Integrability conditions for differential systems

In the mathematical subfield of differential geometry called Pfaffian form ( after Johann Friedrich Pfaff ) or differential form of degree 1 or short 1 - form an object that is dual to a vector field in some way. Pfaffian forms are the natural integrand for path integrals.

Context

It should be

• An open subset of
• Or more generally an open part of a differentiable submanifold of
• Or generally an open part of a ( abstract ) differentiable manifold.

In each of these cases there is

• The term on the differentiable function; the space of infinitely differentiable functions on is denoted by;
• The notion of tangent space in a point;
• The concept of directional derivative of a tangent vector and a differentiable function;
• The term on the differentiable vector field. The space of vector fields on be denoted by.

Basic definition

A Pfaffian form to assigns each point to a linear form. Such linear forms hot Kotangentialvektoren; they are elements of the dual space of the tangent space. The space is called the cotangent space.

A Pfaffian form is thus an illustration

Other definitions

• A differentiable Pfaffian form is a linear map Continuous or measurable Pfaffian forms are defined analogously.
• The above- given amount is referred to as cotangent bundle. This is nothing else than the dual vector bundle of the tangent bundle. A Pfaffian form can thus be defined as a section of the cotangent bundle.
• The pfaff between shapes are exactly the covariant tensor of first order.

Total differential of a function

The total differential or the outer discharge of a differentiable function is Pfaff form which is defined as follows: If a tangent so is thus equal to the directional derivative in the direction of.

Thus, if a path with and so is

The following applies:

• If a constant function;
• For differentiable functions.

If applied to a scalar, then the total differential of using the gradient can be represented:

Coordinate representation

Either on a co-ordinate system. The coordinates can be used as functions

Be construed that assign a point its - th coordinate. The total differentials of these functions form a local basis. That is, for each point

A basis of.

This allows each Pfaffian form in a unique way as

Write functions.

The exterior derivative of any differentiable function has the representation

Definition of the line integral

It is a continuously differentiable way in and a 1-form. Then the integral of along is defined as:

The derivation of the parameter designated by.

Geometric interpretation of the line integral

A continuously differentiable function, the parametrization of a space curve dar. The parameter can be interpreted as a time parameter. At the time you find yourself at the site. Then, it is driven along a certain path or curve to the place. Thus, at the time of the end point of the curve is reached. Will be recorded for each time the place traveling over, the result is the picture.

It is intuitively clear that the same curve can be traversed in different ways. Such constant speed is one possibility. Another arises from a slow start and with subsequent acceleration. For the same curve, there are different parameterizations. The term " integral curve " is justified because it can be shown that the value of the integral is independent of the chosen parametrization of the curve. With one exception: If the start and end points of the curve reversed, so the movement is from the end point back to the beginning point of the curve, so the sign of the integral changes.

In the space of intuition, tangential and Kotangentialvektoren can be identified with each other by means of the scalar product: A Kotangentialvektor corresponds to the vector for which

Applies. Thus, 1-forms are identified with vector fields.

The integral of a 1-form corresponding to the (ordinary ) integral of the scalar product with the tangent vector:

, The curve is parametrized by the arc length, as the integrand the ( directed) length of the projection of the vector on the tangent to the curve:

Exact and closed forms

A continuously differentiable function is called a primitive function of the 1-form if:

A 1-form is called exact if it has a primitive function.

1 is a form - closed, if the following applies:

More generally, a total differential can be defined that maps each 1-form a 2- form. One form is called closed if and only if the following holds. It follows from the set of black that every exact form is closed.

Line integral of the total differential

For the line integral of the total differential along a path following applies:

The integral of the total differential therefore is not dependent on the waveform, but only by the endpoints of the curve. The integral over a closed curve, so therefore is equal to zero:

In the special case and there is the Fundamental Theorem of Calculus, since the integral on the left

Is. The above statements can be directly attributed to the fundamental theorem.

Existence of a primitive function

• As already mentioned, unity is a necessary condition for exactness. The Poincaré Lemma states that the barriers to the inversion of a global nature: In a simply connected, in particular in each star-shaped area has every closed Pfaffian form a primitive function. In particular, every closed Pfaffian form is locally exact.

• A steady Pfaffian form on a field if and only has a primitive function, if the integral of closed along each curve disappears.

Physical examples of Pfaffian forms

First Example " force field "

A force field describes the force which is applied to an article at any desired location. For example, the earth moves in the force field of the sun. The force field assigns each point to a force vector. Each force vector can be assigned to a linear mapping that maps the scalar product by an arbitrary vector linearly to the number field. According to this interpretation, the force field can be understood as a Pfaffian form or differential form of first order.

If the force field shown in Cartesian coordinates, with i = 1,2 or 3 are the unit vectors in Cartesian coordinates, then for the coordinate representation of the Pfaffian form:

.

The differentials are simply the corresponding basis vectors of the dual space, namely:

.

It must be made work to move an object in a field of force along a path from one place to place. The size W of the work done is given by the line integral along the way:

In a conservative force field the size W of the work done is path independent. A conservative force does no work on a closed road.

The antiderivative of a conservative force field is called potential or potential energy of the force. So is the total differential of the potential turn the power dar. applies:

The sign is the only convention.

Second Example " entropy "

Another important application of the theory of differential forms is in the range of thermodynamics. According to the Clausiuschen inequality holds:

Represents the temperature of the thermodynamic system and the contact of the heat exchange system with its environment Represents the thermodynamic system, for example, be a gas, the state variables are independent of temperature, pressure and volume of the gas. The coordinate representation of the heat exchange contact is thus given by:

.

The above integral is formed along a closed path C in three-dimensional state space (P, V, T). A closed path in the state space is called c in the thermodynamic cycle. The differential form then has exactly one ordinary function if each cycle is reversible:

In this case, has the Pfaffian form a primitive function, the entropy is called. For reversible cycles applies:

1 / T is an integral factor of the generated from the differential form a total differential.

It follows from the second law of thermodynamics:

Or

In an isolated system, there is no heat exchange with the environment, and therefore is considered. It follows from the second law that the entropy of an isolated system can not decrease.