Spherical coordinate system
In spherical coordinates or spatial polar coordinates, a point is specified by its distance from the origin and two angles. For points on a spherical surface ( sphere ) is the distance from the origin ( center of the sphere ) is constant. The number of variables is reduced to the two angles, which are referred to as spherical coordinates. The term " spherical coordinates " can be regarded as a generic term for the general case and the spherical coordinates.
For polar coordinates in the plane (a distance, an angle ) and cylindrical coordinates (two intervals, an angle ), see the article in polar coordinates.
- 3.1 Jacobian
- 3.2 differentials, volume element line element
- 3.3 Metrics and rotation matrix
- 4.1 Transformation of the vector space basis
- 4.2 Transformation of a vector field
- 4.3 Transformation of the partial derivatives of
- 4.4 Transformation of the nabla operator
- 4.5 Transformation of the Laplacian
- 5.1 Jacobian
- 5.2 Example
Usual convention
The figure shows a point with spherical coordinates. The transformation equations of the Cartesian coordinates read in spherical coordinates:
In the analysis and its applications, spherical coordinate angles are usually expressed in radians. Is the position vector of the point ( ie, the vector connecting the coordinates of the origin) and the vertical projection of the xy plane, coordinates of the ball have the following physical meaning:
- (Radius) is the distance of the point of, ie the length of the vector
- Or ( polar angle and polar distance angle) is the angle between the positive z -axis and is counted up ( 0 ° to 180 °)
- Or (azimuth angle) is the angle between the positive x- axis, and is counted up ( 180 ° to 180 ° ) counter-clockwise
The reverse transformation is carried out according to the equations ( parameterization ):
The unit vectors of the spherical coordinate loud so:
To obtain these unit vectors, the parameterization according to the needs of each coordinate ( here ) is derived, and are normalized to Euclidean norm. They form a legal system in order.
Applications
The corresponding directions are referred to as " radial ", " meridional ", and " azimuthal " direction. These terms play not only in astronomy and the earth sciences (eg, geography, geology or geophysics ) play a central role, but also in mathematics, physics and various engineering sciences, such as the emission of electromagnetic waves ( " Hertz dipole " ) by a z direction spanned antenna where the radiation in the radial direction occurs while swinging electric or magnetic field in the meridional and azimuthal direction.
In rotational symmetry around the z- axis, the observed size does not depend on the azimuthal angle, which many formulas simplified.
Other conventions
The above choice of coordinates is international consensus in theoretical physics. Sometimes the characters are θ and φ just in the opposite direction used, especially in American literature.
The polar angle θ is the latitude. Rather it is defined as the angle between the equatorial plane and the position vector and takes values between -90 ° and 90 °. If they designated φ, then φ = 90 ° - θ, θ = 90 ° - φ. In contrast, one can used above φ λ east of Greenwich readily with the longitude equate. See article: geographic coordinates.
Furthermore, the above construction is to some extent inconsistent with the structure of the plane polar coordinates. For some problems, it is more practical, the presentation
To use. In this representation corresponds to the latitude.
The inverse transform of the point or vector in the angular components is then carried out with:
Said.
Transformation of differentials
Jacobian matrix
The local characteristics of the coordinate transformation can be described by the Jacobian matrix. For the transformation from spherical coordinates to Cartesian coordinates reads this
In spherical polar coordinates ( only θ, φ ) drops the first column ( and row) away.
The corresponding Jacobian is:
The Jacobi matrix of the opposite transition is defined only for spatial, not spherical polar coordinates; to calculate it the easiest as the inverse of J:
Some of the components of this matrix are openings at whose denominators are at r = 0 and sin θ = 0 (that is, θ = 0 or π ) detects the ambiguity of the polar coordinates. Ungebräuchlicher is a representation in Cartesian coordinates:
Differentials, volume element line element
The Jacobian matrix allows to write the conversion of differentials clearly as a linear map:
Respectively
The volume element is particularly easy with the help of the Jacobian
Convert:
Here applies ( with as a determinant of the metric tensor ) so you can see immediately that:
This plays a role in the case of a generalization in the sense of general scalar products and, for example, the Laplace -Beltrami operator ( g from the general scalar product including ).
