Curvilinear coordinates

Coordinates are curvilinear coordinates system on the Euclidean space, in which the coordinate lines may be curved and the diffeomorphic to Cartesian coordinates. That is, the transformation between curvilinear coordinates and Cartesian coordinates must be locally invertible, wherein the image as well as the inverse mapping must be continuously differentiable.

The curvilinear coordinate systems most commonly used, both of which belong to the orthogonal coordinate systems are:

• Plane polar coordinates ( 2D ) or the 3-dimensional analogue, the cylindrical coordinates
• Spherical coordinates and spherical coordinates called (3D)

Depending on the problem are calculations in curvilinear coordinate systems easier than perform in Cartesian. For example, physical systems with radial symmetry are often easier to treat in spherical coordinates.

The following comments relate specifically to the three-dimensional Euclidean space, much of it can be expanded, however, to the -dimensional case.

• 5.1 components projected on the basis vectors: Orthogonal coordinates
• 5.2 Introduction dual space and dual basis
• 5.3 Dual base
• 5.4 Covariant components: vectors as a linear combination of the contravariant basis vectors
• 5.5 components projected on the basis vectors: general curvilinear coordinates
• 5.6 Dual base and components for orthogonal coordinates
• 5.7 Dual basis in three dimensions
• 5.8 Example for straight, oblique coordinate systems

• 6.1 Tensors second stage
• 6.2 scalar product of two vectors
• 6.3 tensors of third rank

• 7.1 Christoffel symbols
• 7.2 Covariant derivative
• 7.3 Properties of the Christoffel symbols

• 8.1 Vector product and alternating tensor
• 8.2 Coordinate area: interior geometry
• 8.3 Coordinate surface: exterior geometry

• 9.1 curve element
• 9.2 arc element
• 9.3 surface element
• 9.4 volume element

Transformation of Cartesian coordinates

Coordinates of a point in -dimensional space is a tuple of real numbers, which are determined relative to a specific coordinate system. The following are the coordinates in two different coordinate systems are considered for a spot.

The Cartesian coordinates can be used as continuously differentiable functions of new coordinates write ( direct transformation ):

This is a system of equations is invertible ( that is, after the soluble ) ( inverse transformation )

If the inverse Jacobian is non-zero or infinity is:

The inverse transformation must as well as the direct transformation to be continuously differentiable.

For the points in which the transformation is bijective, ie the transformation regularly, otherwise singular. Is given by: If a point with Cartesian coordinates given, using the inverse transformation can clearly the coordinates of the curvilinear coordinate can be calculated. Every regular point of space can be uniquely described by both as well as equivalent by the.

A set of transformation equations with the properties described above together with a Cartesian coordinate system defining a curvilinear coordinate system.

Coordinate surfaces, lines and axes

The terms coordinate surfaces, lines and axes are clearly explained below using the three-dimensional space.

Coordinate surfaces obtained by each held a coordinate ( ) and the other two are varied.

Through each non- singular point is exactly one face of each family of surfaces.

Coordinate lines are obtained by detained two coordinates ( with ) and the third is varied, ie. Than intersection of two coordinate surfaces for different coordinate

The above condition for the Jacobian means that at any point in three -dimensional space can intersect only 3 coordinate lines, otherwise this point has no clear coordinates ( Jacobian equal to zero ).

As an example of the ambiguity - axis counts in spherical coordinates, in which all levels ( the azimuth angle) cut; Thus, the coordinates of points on the axis are not unique ( but arbitrary). Such points are called singular points of transformation.

Intersect the coordinate lines at right angles, it does coordinate system orthogonal.

The coordinate axes are defined as the tangent to the coordinate lines. Since the coordinate lines are curved in general, the coordinate axes are not spatially fixed, as it applies to Cartesian coordinates. This leads to the concept of local basis vectors whose direction depends on the considered point in space - as opposed to global base vectors of the Cartesian or affine coordinates.

Various bases

To be able to define by means of a vector coordinate, a base is needed. In -dimensional space, this consists of linearly independent vectors, the basis vectors. Any vector can be represented as a linear combination of the basis vectors, the coefficients of the linear combination are called the components of the vector.

