Let be one of them and its Jacobian matrix evaluated at the point . Then, for any other vector field ,
measures how varies in the direction at .
We have , where in components. Also, it is obvious that defines a new vector field in which is symbolized as
We can be consider it as a bilinear map
Further, it is easy to see that for any scalar
This is called the standard connection of .
Now, let be a n-dimensional differentiable manifold and let be its tangent bundle. The set of differentiable sections is a differentiable Lie algebra which is endowed with a differentiable inner product via
in each .
It is possible construct a bilinear operator
So it makes sense to speak about the derivatives and since the are tangent which generate at a point , then is also tangent, so there are numbers (functions if one varies position) which enters in the relation
where , are the entries of the matrix and .
Routinely one can check that under a change of coordinates these functions transform as
here we have used Einstein’s sum convention (-sums) and the term
shows that the are not tensors.
For a proof please see the last part in: http://planetmath.org/?op=getobj&from=collab&id=64http://planetmath.org/?op=getobj&from=collab&id=64
Connection with base vectors.
Let us assume that coordinates are referred to a right-handed orthogonal Cartesian system with attached constant base vectors and coordinates referred to a general curvilinear system attached to a local covariant base vectors and local contravariant base vectors , both systems embedded in the Euclidean space . We shall also suppose diffeomorphic the transfomation . Then, by definition
and its inverses
Let us consider differentiation of base vectors , which may be written from (1),(2)
and using the Christoffel symbols this becomes
is easy to see from here that Christoffel symbol enjoy the property
In a similar way we find for the derivative of the contravariant base vectors
Is easy to show the following results:
comma denoting differentiation with respect to the curvilinear coordinates and . When the coordinate curves are orthogonal we have the following formulae for the Christoffel symbols: (repeated indices are not to be summed)
|Date of creation||2013-03-22 15:43:52|
|Last modified on||2013-03-22 15:43:52|
|Last modified by||juanman (12619)|