Spherical polar coordinates
In mathematics and physics, spherical polar coordinates (also known as spherical coordinates) form a coordinate system for the three-dimensional real space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{R}^3} . Three numbers, two angles and a length specify any point in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{R}^3} . The two angles specify the position on the surface of a sphere and the length gives the radius of the sphere.
Spherical polar coordinates are useful in cases where there is (approximate) spherical symmetry, in interactions or in boundary conditions (or in both). In such cases spherical polar coordinates often allow the separation of variables simplifying the solution of partial differential equations and the evaluation of three-dimensional integrals.
Definition
Let x, y, z be Cartesian coordinates of a vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec\mathbf{r}} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{R}^3} , that is,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec\mathbf{r} = (\vec\mathbf{e}_x, \, \vec\mathbf{e}_y, \, \vec\mathbf{e}_z) \begin{pmatrix} x\\y\\z\\ \end{pmatrix} \equiv x\,\vec\mathbf{e}_x + y\,\vec\mathbf{e}_y + z\, \vec\mathbf{e}_z, }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec\mathbf{e}_x, \, \vec\mathbf{e}_y, \, \vec\mathbf{e}_z } are unit vectors along the x, y, and z axis, respectively. The x, y, and z axes are orthogonal and so are the unit vectors along them.
The length r of the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec\mathbf{r}} is one of the three numbers necessary to give the position of the vector in three-dimensional space. By applying twice the theorem of Pythagoras we find that r2 = x2 + y2 + z2.
Let θ be the colatitude angle (see the figure) of the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec\mathbf{r}} . In the usual system to describe a position on Earth, latitude has its zero at the equator, while the colatitude angle, introduced here, has its zero at the "North Pole". That is, the angle θ is zero when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec\mathbf{r}} is along the positive z-axis. The sum of latitude and colatitude of a point is 900; these angles being complementary explains the name of the latter. The colatitude angle is also called polar or zenith angle in the literature.
The angle φ gives the angle with the x-axis of the projection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec\mathbf{r}'} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec\mathbf{r}} on the x-y plane. The angle φ is the longitude angle (also known as the azimuth angle).
Note that the projection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec\mathbf{r}'} has length r sinθ. The length of the projection of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec\mathbf{r}'} on the x and y axis is therefore r sinθcosφ and r sinθsinφ, respectively. In summary, the spherical polar coordinates r, θ, and φ of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec\mathbf{r}} are related to its Cartesian coordinates by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} x &= r \sin\theta\cos\phi \\ y &= r \sin\theta\sin\phi \\ z &= r \cos\theta \end{align} }
Given a spherical polar triplet (r, θ, φ) the corresponding Cartesian coordinates are readily obtained by application of these defining equations.
The figure makes clear that 00 ≤ φ ≤ 3600, 00 ≤ θ ≤ 1800, and r > 0. The coordinate surfaces are:
- r constant, all θ and φ: surface of sphere.
- θ constant, all r and φ: surface of a cone.
- φ constant, all r and θ: plane.
The computation of spherical polar coordinates from Cartesian coordinates is somewhat more difficult than the converse, due to the fact that the spherical polar coordinate system has singularities, also known as points of indeterminacy. The first such point is immediately clear: if r = 0, we have a zero vector (a point in the origin). Then θ and φ are undetermined, that is to say, any values for these two parameters will give the correct result x = y = z = 0. Compare this to the case that one of the Cartesian coordinates is zero, say x = 0, then the other two coordinates are still determined (they fix a point in the yz-plane). Two other points of indeterminacy are the "North" and the "South Pole", θ = 00 and θ = 1800, respectively (while r ≠ 0). On both poles the longitudinal angle φ is undetermined.
So, when going from Cartesian coordinates to spherical polar coordinates, one has to watch for the singularities, especially when the transformation is performed by a computer program. Given x, y and z, the consecutive steps are
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} r &= \sqrt{x^2+y^2+z^2} \\ \theta &= \arccos(z/r), \quad r \ne 0 \\ r' & = r \sin\theta = \sqrt{x^2+y^2} \\ \phi & = \begin{cases} \arccos(x/r')&\quad \hbox{if}\quad y \ge 0,\quad r' \ne 0 \\ 360^0 - \arccos(x/r')&\quad \hbox{if}\quad y < 0,\quad r'\ne 0 \\ \end{cases} \end{align} }
Other convention
The convention introduced above (θ for the colatitude angle, φ for the azimuth angle) is used universally in physics. In mathematics it is very widespread, too. To quote a few mathematics books that apply it: Abramowitz and Stegun[1] p. 332, Whittaker and Watson[2] p. 391, Courant and Hilbert[3] p.195, and Kline[4] p. 527.
Unfortunately, some American mathematical textbooks, e.g. Ref.[5], interchanged φ and θ, apparently for pedagogical reasons, as can be gathered from the following quotation of Eric Weisstein:
In this work, following the mathematics convention, the symbols for the radial, azimuth, and zenith coordinates are taken as r, θ, and φ, respectively. Note that this definition provides a logical extension of the usual polar coordinates notation, with θ remaining the angle in the xy-plane and φ becoming the angle out of that plane. The sole exception to this convention in this work is in spherical harmonics, where the convention used in the physics literature is retained (resulting, it is hoped, in a bit less confusion than a foolish rigorous consistency might engender).
Most of these textbooks do not apply the spherical polar coordinates very intensively. At least they do not cover the field of spherical functions. Especially for the workers in that field the interchange of θ and φ would be extremely confusing as rightly remarked by Weisstein.
The American textbook convention is followed by the Maple algebraic program package and also by the numerical package Matlab. (Matlab also redefines the zero of the colatitude angle to be on the equator). The Mathematica package follows the physics convention.
Unit vectors
We will define algebraically the orthogonal set (a coordinate frame) of spherical polar unit vectors depicted on the right. In order to do this, we note that the spherical polar angles are two of the three Euler angles that describe any rotation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{R}^3} .
Indeed, start with a vector along the z-axis, rotate it around the z-axis over an angle φ. Perform the same rotation on the x, y, z coordinate frame. This rotates the x and y axis over a positive angle φ. The y axis goes to the y'-axis. Rotate then the vector and the new frame over an angle θ around the y'-axis. The vector that was initially on the z-axis is now a vector with spherical polar angles θ and φ with respect to the original (unrotated) frame. In formula
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \vec\mathbf{r} &= (\vec\mathbf{e}_x, \, \vec\mathbf{e}_y, \, \vec\mathbf{e}_z) \begin{pmatrix} r\cos\phi\sin\theta \\ r\sin\phi\sin\theta \\ r\cos\theta \\ \end{pmatrix} \\ &= (\vec\mathbf{e}_x, \, \vec\mathbf{e}_y, \, \vec\mathbf{e}_z) \mathbb{R}_z(\phi) \mathbb{R}_y(\theta) \begin{pmatrix} 0 \\ 0 \\ r \\ \end{pmatrix}, \end{align} }
where the rotation matrices are defined by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}_z(\phi) \equiv \begin{pmatrix} \cos\phi & - \sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}, \qquad \mathbb{R}_y(\theta) \equiv \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \\ \end{pmatrix}. }
By direct matrix multiplication the matrix expression for the spherical polar coordinates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec{\mathbf{r}}} is easily verified—it could have been postulated without reference to Euler rotations and proved by verification.
We now introduce the coordinate frame depicted in the figure on the right:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \vec\mathbf{r} &= (\vec\mathbf{e}_x, \, \vec\mathbf{e}_y, \, \vec\mathbf{e}_z) \mathbb{R}_z(\phi) \mathbb{R}_y(\theta) \begin{pmatrix} 0 \\ 0 \\ r \\ \end{pmatrix} \equiv (\vec\mathbf{e}_\theta, \, \vec\mathbf{e}_\phi, \, \vec\mathbf{e}_r) \begin{pmatrix} 0 \\ 0 \\ r \\ \end{pmatrix}. \end{align} }
That is, the new frame, depicted in the figure, is related to the old frame along the x, y, and z axes by rotation,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\vec\mathbf{e}_\theta, \, \vec\mathbf{e}_\phi, \, \vec\mathbf{e}_r) \equiv (\vec\mathbf{e}_x, \, \vec\mathbf{e}_y, \, \vec\mathbf{e}_z) \mathbb{R}_z(\phi) \mathbb{R}_y(\theta). }
Apparently Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec{\mathbf{r}}}
is along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec{\mathbf{e}}_r}
. Since the two rotation matrices are orthogonal (have orthonormal rows and columns), the new frame is orthogonal. Since the two rotation matrices have unit determinant (are proper rotations), the new frame is right-handed.
Recall, parenthetically, that free parallel vectors have the same coordinate triplet without respect to a given coordinate frame. Or, equivalently, coordinate frames may be freely translated in a parallel manner. That is, the frame in the figure could have been drawn equally well with its origin in the crossing of the x, y, and z axes, which, however, would have obscured the fact that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec{\mathbf{e}}_\phi} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec{\mathbf{e}}_\theta} are tangent to the surface of the sphere.
g-tensor
(To be continued)
References
- ↑ M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.
- ↑ E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge UP, (1965).
- ↑ R. Courant and D. Hilbert, Methoden der mathematischen Physik I, Springer Verlag Berlin (1968).
- ↑ M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford UP, New York (1972)
- ↑ T. M. Apostol, Mathematical Analysis, Addison-Wesley Reading Mass, (1974) p.419