the exteriorderivative take the following expressions:
(2) | ||||
(3) | ||||
(4) |
Note that some authors prefer a different definition of the componentsof a differential. According to this alternate convention, a factorof placed before the summation sign in(1), and the leading factors are removed from(3) and (4).
Title | differential form |
Canonical name | DifferentialForm |
Date of creation | 2013-03-22 12:44:46 |
Last modified on | 2013-03-22 12:44:46 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 28 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 58A10 |
Defines | exterior derivative |
Defines | 1-form |
Defines | exterior product |
Defines | wedge product |
Defines | interior product |
Defines | tensorial |