common formulas in calculus of differential forms
1 Euclidean forms
To begin with we have the total differential for scalars where is a domain in :
or by the Einstein summation convention
being the components of .
this coincides with the calculation .
For a pair of functions we can check Leibniz’s rule
Let be the set of 0-forms in and let (where ) be the set of 1-forms in .
Then the operator can be seen as a linear operator .
This can be generalized by defining to be the set of k-forms; that is, expressions of the type:
where are in i.e. they are scalars and they are multi-indexed sums. Further, the symbols are the wedge products of the .
So is calculated by
For example, if then , hence
which is rearranged as
and for two forms, if then
Now if we have a map between two domains and , we can pullback forms as , beginnig with the observation that at basics , we pullback it as
then, if we want , where , we are going to receive
Here the -sums must be taken between all indexes obeying .
So if ,
We also have
Obviously there are no forms in and usually one set if .
2 The de Rham complex.
The collection of mappings
give us a chain complex due that , so one can measure how much this differs from exactness via its homology
called the cohomological -group for .
Some with the fear of being confused with the giving of the same name to the operator , would like to write
and then one should modify the above conventions with
3 Manifold’s Forms.
One had seen that for mappings between ’s domains behave as . Then we can assign k-forms in each chart of a n-manifold by means of the coordinated functions on the neighborhood . Then
which will be the duals of the derivations .
Observe that if then is a scalar in .
where is a -form in .
4 Forms and connections
The Chistoffel symbols are the components of through the equation
where the are the coordinated tangent vectors.
The curvature tensor is defined as
which is a tri-linear map , so the Riemann-Chistoffel symbols are defined by the components of
With these one define the connection forms and the curvature forms as
these and define a 1-form and a 2-form viewed as a sections and respectively.
Observe that which compared with , it implies and for an arbitrary vector field (in the tangent coordinated basis)
a an-holonomic connection coefficients
as the an-holonomic.
Remember that in the coordinated frame field , but since this define the structural ”constants”
and the give relation
5 Cartan Structural Equations
The connection and the curvature forms satisfy the premiere , where the are the 1-forms dual to the and the deuxieme where the corresponding connection forms are calculated by i.e.
All that fits perfectly to give
This shows that the calculations of are very easy objects to put into an algorithm (Debever).
|Title||common formulas in calculus of differential forms|
|Date of creation||2013-03-22 15:51:28|
|Last modified on||2013-03-22 15:51:28|
|Last modified by||juanman (12619)|