For real curvilinear (ie non- linear ) coordinates vary basis vectors and components from point to point, and therefore the base is referred to as a local basis. The spatial dependence of a vector field is distributed to the coordinates as well as on the basis vectors. To this end, in contrast to global bases are characterized by the fact that the basis vectors are identical at each point, which is only for linear or affine coordinates ( the coordinate lines are straight, but generally oblique ) is possible. The spatial dependence of a vector field lies in rectilinear coordinate systems alone in the coordinates.

To link basis vectors with a coordinate system, there are two common methods:

• Covariant basis vectors: tangential to the coordinate lines, that is collinear with the coordinate axes
• Contravariant basis vectors: normal to the coordinate surfaces

The two classes of basis vectors are dual or reciprocal. These two bases are called holonomic bases. They differ in their transformation behavior under change of coordinates. The transformations are inverse to each other.

Simultaneously at each point of the observed diversity exist both bases. Thus, any vector can be represented as a linear combination of either the covariant basis vectors or the contravariant basis vectors. Here contravariant components are always combined with covariant basis vectors and covariant components with contravariant basis vectors.

This cross- pairing ( counter * ko and ko * counterproductive ) ensures that the vector is invariant under coordinate transformation, because the transformations of components and basis vectors are inverses of each other and cancel each other out. This property is essential for the concept of a vector in physics: In physics, laws must apply regardless of the particular coordinate system. From a physical point of view must be a vector such as the velocity of a particle describes, be independent of the chosen coordinate system.

It is called a contravariant vector ( better: contravariant coordinate vector ) when the components contravariant and covariant basis vectors are. Analog is called a covariant vector when the covariant components and the basis vectors are contravariant.

Covariant base

The covariant basis vectors are nestled in each point tangent to the coordinate lines on.

Normalized and natural basis vectors

The unit tangent vectors to the coordinate lines form a basis (consisting of covariant basis vectors, since collinear to the coordinate lines):

This unit vectors have a direction dependent on the place in general. Since they are tangent vectors to the coordinate lines, tighten the covariant base vectors on the so-called tangent space. The tangent space in the sense of differential geometry is a vector space, which approximates the differentiable manifold at point linearly.

One defines the so-called scale factors.

The un-normalized vectors form the natural basis or unitary basis (denoted by vector arrow instead of roof for the normalized basis):

Contravariant components: vectors as a linear combination of the covariant basis vectors

With the new base can now be all vectors by the basis vectors of the covariant basis (normalized ) or (not standardized = natural basis vectors ) can be expressed:

Here, or ( contravariant ) vector component pointing in the direction of the coordinate line, with respect to the base and with respect to the normalized natural basis. In the tensor is written with superscript index written.

The length of a vector component corresponding to the case of the base to the amount of normalized coordinate, in the case of the natural base to the product of the magnitude of the coordinate and the base length of the vector:

Describes a vector physical quantity, so is in the unnormalized case, not only the length but also the physical dimension partly in the coordinates and partly in the natural basis vectors, which can be cumbersome in concrete calculations. When normalized basis, however, the physical dimension is purely limited to the coordinate. Therefore, the coordinates are called physical coordinates and the normalized basis vectors are also called physical basis vectors.

To distinguish the coordinates therefore holonomic coordinates and the natural basis vectors are called are called holonomic basis vectors or simply contravariant coordinates and covariant basis vectors.

Transformation properties of basis vectors and coordinates, Jacobian

From the definition follows the natural basis vectors for the transformation of the coordinates according to a simple transformation formula:

The natural basis vectors show a very simple transformation behavior. For the normalized basis vectors of the transformation formula contains additional factors:

An arbitrary vector has both the old as well as the new coordinates to be displayed:

Thus we obtain the transformation properties of the coordinates:

While the transformation of the ( covariant ) basis vectors is carried out by means of the Jacobian matrix, the inverse Jacobian matrix has to be used when transforming the ( contravariant ) coordinates.

In the tensor analysis to define a vector about the above transformation behavior. In this respect, the position vector itself is not a vector, the position vector differential is however already.

