Stokes' theorem
The set of Stokes or Stokes' integral theorem is named after Sir George Gabriel Stokes set of differential geometry. In the general version is a very deep theorem about the integration of differential forms, which extends the main theorem of differential and integral calculus, opening up a connection line from the differential geometry of algebraic topology. This relationship is described by the set of de Rham, for which the Stokes' theorem is fundamental.
The point is to convert n- dimensional volume integrals over the interior in (n-1 )-dimensional boundary integrals over the surface of the volume piece. Often, only special variants of the general set are considered, from which the general principle is well seen more or less, but which are important for the respective applications. The two most important special cases of the Gaussian integral theorem and the special Stokes' integral theorem (see below) are taken from the vector analysis. In physics and electrical engineering allows the special Stokes' theorem or the elegant Gauss spellings of physical relationships, for example, in the integrated forms of Maxwell's equations.
- 3.1 Integration on chains
- 3.2 statement
- 3.3 Application
- 5.1 Fundamental Theorem of Calculus
- 5.2 Gaussian integral theorem
- 5.3 Classical integral theorem of Stokes 5.3.1 statement
- 5.3.2 Notes
- 5.3.3 example
Integral theorem of Stokes
Statement
Be an oriented n-dimensional compact differentiable manifold with smooth boundary sections with induced orientation. This is for the most vivid examples of how the full sphere with boundary ( sphere ) or the torus (rubber ring ), if.
Further, let a defined on (or in a sufficiently large open environment ) alternating differential form of degree, which is assumed to be continuously differentiable.
Then the following statement, which was named after Stokes applies:
The Cartan derivative referred. The right-hand integral can be understood as a surface integral, or more generally as an integral over the manifold.
The Cartan Derivation " d Ⓜ " here is in a sense " dual" to the topological operation whereby the cross- relation contained in this formula between aspects of analysis and topological - algebraic aspects results.
Comments
Under the very general condition that applies, - with the (n- 1 )-dimensional base forms, for example, and the "wedge - product ", which among other things satisfy the condition of the anti- symmetry, the outer discharge means specifically to the following:
Is particularly simple proof of the " fundamental theorem " when as in the example on a normal area, the integration of diversity (in the drawing D called ) into vertical strips can be segmented ( in xn- direction) so that only the yellow marked "top" and occur at the red marked " bottom " non-trivial contributions and also because of the marked orientation ( the direction of the arrows ) with the opposite sign.
Conclusion
Be open and a continuously differentiable (k -1) - form. Then for any compact oriented rimless k- dimensional submanifold the statement:
Applications
The ( general ) set of Stokes is used mainly in mathematics. he
- Includes as special cases for physicists and electrical engineers the set of Gauss and the special Stokes' theorem (see below), and
- Secondly, forms a physical link between differential geometry and algebraic aspects of topology by two different routes and emanating from the same starting point and lead to the same end point, are as defined topologically homologous if ω for certain single-stage differential forms the contour integral vanishes approximately in a differentiable manifold. Corresponding concepts of algebraic topology can be built also with the higher-dimensional Stokes' theorem.
Integral Stokes' theorem for chains
Integration over chains
Be a smooth p- simplex and a smooth, closed differential form on the smooth manifold. Then the integral over is defined by
In this case, the return referred to by respect. The definition makes sense, since a smooth submanifold with boundary and induced orientation is. ( Or is understood simply as a closed subset of. ) In the case of the definition corresponds to the usual line integral. Is a smooth p- chain of the singular complex, then the integral of over is defined as
In the event you can find the definition and other information in the article cycle (function theory).
Statement
Is a smooth p- chain of the singular complex and a smooth (p -1) differential form on the smooth manifold. Then we have
With the boundary operator of the singular complex is called.
Application
This set shows a link between differential geometry and topological properties of a smooth manifold. For, if the De Rham cohomology and singular homology, we obtain by
With a homomorphism. Because the set of Stokes this homomorphism is well defined and it is not on the choice of the representative of the homology class. Let and be two representatives of the same singular homology class, then apply for two representatives differ only by an element of the boundary. Hence it follows by Stokes' theorem
The last equality holds, since an element of de Rham cohomology, and therefore applies. Is an exact differential form then applies
After the central set of the de Rham homomorphism is actually an isomorphism.
Underlying topological principle
Behind the Stokes' sentence is a general topological principle, which states in its simplest form, that the way pick up at " oriented paving a surface element " inside "because of oncoming traffic " in pairs, so that only the boundary curve remains.
Links in the sketch you can see four small equal -oriented " cobblestones ". The drawn in the middle of the "inner way " in pairs through in the opposite direction; therefore their contributions to the line integral cancel each other, so that only the contribution of the boundary curve remains. It is therefore sufficient to prove the integral theorems only for small as possible " cobblestones ".
With sufficient refinement of the pavement which is almost elementary in general.
Special cases
Several special cases of the general set of Stokes are in classical vector analysis of meaning. This naturally includes the classical Stokes' theorem. It follows from the general set of ω: = F1dx1 F2dx2 F3dx3. In addition, also the main theorem of differential and integral calculus, the set of Green and the Gaussian integral theorem special cases of the general Stokes' theorem.
Main theorem of differential and integral calculus
Be an open interval and a continuously differentiable function. Then a 1-form (so-called Pfaff'sche form), and the general Stokes' integral theorem degenerates to
This is the main theorem of differential and integral calculus.
Gaussian integral theorem
For a compact subset of and an n-dimensional vector field is obtained as a further special case of the Gaussian integral theorem.
Here, the n-dimensional unit vector normal and the integrals are now n and ( n-1) -dimensional, where the size is written as well. If one chooses
We obtain the Gaussian integral theorem from the stoke between.
One can use to define the divergence of a vector field that set also, which definition is independent of the used coordinates.
Classic integral theorem of Stokes
The classical integral theorem of Stokes is also known as a set of Kelvin -Stokes or rotation rate. He finds at physicists and electrical engineers use primarily in connection with Maxwell's equations. He states that a surface integral can be transformed via the rotation of a vector field in a closed line integral of the tangential component of the vector field. This is helpful because the line integral contains the vector field alone and easier to calculate is usually as surface integrals, especially if the considered area is then curved. In addition, the line integrals are directly affected in many applications - and only secondarily the associated surface integrals - for example, when Faraday 's law of induction. Is especially given so, the fact that many different manifolds into a single closed boundary manifold can be " squeezed " to the gauge invariance of theories like that of Maxwell.
Statement
It should be an open subset of the three-dimensional space and on a defined once continuously differentiable vector field. This is required so that the expression can be formed. Next one in two-dimensional regular surface contained, which is oriented by a unit normal field was (that is, it defines what the "top" of the surface). In addition, the unit tangent vector of the edge curve. With the property regularly to ensure that the margin is sufficiently smooth.
The boundary of is denoted by. Hereinafter, this edge is always identified by a closed curve. With all of these conditions apply
In the applications we also write
With and. Furthermore, the rotation and (or ) the scalar product of two vectors. The form is the volume form of the two-dimensional surface, and is the element of length of the boundary curve.
Comments
In the case, that a flat subset applicable in appropriate coordinates. Is not flat, it can be provided that the two-dimensional surface parameterization
Can be broken down into segments, the volume form for fixed by
Calculate. Also, the vector can be calculated analogously, and indeed is the unit vector formed from the three components of the vector product, ie
Example
It is a designated area as a normal flat manifold, which meets the requirements of the sentence, and the vector field F given by. The unit normal field is given by Then
After applies Stokes' theorem
This example shows that the Green's theorem is a special case of the Stokes integral theorem.