Line integral

The curves, line or path integral extends the ordinary integral term for the integration in the complex plane ( theory of functions ) or in multi-dimensional space ( vector analysis ).

Path integrals over closed curves are also referred to as a ring integral, contour integral or circulation and written with the symbol.

Real path integrals

Line integral of the first kind

The path integral of a continuous function

Along a piecewise smooth path

Is defined as

It refers to the derivation of after and the Euclidean norm of the vector.

A special case is the length of the curve parameterized by:

Line integral of the second kind

The path integral over a continuous vector field

With a likewise so parameterized curve is defined as the integral of the scalar product and:

Influence of the parameterization

Are and simple (ie, and are injective ) and paths with the same picture, so they parametrize the same curve in the same direction and they go through the curve exactly once, so vote the integrals along and agree. This justifies the name line integral; is the integration direction from the context or irrelevant can be seen, therefore, the way the notation is suppressed.

Path element length and element

The integrals occurring in the curve of the first kind expression

Called scalar path element or element length. The integrals occurring in the curve of the second kind expression

Ie vectorial path element.

Calculation rules

Be, line integrals of the same type (ie either both first or both of the second kind ), was the archetype of the two functions and of the same dimension and is. Then apply, and the following calculation rules:

Notation for line integrals of closed curves

Is a closed curve, one writes

The circle in the integral one would like to make it clear that is closed. The only difference here is in the notation.

Examples

• If the graph of a function, then the graph is

• An ellipse with semi-major axis and the minor semiaxis is parameterized by for. Its scope is therefore

Path independence

If a vector field is a gradient field, i.e., is the gradient of a scalar field, with

Then for the derivation of the concatenation, and

Which corresponds precisely to the integrand of the path integral over on. Consequently, for a given path

This means that the integral of depending solely on the points, and the route between them is not relevant for the result. For this reason, the integral of the gradient field is called " path-independent ".

In particular, for the ring integral over the closed curve, with any two paths and

This is in particular in the Physics of great importance, as for example, the gravity has these properties. Since the energy in these force fields is always a conserved quantity, they are referred to in physics as a conservative force fields. The scalar field is the potential or the potential energy; this is according to the last relationship over a closed path is zero.

Path independence can be demonstrated with the help of integrability.

Complex path integrals

Is a complex-valued function, then it is called integrable if and integrable. We define

The integral is therefore linear. Is continuous and an antiderivative of, shall apply as in the real

The integral term is now extended to the complex plane as follows: If a complex-valued function on a field, and is a piecewise continuously differentiable path in, the path integral along the path is defined as

The Malpunkt called complex multiplication here.

The central statement about integrals of complex functions, the Cauchy integral theorem: For a holomorphic function from the path integral depends only on the homotopy class of. Is simply connected, then the integral depends therefore not of, but only the starting and ending point.

Analogous to the real case, to define the length of the path through

For theoretical purposes following inequality, called the standard assessment, is of particular interest:

As in the real case the path integral is independent of the parametrization of the path, ie it is not absolutely necessary to choose as the parameter range, as can be shown by substitution.