The Jacobian of the coordinate transformation from Cartesian to curvilinear coordinates is identical to the matrix, which is made ​​of natural -based vectors as columns:

The condition for the inverse Jacobian can be explained by the following relationship:

This corresponds to an inhomogeneous linear equation for the vector. That is, the unknowns are the basis vectors of the curvilinear coordinates. The system of equations is only a unique solution when the core of the matrix is ​​zero-dimensional or the row or column vectors are linearly independent. This is equivalent to that the determinant is non-zero. Then the unknowns are uniquely determined, ie, at each point there is exactly one defined basis.

Metric Tensor and Gram determinant

The scalar products between the natural basis vectors define the components of the metric tensor or the fundamental tensor:

Note that the metric tensor is symmetric because of the scalar product of the commutative:

Because of the symmetry of the metric tensor has independent elements (instead ). In three dimensions, ie six coefficients: three lengths of the natural basis vectors and three angles between the basis vectors.

The metric tensor can be written as a product of the Jacobian matrix and its transpose:

The sizes are called metrics and Maßkoeffizienten, since these are required to calculate the length of a vector from the contravariant coordinates. To this end, the scale factors are required.

The scale factors are represented by the diagonal elements, as the following applies:

The determinant of the metric tensor is called the Gram determinant:

It follows from that the determinant of the Jacobian matrix (the Jacobian ) is equal to the square root of the Gram determinant.

And the determinant of the normalized basis vectors yields (due to the multi -linearity of determinants):

For the inverse of the metric tensor is considered by Cramer's rule

The adjoint ( transpose of the cofactor matrix which, whose entries are the signed minors ) and is designated Gram determinant. from

Follows for the inverse metric tensor:

Special case: Orthogonal coordinates

Cut in -dimensional space at each point in space the coordinate lines in pairs vertically, it is called an orthogonal coordinate system. Thus the unit vectors form an orthonormal basis of:

For the natural basis vectors applies:

Thus, for diagonal orthogonal basis vectors of the metric tensor.

The inverse metric tensor is orthogonal coordinate equal to:

The Gram determinant is simplified for orthogonal coordinates to:

Applies here to the determinants of natural or normalized basis vectors:

Special case: Orthogonal coordinates in three dimensions

Form the orthonormal basis vectors of a right-handed basis (positive orientation), the following relationships apply:

Written out:

Special case: Rectilinear coordinate systems

General curvilinear coordinate the coordinate lines are curved and the base vectors vary from point to point. In the special case of the straight, but skewed quite, coordinate systems, the coordinate lines are straight and the basis vectors are thus independent of location. The coordinate surfaces are planes, a group of coordinate surfaces forming parallel planes.

The transformation equations can in this case be written as:

And said constant. The Jacobi matrix corresponds to the transformation matrix. Thus correspond to the natural unit vectors of the th column of the matrix.

Example of straight, oblique coordinate systems

As an example of a rectilinear, oblique coordinate system is considered a Minkowski diagrams with two reference systems that move to one another with uniform speed. About append the parameters of relative speed, rapidity and angle with the value ranges along with and well. The Lorentz transformation transforms the reference systems into one another:

Because the coordinate transformation is linear, the following applies:. The unit vectors in the direction of be in Cartesian coordinates:

If one interprets the Euclidean Minkowski diagram ( using the Standardskalarprodukts and not the Minkowski scalar product ), we obtain the metric tensor

And the Gram determinant

Since occur for off-diagonal elements, the coordinate lines do not form a right angle:

There are unequal to the one diagonal elements, which are not natural base vectors are unit vectors, ie, the scale of the coordinates tilted lines is extended:

Side note: With the scalar product of the special theory of relativity, the non-Euclidean Minkowski metric is one the invariance of the scalar product is replaced under Lorentz boosts.

Dual basis: contravariant basis

The contravariant basis vectors are perpendicular to the coordinate surfaces at each point. They are dual to the covariant basis vectors. The contravariant components of a vector can be obtained by projection onto contravariant basis vectors.

