1 Notation and Preliminaries.
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
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
3 Local coordinates.
Let be a system of local coordinates on , and let denote the corresponding frame of coordinate vector fields. In other words,
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
More generally, locally is a freely generated by the differential -forms
Thus, a differential form is given by
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:
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).
|Date of creation||2013-03-22 12:44:46|
|Last modified on||2013-03-22 12:44:46|
|Last modified by||rmilson (146)|