Īfter the substitution, equation (4) looks like the following We can do this by substituting the following values (which are easily derived from (2)) in their respective places in the above three equations The next step is to convert the right-hand side of each of the above three equations so that it only has partial derivatives in terms of r, θ and ϕ. We also know that the partial derivatives in rectangular coordinates can be expanded in the following way by using the chain rule A more rigorous approach would be to define the Laplacian in some coordinate free manner. Now, we know that the Laplacian in rectangular coordinates is defined 1 1Readers should note that we do not have to define the Laplacian this way. R = x 2 + y 2 + z 2, θ = arccos ( z r ), ϕ = arctan ( y x ). X = r sin ( θ ) cos ( ϕ ), y = r sin ( θ ) sin ( ϕ ), z = r cos ( θ ) ,Īnd conversely from spherical to rectangular coordinates "Spherical Coordinates."įrom MathWorld-A Wolfram Web Resource.We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that ϕ is used to denote the azimuthal angle, whereas θ is used to denote the polar angle) Referenced on Wolfram|Alpha Spherical Coordinates Cite this as: Standard Mathematical Tables and Formulae. "Tensor Calculations on Computer: Appendix." Comm. Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Orlando, FL: Academic Press, pp. 102-111, "Spherical Polar Coordinates." §2.5 in Mathematical To Differential Equations and Probability. Apostol,Ģnd ed., Vol. 2: Multi-Variable Calculus and Linear Algebra, with Applications Spherical coordinates of vector (1, 2, 3) Extreme care is therefore needed when consulting the literature. The following table summarizes a number of conventions The symbol is sometimes also used in place of, instead of, and and instead of. Typically means (radial, azimuthal, polar) to a mathematician but (radial, polar,Īzimuthal) to a physicist. This is especially confusing since the identical Unfortunately, the convention in which the symbols and are reversed (both in meaning and in order listed) is alsoįrequently used, especially in physics. Used in the physics literature is retained (resulting, it is hoped, in a bit lessĬonfusion than a foolish rigorous consistency might engender). Is in spherical harmonics, where the convention The sole exception to this convention in this work ![]() Remaining the angle in the - plane and becoming the angle out of that Note that this definition provides a logicalĮxtension of the usual polar coordinates notation, In this work, following the mathematics convention, the symbols for the radial, azimuth, and zenith angleĬoordinates are taken as, , and, respectively. This is the convention commonly used in mathematics. To be distance ( radius) from a point to the origin. Is the latitude) from the positive z-axis Known as the zenith angle and colatitude, When referred to as the longitude), to be the polar angle (also Define to be the azimuthal angle in the - plane from the x-axis That are natural for describing positions on a sphere Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates
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