Components projected on the basis vectors: Orthogonal coordinates

The vector component ( contravariant component ) of the vector can be used for an orthonormal basis ( ) simply determined using the following projection:

For non- orthogonal coordinate systems ( oblique ) is obtained by the projection of a vector on a covariant basis vector the covariant component ( covariant component - in the tensor written with subscript ) and not the contravariant component, since not apply the relation, or the metric tensor is not diagonal. This requires the concept of the dual space and the dual basis.

Introduction of dual space and dual basis

The dual space to the vector space of tangent vectors is formed from the linear functionals (also 1-forms ), represent the vectors in the underlying body. A basis of the dual space are dual to the basis vectors. These are defined so that the following applies.

Furthermore, we define the following bilinear form, the so-called dual pairing. Thus, the effect of dual basis vectors to basis vectors written as:

For finite-dimensional is isomorphic to so. In Euclidean spaces (the one with the standard scalar product ) will enable the dual pairing with the scalar product

Identify and thus dual vectors also represent as vectors ( applies here: and as well ).

Dual basis

The dual basis is thus defined so that for the scalar product of the basis vectors ( covariant basis vectors ) and dual basis vectors ( contravariant basis vectors ) applies ( here for the normalized basis vectors ):

Respectively. analogously to the natural basis vectors and their dual basis vectors:

For the natural basis vectors and their dual basis vectors in matrix notation applies:

Since the matrix corresponds to the covariant basis vectors as column vectors of the Jacobian matrix, therefore, the matrix of the basis vectors must correspond contravariant as row vectors of the inverse Jacobian matrix:

To obtain the dual basis vectors, so the inverse of the Jacobian matrix to be determined.

The Gram determinant of the contravariant basis vectors must correspond to the inverse of the determinant of the covariant basis vectors:

Covariant components: vectors as a linear combination of the contravariant basis vectors

With the new base can now be all vectors by the basis vectors of the contravariant basis (normalized ) or (not standardized = natural basis vectors ) can be expressed:

It is or the ( covariant ) vector component pointing in the direction of the normal to the coordinate surface. In the tensor is written subscripted.

Components projected on the basis vectors: general curvilinear coordinates

The contravariant component of a vector obtained by projection onto the dual basis vector ( contravariant base - written in tensor analysis with superscript index).

In the orthonormal basis vectors ko and contravariant basis vectors are consistent and well co- and contravariant components of a vector.

General can be any vector contravariant or covariant basis vectors represent:

Thus contravariant components with covariant basis vectors or covariant components with contravariant basis vectors are combined. This property leads to the invariance of the vectors of a change of the coordinate system.

Provides multiplication on both sides with

Thus, with the help of the metric tensor and its inverse contravariant components in covariant and vice versa transfer ( in Tensorsprache: raising and lowering of indices).

Dual base and components for orthogonal coordinates

In orthogonal coordinates vote in the normalized form consistent basis vectors and dual basis vectors. For the natural base, this means that two mutually dual basis vectors are collinear, that is, one is a multiple ( factor ) of the other:

Thus, the components agree with respect to the normalized base also agree:

Dual basis in three dimensions

The dual basis vectors can be divided by their scalar triple or express in three-dimensional cross- products of the basis vectors:

In more compact notation for the normalized basis vectors

Or for the natural base vectors:

While the ( covariant ) basis vectors are tangent to the coordinate lines, the dual ( contravariant ) basis vectors are perpendicular to the coordinate surfaces. For example, while the vectors and are in the coordinate surface is perpendicular to this.

Conversely, the contravariant basis vectors can be in three-dimensional cross- products of the covariant basis vectors divided by their scalar triple or express:

Form the covariant basis vectors a legal system ( positive Jacobian ), then also form the contravariant basis vectors a legal system (inverse Jacobian is positive). The product of the two determinants must result in one namely.

Example of straight, oblique coordinate systems

As an example of a rectilinear, oblique coordinate system serves as a continuation of the above example, a Minkowski diagram. The Lorentz transformation was given by, so is the inverse transformation ::

Because the coordinate transformation is linear, the following applies:. Thus, the dual unit vectors are denominated in Cartesian coordinates:

These fulfill the duality conditions: orthogonality and normalization for and for and.

