differential form
1 Notation and Preliminaries.
Let be an -dimensional differential manifold. Let denote
the manifold’s tangent bundle, the algebra of smooth
functions, and the Lie algebra of smooth vector fields. The
directional 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 all for 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 mapping
from to . More generally, for we
define a differential -form to be a tensorial multilinear,
antisymmetric, mapping from (
times) to . Using slightly fancier language
, the above amounts
to 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 of
differential -forms. In particular, a differential -form is the
same 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 natural
operator, called the exterior product, that endows with
the structure of a graded algebra. We describe this operation below.
Exterior and Interior Product.
Let be a vector field and a differential form. We define , the interior product of and , to be the differential form given by
The interior product of a vector field with a -form is defined to be zero.
Let and be differential forms. We define the exterior, or wedge product to be the unique differential form such that
for all vector fields . Equivalently, we could have defined
where the sum is taken
over all permutations of such that and , and where
according to whether is an even
or odd permutation
.
Exterior derivative.
The exterior derivative is a
first-order differential operator , that can be defined as the unique linear mapping
satisfying
3 Local coordinates.
Let be a system of local coordinates on , and let denote the corresponding frame of coordinate vector fields. In other words,
where the right hand side is the
usual Kronecker delta symbol. By the definition of the
exterior derivative,
In other words, the 1-forms form the dual coframe.
Locally, the freely generate , meaning that every vector field has the form
where the coordinate components are uniquely determined as
Similarly, locally the freely generate . This means that every one-form takes the form
where
More generally, locally is a freely generated by the differential -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 index notation, the interior and exterior product, and the exterior derivative take the following expressions:
(2) | ||||
(3) | ||||
(4) |
Note that some authors prefer a different definition of the components of a differential. According to this alternate convention, a factor of 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 |