Exterior derivative
The exterior derivative or Cartan derivative is a term used in the fields of differential geometry and analysis. It generalizes the well-known from calculus derivative of functions on differential forms. The name Cartan derivative explained by the fact that Élie Cartan is the founder of the theory of differential forms.
- 3.1 Loading and Cross (Flat and Sharp ) isomorphism
- 3.2 Cross Product
- 3.3 gradient
- 3.4 rotation
- 3.5 divergence
- 4.1 Definition
- 4.2 Properties
Exterior derivative
Definition
Be an n-dimensional smooth manifold and an open subset. With here is the space of k- forms is called on the manifold M. So is there for all exactly one function, so that the following properties are valid:
It must of course be proved that such an operator exists and is unique. This is called Cartan exterior derivative or derivative and is usually denoted by. Thus, it dispenses with the index which indicates the degree of differential form, which is applied to the operator.
Formula for the exterior derivative
One can also derive the outside using the formula
Represent, it means the caret ^ in that the corresponding argument is omitted, denotes the Lie bracket.
Coordinate representation
Be a point on a smooth manifold. The exterior derivative of this point has the representation
It has the local representation
Representation on Antisymmetrisierungsabbildung
The exterior derivative of forms is simply given by the total derivative and always covariant (see also covariant derivative ) and antisymmetric. The outer shape of a derivative can be considered up to a multiple than the formal antisymmetrization tensor of the form:
In index notation:
Repatriation
Be two smooth manifolds and a once continuously differentiable function. Then the return transport is a homomorphism such that
Applies.
In words, it is also said: product formation and external differentiation are compatible with the " pullback " relation.
Adjoint exterior derivative
Be in this section a pseudo- Riemannian manifold with index. The Hodge star operator is referred to below. The operator
Is defined by and through
He is referred to as adjoint exterior derivative or Koableitung.
This operator is linear and it is true. In fact, the adjoint operator to. If the manifold in addition compact, then for the Riemann metric and the relation
For this reason, also listed as because this really is the adjoint operator. Similar duality relations can be also defined for pseudo - Riemannian metrics, for example, the Minkowski metric of special relativity and the Lorentz metric of general relativity.
Another generalization of differential operators
Known from the vector analysis differential operators can be extended by means of the exterior derivative and the Hodge star operator on Riemannian manifolds. In particular one obtains a formula for the rotation, which operates on n-dimensional space. In the following it is always a smooth Riemannian manifold.
Loading and Cross (Flat and Sharp ) isomorphism
These two isomorphisms are induced by the Riemannian metric. They form on tangent vectors from Kotangentialvektoren and vice versa. It is sufficient to understand at this point the action of the three-dimensional space to show isomorphism. Be a vector field, then for the flat operator in standard coordinates of
So the flat operator is vector fields from their dual space. The Sharp - operator is the inverse to the operation. Be a Kovektorfeld (or a 1-form ), then ( also standard coordinates)
Cross product
The cross product is not a differential operator and is also defined in the vector analysis for three-dimensional vector spaces. Nevertheless, it is, in particular for defining the rotation is very important: Let a vector space, and two elements of an external power, then the generalized cross-product is defined by
For a justification of this definition, see under external algebra.
Gradient
It is a partially differentiable function and the standard scalar product is given. The gradient of the function at the point is for any of the claim by the
Uniquely determined vector. Using the differential forms calculus can be the gradient on a Riemannian manifold by
Define. Since the amount of 0 - forms according to the definition is equal to the amount of the infinitely differentiable functions, this generalized definition of the gradient function. This can quickly see through a short statement. Is a smooth function, then applies
In Euclidean vector spaces is often quoted this as follows:
Rotation
In vector analysis, the rotation is a picture. For general vector fields
The following calculation shows that one obtains the well-known expression for the rotation for the dimension:
This formula is obtained immediately by substituting the definition of the gradient in the the cross product.
Divergence
Similarly, there is a generalization of the divergence, this is
Hodge Laplacian
Hodge the Laplacian is a special generalized Laplace operator. Such operators have an important role in differential geometry.
Definition
Let be a smooth Riemannian manifold, the Hodge -Laplace operator is defined by
A function is called harmonic if it satisfies Laplace's equation. Analogously one defines the harmonic differential forms. A differential form is balanced if the Hodge -Laplace equation is satisfied. With the set of all harmonic forms is listed on. This space is isomorphic to the corresponding de Rham cohomology group due to the Hodge decomposition.
Properties
The Hodge - Laplace operator has the following properties:
Dolbeault operator
Two other differential operators, which are related to the Cartan derivative of the Dolbeault and the Dolbeault - cross - operator on manifolds. Thus, one can introduce the spaces of differential forms of degree that are listed by, and receives in a natural way the pictures
And
With. In local coordinates, these differential operators have the representations
And