Tensors

Tensors -th stage can generally be represented as -fold tensor product of vectors:

The tensor product of vectors is not kummutativ, so the order of the (basis ) vectors must not be interchanged.

Here are scalars ( functions of the coordinates in the body, ie, the coordinate transformation under its function value at each point do not change) zero-order tensors and vectors are tensors of first order.

Since vectors in two different ways, namely let covariant or contravariant, represent, there is a tensor -th display options. Through the representation by means of vectors, the properties of the vector are inherited by tensors. Thus, for example, lift with the help of the metric tensor indices and lower, ie in co- convict contravariant components, or vice versa. Tensors that by raising and lowering (ie, inner products with the metric tensor ) result, hot -associated tensors. Likewise, the transformation properties of vectors is assumed for tensors, ie covariant shares of a tensor transform like covariant vectors, ie by means of the Jacobian matrix, and contravariant components with the inverse Jacobian matrix, as contravariant vectors.

Tensors of second stage

A second order tensor can be represented in four different ways:

The four cases are: (in) contravariant, (in) covariant, mixed contra - covariant, mixed co- contravariant.

The unit tensor defined by is given by:

Scalar product of two vectors

The scalar product of two vectors in curvilinear coordinates is given by:

This corresponds to the contraction of the second stage tensor to a tensor of zeroth order.

Tensors of third rank

A tensor of the third stage can be represented in eight different ways:

In three dimensions, the totally antisymmetric tensor is given by:

The first relation is the Cartesian notation, the following two of eight spellings of the curvilinear version of the tensor.

Derivatives of the basis vectors

The derivatives of vectors that are represented in curvilinear coordinates, have over the Cartesian following feature. Since the coordinate lines in general are not straight and therefore the basis vectors have a direction dependent on the location, the basis vectors need to be differentiated ( applying the product rule ):

Respectively. with respect to the natural basis

Christoffel symbols

The derivation of the basis vectors according to a coordinate can be written as a linear combination of all basis vectors.

The coefficients are called Christoffel symbols of the second kind

The sizes are called Christoffel symbols of the first kind, the total differential of a natural basis vector is:

The Christoffel symbols are now used for the derivation of a vector ( the second equal sign, the indices and swap, which is possible because over both is summed, and are excluded ):

Covariant derivative

Based on this, we define the covariant derivative by ( an excellent connection on a Tensorbündel, a special vector bundle ) of a vector:

The first term describes the change of the vector components of the field along the coordinate axis, the second change of the field, which is due to the change of the coordinate system state. In the rectilinear coordinate system (in this case the metric tensor is a constant ) to disappear, the Christoffel symbols and the covariant derivative is identical to the partial derivative.

The covariant derivative introduces an additional geometric structure on a manifold which allows to compare vectors from different vector spaces and that of neighboring tangent spaces. Thus, the covariant derivative establishes a link between different vector spaces. This is, for example, needed to calculate the curvature of a curve - this is the derivative of the vectors and form who live in different vector spaces.

The covariant derivative of a tensor -th coordinate gives the coordinates of a Level tensor, as a covariant index is added. For tensors of step applies: The partial derivative of a Tensorkoordinate by curvilinear coordinates is, in contrast to the covariant derivative, no Tensorkoordinate.

The covariant derivative of the coordinates of the metric tensor vanish.

With the covariant derivative can be generalized the directional derivative:

Example: The geodesic on a Riemannian manifold, the shortest curve between two points, can be expressed by the geodesic equation. This equation means that the velocity vector field (or tangent vector field ) of the curve along the curve (that is, parallel to ) is constant. This definition is based on the consideration that the geodesics of the straight lines are. The curvature of the curve must therefore vanish, and thus be the directional derivative of the tangent vectors along the curve to zero. As reported in local coordinates is the geodesic differential equation:

There are basically two ways the Christoffel symbols, ie the coefficients of the affine connection specify: Either you are in front of the coefficients, ie you specifies how to change the coordinate system from point to point on the manifold, or you have more information about the viewing room than that it is a differentiable manifold is (eg a notion of distance ) and thus knows what has to be understood by the covariant derivative, so that the Christoffel symbols are also defined. This latter case is realized because the manifolds are Riemannian manifolds considered here, and hence for each tangent space of the manifold a scalar and thus induces a metric, ie, a notion of distance exists.