By differentiating dV / dr is obtained for a surface element dA on a sphere with radius r
A line element ds is expected according to
To ( i, j = 1 to 3). The components of the metric tensor that defines the multiplication in non-trivial spaces are (Cartesian otherwise), or is used to raise indices and lower, that is covariant tensors in contra- variant and conversely to transform so, or a particularly important role in the general and in the special theory of relativity, ie, in the standard model of cosmology and elementary particles, were playing.
Metric and rotation matrix
In the absence of mixed terms in the line element ds reflects the fact that the metric tensor
In spherical coordinates has no off-diagonal elements.
The metric tensor is obviously the square of the diagonal matrix
Using this matrix, the Jacobian matrix can be written as J = Sh, where S is the rotation matrix
Is.
Transformation of vector fields and operators
In the following the transformation of vectors and differential operators will be derived as an example. The results are preferably written in a compact form using transformation matrices.
Transformation of the vector space basis
The basic vector eφ to coordinate φ indicates the direction in which a point P ( θ r, φ ) ( often called event ) when the coordinate φ is changed by an infinitesimal amount d.phi:
From this one obtains
To obtain an orthonormal basis, must be eφ normalized to the length 1
Similarly, the basis vectors are obtained and he e?. To write the following transformations in a compact form, we use the established above rotation matrix S. The matrix is orthogonal, which means that. The normalized basis vectors of the spherical coordinate system can then notify summarized as follows:
Accordingly, the transformation is in the opposite direction
Transformation of a vector field
A vector, as a geometric object, must be independent of the coordinate system:
This condition is achieved by
Respectively
Transforming the partial derivatives of
The partial derivatives transform as the basis vectors, but without normalization. You can expect the same as above, but can you point P in the numerator away (actually, the coordinate basis vectors of the tangent space and the partial derivatives are in the modern formulation of differential geometry equated ) and uses the Jacobian matrix J = Sh instead of the rotation matrix S. The transformation is:
And in the opposite direction
Transformation of the nabla operator
The nabla operator only in Cartesian coordinates the simple form
Both the partial derivatives and the unit vectors must be transformed in the derived above manner. One finds:
In this form, the transformed nabla operator can be directly used to calculate the gradient of a given scalar field in spherical coordinates.
To calculate the divergence of a vector field shown in spherical coordinates A, is taken into account, however, that not only the coefficients Ar, ... acts, but also on the basis vectors implicitly contained in A it
To calculate the rotation of a given in spherical coordinates vector field A, the same is to be considered:
Transformation of the Laplacian
As a vector field A Substituting the gradient operator in the divergence formula, we find the Laplace operator
Or
Derive a particularly simple form to this form of the Laplace operator obtained using the formula for the Laplace -Beltrami operator. The following applies:
Where, in other words.
The Laplace -Beltrami operator is then defined the same as the ordinary Laplace operator, but applies to more general Riemannian manifolds:
The metric tensor in spherical coordinates and is diagonal of the form
Thus, the sum accounts for over and which are the reciprocals of the components of the metric tensor, and thus
The root of the sum of the determinant is thus equal to the Jacobian.
Substituting the various parameters in the formula of the Laplace -Beltrami operator, then results
Adding to, then the above formula for the Laplace operator in spherical coordinates yields.
Generalization to n-dimensional spherical coordinates
A generalization of the spherical coordinates on dimensions:
The angular develop according to:
By renumbering we obtain a recursion formula for the angle:
From which the following angles can be calculated:
With and
The radius is:
A case distinction provides the appropriate means of arctangent angle to the given Cartesian coordinate, wherein:
It is noteworthy that getting a two-dimensional vector is for.
Jacobian matrix
The Jacobi matrix of the spherical coordinates is given as the upper with respect to the numbering:
Its determinant is:
Example
Assignment in the example of the common coordinate axes:
The angles are then: