Let x, y, z be Cartesian coordinates of a vector in , that is. T changes with each of the coordinates. Or, equivalently, coordinate frames may be freely translated in a parallel manner. Rotate then the vector and the new frame over an angle θ around the y'-axis. φ {\displaystyle \mathbf {r} } and x̂, ŷ, and ẑ are the unit vectors in Cartesian coordinates. The Laplace operator of the scalar function Φ is. Consider the following expressions between differentials, obtained by application of the chain rule. It is possible to derive general expressions for these operators that are valid in any coordinate system and are based on the metric tensor associated with the coordinate system. , Recall, parenthetically, that free parallel vectors of equal length have the same coordinate triplet with respect to a given coordinate frame. − In doing this, we first wish to point out that the spherical polar angles can be seen as two of the three Euler angles that describe any rotation of . , {\displaystyle (r,\theta ,\varphi )} The Mathematica package follows the convention that has θ as the angle between the vector and the z-axis. The line element for an infinitesimal displacement from (r, θ, φ) to (r + dr, θ + dθ, φ + dφ) is. Also, these coordinates are determined by the help of Cartesian coordinates (x,y,z). According to the conventions of geographical coordinate systems, positions are measured by latitude, longitude, and height (altitude). A number of polar plots are required, taken at a wide selection of frequencies, as the pattern changes greatly with frequency. gives the radial distance, polar angle, and azimuthal angle. The angles are typically measured in degrees (°) or radians (rad), where 360° = 2π rad. for physics: radius r, inclination θ, azimuth φ) can be obtained from its Cartesian coordinates (x, y, z) by the formulae, An infinitesimal volume element is given by. ) It is also possible to deal with ellipsoids in Cartesian coordinates by using a modified version of the spherical coordinates. Since the two rotation matrices are orthogonal (have orthonormal rows and columns), the new frame is orthogonal. Two other points of indeterminacy are the "North" and the "South Pole", θ = 00 and θ = 1800, respectively (while r ≠ 0). One can add or subtract any number of full turns to either angular measure without changing the angles themselves, and therefore without changing the point. In many mathematics books, To a first approximation, the geographic coordinate system uses elevation angle (latitude) in degrees north of the equator plane, in the range −90° ≤ φ ≤ 90°, instead of inclination. The azimuth angle (longitude), commonly denoted by λ, is measured in degrees east or west from some conventional reference meridian (most commonly the IERS Reference Meridian), so its domain is −180° ≤ λ ≤ 180°. 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). , θ {\displaystyle (r,\theta ,\varphi )} The two angles specify the position on the surface of a sphere and the length gives the radius of the sphere. 180 To make the coordinates unique, one can use the convention that in these cases the arbitrary coordinates are zero. is equivalent to To apply this to the present case, one needs to calculate how J The correct quadrants for φ and θ are implied by the correctness of the planar rectangular to polar conversions. r Citizendium - a community developing a quality, comprehensive compendium of knowledge, online and free, https://en.citizendium.org/wiki?title=Spherical_polar_coordinates&oldid=100709417, Creative Commons-Attribution-ShareAlike 3.0 Unported license. These formulae assume that the two systems have the same origin and same reference plane, measure the azimuth angle φ in the same senses from the same axis, and that the spherical angle θ is inclination from the cylindrical z axis. θ Given x, y and z, the consecutive steps are. {\displaystyle \mathbf {r} } Longitude is always within the range −180° to +180°. ( θ φ ( ∘ Latitude λ is conventionally measured as angles north and south of the equator, with latitudes north of the equator taken as positive, and south taken as negative. The Cartesian metric tensor is the identity matrix and hence in Cartesian coordinates. It can be seen as the three-dimensional version of the polar coordinate system. conflicts with the usual notation for two-dimensional polar coordinates and three-dimensional cylindrical coordinates, where θ is often used for the azimuth.[3]. . 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. , Alternatively, the conversion can be considered as two sequential rectangular to polar conversions: the first in the Cartesian xy plane from (x, y) to (R, φ), where R is the projection of r onto the xy-plane, and the second in the Cartesian zR-plane from (z, R) to (r, θ). , ( The spherical coordinates of a point P are then defined as follows: That is, the angle θ is zero when is along the positive z-axis. The metric tensor in the spherical coordinate system is The use of Latitude is either geocentric latitude, measured at the Earth's center and designated variously by ψ, q, φ′, φc, φg or geodetic latitude, measured by the observer's local vertical, and commonly designated φ. The standard convention ( We now introduce the coordinate frame depicted in the figure on the right: That is, the new frame, depicted in the figure, is related to the old frame along the x-, y-, and z-axes by rotation. Because hr = 1, it so happens that the weight of the volume element is equal to the weight of the surface element perpendicular to r. As an example of the use of dV, the volume V of a sphere with radius R is, We will express the velocity of a particle in spherical polar coordinates. r and similarly the time derivatives of y, z , θ, φ, and r are given in Newton's fluxion (dot) notation.