Since the considered manifolds (semi-) Riemannian manifold are (here the Torsionstensor disappears), the relationship is a so-called Levi- Civita connection, ie it is torsion or symmetrically and also a metric connection. Here, since the connection is torsion, corresponding to the antisymmetrized directional derivative exactly the Lie derivative. While the directional derivative linear in the direction field ( the directional derivative depends on the direction field at only one point off ), the Lie derivative is no argument linear ( for the Lie derivative of both vector fields must be known in an open environment ).

Properties of the Christoffel symbols

From the set of black (or from the torsional freedom of the link ), it follows that the Christoffel symbols in the lower two indices are symmetric:

Therefore, the Christoffel symbols by deriving the metric coefficients can be determined:

This follows from the following relation

And two permutations of, namely and.

For the derivation of the dual basis vectors we obtain the following relation with the negative Christoffel symbol:

This follows the covariant derivative of the covariant components:

It is important to note that the Christoffel symbols with their three indices do not describe a tensor of the third stage, since they do not show the required transformation properties of tensors.

The occurrence of the second summand in the transformation formula shows that it is not a tensor. Therefore, the Christoffel symbols in the literature are sometimes listed with symbols that can not be confused with tensors:

The statement on the transformation behavior can be generalized: the index () a partial derivative of a tensor transforms like a covariant index (). In contrast, the two indices () does not transform a second partial derivative as tensor indices. As a solution the covariant derivative is available: The indices of a -th covariant derivative of a Tensorkoordinate are again Tensorkoordinaten, they transform as covariant indices. For example, are in the indices and covariant indices.

Other characteristics of curvilinear coordinates in three dimensions

Vector product and alternating tensor

In Cartesian coordinates, the cross or vector product is the Levi- Civita symbol

In curvilinear coordinates, this is using the alternating tensor

Be replaced by:

This can be derived by:

In the following calculation we see that the correct transformation properties of a tensor is (in this case the covariant version of the tensor ):

Bzgl. the normalized basis is the vector product:

Coordinate space: interior geometry

We consider without loss of generality the surface. A ( unnormalized ) normal vector of the surface is collinear with the contravariant basis vector:

One defines a surface in the convention, the following quantities of "internal geometry ", which can be determined by linear and angular measurements within the area (see first fundamental form ):

For orthogonal coordinates.

The metric tensor of the surface and the Gram determinant

The Jacobian of the surface is, the normalized normal vector of the surface is:

The inverse metric tensor of the surface is:

Coordinate surface: exterior geometry

Greek indices run over the range below 1.2 and thus characterize coordinates and base vectors in the area.

The partial derivative of the normalized normal vector on the coordinate can be represented as a linear combination of the basis vectors of the surface. This follows from the normalization condition by discharging. Thus, it is orthogonal to the surface normal and therefore should be in the area. It introduces a new size, which is a second -rank tensor:

The tensor is called in the literature partly second stage, curvature or Haupttensor Flächentensor. The covariant coordinates can be calculated as follows, where:

This can also be rewritten as (see second fundamental form ):

The can be reconciled with the Christoffel symbols of the second kind in connection. It applies in the following:

From this follow the Gauss -Weingarten equations:

The second fundamental form depends on the position of the surface in the surrounding space and is used for calculation of curvature. Using the mixed contravariant - covariant tensor

The principal curvatures ( eigenvalues ​​), the mean curvature and Gaussian curvature of the surface to be defined.

The Riemann curvature tensor can be expressed by the tensor product. More integrability conditions are the Mainardi - Codazzi equations.

Integration elements in three dimensions

Curve element

A vectorial path element or cam element can be represented as the total differential of the position vector.

The differentials in the direction of coordinate lines can be identified:

It should be noted that the index in not covariant index.

With the help of the vectorial path element arc, area, and volume element can now be determined.

Arc element

The scalar path element or element or arc length element is defined by

With physical ( normalized ) basis vectors applies:

For orthogonal coordinates applies:

Special case: If the curve in the plane, then apply the first fundamental form

Surface element

The vectorial surface element of a coordinate surface is

The sign depends on the orientation of the surface element. The size is called scalar surface element.

We consider without loss of generality the area:

With physical ( normalized ) basis vectors applies:

For orthogonal coordinates applies:

Volume element

The volume element is, with the amount of the Jacobian can be identified:

With physical ( normalized ) basis vectors applies:

For orthogonal coordinates applies:

Differential operators in three dimensions

Special case: Orthogonal coordinate systems

There are the differential operators gradient, divergence, curl and Laplacian given for orthogonal coordinate systems:

• Gradient of a scalar function (actually a box function):

Note that not only, but all these sizes, the basis vectors and the coefficient h may depend on u.

• Divergence:

• Rotation:

• Laplace operator:

Here you should not simply set, but use the definition. The above given results are obtained, in fact, easier to visuality way than from the usually quite complicated bill, if you like of the existing kordinatenunabhängigen definitions of the variables emanates.

General curvilinear coordinate systems

Finally, in general curvilinear coordinate system, the differential operators are given.

The following are the natural basis is used and the correct notation of tensor ( contravariant = superscript, subscript = covariant ) used. is a scalar field and a vector field.

It is used the notation. Furthermore, the Christoffel symbols defined by and the covariant derivative be used. The covariant derivative of a scalar and the covariant derivative of a vector or.

• Gradient of a scalar field

• Gradient of a tensor field

• Gradient of a vector field

• Gradient of the level two tensor

• Divergence of a vector field

• Divergence of a tensor field

• Divergence of a second rank tensor

• Rotation of a tensor field

• Rotation of a vector field

• Laplacian of a scalar field

Gradient and total differential

In the following, the gradient is to be derived in curvilinear coordinates. The total differential of the position vector can be represented as:

Consider now an arbitrary scalar field. Its total differential is ( being above representation of is used):

The gradient is defined by

And can thus be identified as:

For orthogonal coordinate a covariant basis vector is the same and the corresponding dual contravariant base vector. Thus, the gradient for orthogonal coordinates:

For obtained as the gradient of the contravariant basis vector, ie the normal vector to the coordinate surface.

Special Christoffel symbols

In the calculation of the divergence Christoffel symbol is needed. This can be expressed by the determinant of the metric tensor:

What and the following relation follows:

Thus we obtain for the divergence and Laplace:

Divergence in coordinate -free representation

The coordinate- free definition of the divergence leads them as " source density " field:

It is an arbitrary volume, the flow is integrated on the edge of this volume element. Hereinafter, this volume is a ( infinitesimally ) small parallelepiped at point in space which is spanned by the vectors in the direction of coordinate lines, that is, each coordinate is running in the interval. The particle size is, the edges are not necessarily orthogonal to each other. The volume is calculated to be in general:

The parallelepiped is for each coordinate bounded by the surfaces and. The surface element for a coordinate surface is in three dimensions

And the local flux through this surface element of the vector field is:

Thus, the flow is through the surface ( pointing outwards vectorial surface element, because of it)

And the flow through the area

In this case, the integrand has been developed at the location in first order. Taking stock of the two obtained

Similarly for the other two coordinates

Thus, the divergence is in the natural or physical coordinates:

For orthogonal coordinates applies:

Rotation in coordinate- free representation

The coordinate- free definition of the rotation is given by

Here, an arbitrary surface normal unit vector, the line integral runs over the edge of this surface.

First, here is an area with considered. The left side is then to:

In the following, the surface is a ( infinitesimally ) small parallelogram at point in space which is spanned by the vectors and. The area is.

As a ( closed ) path of integration are the edges of the parallelogram:

With valid for and for and for and for.

The path integrals over 1 and 3 can be summarized:

Expanding the integrand at the point in the first order in, the approximate integrand only depends on, and is therefore independent of and one can easily evaluate the integral:

An analogous procedure for the integrals over path 2 and 4 yields:

Overall, we obtain the circulation in the area around the parallelogram.

For be exact relations of the approximations. Substituted into the above defining equation for the rotation (all sizes at point evaluated)

Similarly for the other two coordinates under cyclic permutation. Thus, the rotation is with:

The (natural ) covariant coordinate is calculated from the ( natural ) contravariant means. Furthermore, if physical coordinates as well.

If the coordinates are orthogonal, applies because of the relationship as well. So For orthogonal physical coordinates and the rotation is in this special case:

Rotation as an anti- symmetric tensor

In terms of the form of the rotary contact, which can be simplified to partial discharges, as the Christoffel symbols are symmetrical with the lower indices of:

This size represents an antisymmetric tensor of second order, the rotor of the vector.

Examples of curvilinear coordinate systems

Orthogonal coordinate systems

• Cylindrical coordinates:

• Spherical coordinates:

• Parabolic cylindrical coordinates:

• Paraboloid coordinates:

• Elliptic cylindrical coordinates:

• Stretched spheroid coordinates:

• Oblate spheroid coordinates:

• Bipolar coordinates:

• Ellipsoidal coordinates:

Non- orthogonal coordinate systems

• Alternative elliptic cylindrical coordinates:

Differential Geometry

Curvilinear coordinates can be view as an application of differential geometry, especially as a map on a differentiable manifold. The following connections are made ​​to the calculus of differential forms, as can be shown independently coordinate with these calculations.

Differential forms - general

Let be a -dimensional differentiable manifold. A - form assigns each point to a smooth alternating multi - linear form on the tangent space. This is a real-valued linear functional, the tuples of vector fields assigns real numbers:

This itself is an element of the external potency of the tangent space, ie (as evidenced here and ). The set of all - forms, ie the bundle or the disjoint union, is the vector space. These forms have the great advantage that you can integrate cards independently on a manifold with their help.

In a tensor antisymmetric (due to alternating) is covariant tensor -th order (due to multi - linear form ).

Differential forms - coordinate representation

Be part of an open and a local coordinate system ( map ) with local coordinates. Then form at the site

The local base of the tangent space and

The corresponding dual basis (the duality is expressed through out ), so the base of the Kotangentialraums, these are 1-forms on the vector space.

Times the outer product of the 1-forms (where associative, bilinear and anticommutative ) is a form, wherein

Is a base of the outer algebra over the cotangent space. Each differential form has a unique representation on all cards:

2- form is, for example

Which corresponds to an antisymmetric covariant tensor field of second stage. So for one obtains:

Connection between scalar and vector fields with differential forms

For differentiable scalar fields, the identity holds: Smooth functions are identical to 0 - forms:

By following isomorphism is a differentiable vector field is still identified with a 1-form (where denote the scalar product ) ( it is the Einstein summation convention is used):

Using the Hodge star operator ( see below) can also be assigned a form and a vector field a form a scalar field.

Link Tangential-/Kotangentialvektoren

The musical operators (flat and sharp) describe isomorphisms induced by the Riemannian metric, and tangent vectors on Kotangentialvektoren factor or vice versa:

In tensor notation this corresponds to the raising and lowering of indices.

Hodge star operator

For -dimensional vector spaces, the Euclidean oriented and are ( thus an oriented Riemannian manifold must match ), a canonical isomorphism of alternating multilinear forms complementary degree exists (ie and ) successive maps. This is the so-called Hodge star operator:

Both vector spaces have dimension.

In three-dimensional space is thus a 0 - form can be assigned a 3- form

And a 1-form a 2- form

Thus, a differentiable vector field can not just assign a 1-form but also a 2- form. And a differentiable scalar function can be both a 0 - form as well as a 3- form to be assigned.

Due to the exterior derivative of a form creates a form. With the musical operators and the Hodge star operator of the de Rham complex is formed. The concatenation of two outer derivations is identically zero. From this, the integral theorems of vector analysis ( Stokes, Gauss and Green ) can be derived.

Swell