area element in spherical coordinates

It can also be extended to higher-dimensional spaces and is then referred to as a hyperspherical coordinate system. $$ flux of $\langle x,y,z^2\rangle$ across unit sphere, Calculate the area of a pixel on a sphere, Derivation of $\frac{\cos(\theta)dA}{r^2} = d\omega$. The polar angle, which is 90 minus the latitude and ranges from 0 to 180, is called colatitude in geography. \[\int\limits_{all\; space} |\psi|^2\;dV=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{0}^{\infty}\psi^*(r,\theta,\phi)\psi(r,\theta,\phi)\,r^2\sin\theta\,dr d\theta d\phi=1 \nonumber\]. When using spherical coordinates, it is important that you see how these two angles are defined so you can identify which is which. This page titled 10.2: Area and Volume Elements is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Marcia Levitus via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. (g_{i j}) = \left(\begin{array}{cc} Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (radius r, inclination , azimuth ), where r [0, ), [0, ], [0, 2), by, Cylindrical coordinates (axial radius , azimuth , elevation z) may be converted into spherical coordinates (central radius r, inclination , azimuth ), by the formulas, Conversely, the spherical coordinates may be converted into cylindrical coordinates by the formulae. Often, positions are represented by a vector, $\vec{r}$, shown in red in Figure $\PageIndex{1}$. Lets see how this affects a double integral with an example from quantum mechanics. Spherical Coordinates In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. In spherical polar coordinates, the element of volume for a body that is symmetrical about the polar axis is, Whilst its element of surface area is, Although the homework statement continues, my question is actually about how the expression for dS given in the problem statement was arrived at in the first place. Converting integration dV in spherical coordinates for volume but not for surface? Then the area element has a particularly simple form: Planetary coordinate systems use formulations analogous to the geographic coordinate system. {\displaystyle (r,\theta ,\varphi )} The azimuth angle (longitude), commonly denoted by , is measured in degrees east or west from some conventional reference meridian (most commonly the IERS Reference Meridian), so its domain is 180 180. {\displaystyle (r,\theta ,\varphi )} { "32.01:_Complex_Numbers" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "32.02:_Probability_and_Statistics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "32.03:_Vectors" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "32.04:_Spherical_Coordinates" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "32.05:_Determinants" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "32.06:_Matrices" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "32.07:_Numerical_Methods" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "32.08:_Partial_Differentiation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "32.09:_Series_and_Limits" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "32.10:_Fourier_Analysis" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "32.11:_The_Binomial_Distribution_and_Stirling\'s_Appromixation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_The_Dawn_of_the_Quantum_Theory" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_The_Classical_Wave_Equation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_The_Schrodinger_Equation_and_a_Particle_in_a_Box" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Postulates_and_Principles_of_Quantum_Mechanics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_The_Harmonic_Oscillator_and_the_Rigid_Rotor" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_The_Hydrogen_Atom" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Approximation_Methods" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Multielectron_Atoms" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Chemical_Bonding_in_Diatomic_Molecules" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Bonding_in_Polyatomic_Molecules" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_Computational_Quantum_Chemistry" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_Group_Theory_-_The_Exploitation_of_Symmetry" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "13:_Molecular_Spectroscopy" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "14:_Nuclear_Magnetic_Resonance_Spectroscopy" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15:_Lasers_Laser_Spectroscopy_and_Photochemistry" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "16:_The_Properties_of_Gases" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "17:_Boltzmann_Factor_and_Partition_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "18:_Partition_Functions_and_Ideal_Gases" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "19:_The_First_Law_of_Thermodynamics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "20:_Entropy_and_The_Second_Law_of_Thermodynamics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "21:_Entropy_and_the_Third_Law_of_Thermodynamics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "22:_Helmholtz_and_Gibbs_Energies" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "23:_Phase_Equilibria" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "24:_Solutions_I_-_Volatile_Solutes" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "25:_Solutions_II_-_Nonvolatile_Solutes" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "26:_Chemical_Equilibrium" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "27:_The_Kinetic_Theory_of_Gases" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "28:_Chemical_Kinetics_I_-_Rate_Laws" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "29:_Chemical_Kinetics_II-_Reaction_Mechanisms" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "30:_Gas-Phase_Reaction_Dynamics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "31:_Solids_and_Surface_Chemistry" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "32:_Math_Chapters" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Appendices : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "showtoc:no", "license:ccbyncsa", "Spherical Coordinates", "autonumheader:yes2", "licenseversion:40" ], https://chem.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FPhysical_and_Theoretical_Chemistry_Textbook_Maps%2FPhysical_Chemistry_(LibreTexts)%2F32%253A_Math_Chapters%2F32.04%253A_Spherical_Coordinates, $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}$ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$, status page at https://status.libretexts.org. , , (b) Note that every point on the sphere is uniquely determined by its z-coordinate and its counterclockwise angle phi, $0 \leq\phi\leq 2\pi$, from the half-plane y = 0, Solution We integrate over the entire sphere by letting [0,] and [0, 2] while using the spherical coordinate area element R2 0 2 0 R22(2)(2) = 4 R2 (8) as desired! $$y=r\sin(\phi)\sin(\theta)$$ , For positions on the Earth or other solid celestial body, the reference plane is usually taken to be the plane perpendicular to the axis of rotation. As we saw in the case of the particle in the box (Section 5.4), the solution of the Schrdinger equation has an arbitrary multiplicative constant. (26.4.7) z = r cos . $$, So let's finish your sphere example. If the inclination is zero or 180 degrees ( radians), the azimuth is arbitrary. There are a number of celestial coordinate systems based on different fundamental planes and with different terms for the various coordinates. ( @R.C. 1. Coming back to coordinates in two dimensions, it is intuitive to understand why the area element in cartesian coordinates is $dA=dx\;dy$ independently of the values of $x$ and $y$. A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis (axis L in the image opposite), the direction from the axis relative to a chosen reference direction (axis A), and the distance from a chosen reference plane perpendicular to the axis (plane containing the purple section). :URn{\displaystyle \varphi :U\to \mathbb {R} ^{n}} In space, a point is represented by three signed numbers, usually written as $(x,y,z)$ (Figure $\PageIndex{1}$, right). The answer is no, because the volume element in spherical coordinates depends also on the actual position of the point. The answers above are all too formal, to my mind. To conclude this section we note that it is trivial to extend the two-dimensional plane toward a third dimension by re-introducing the z coordinate. {\displaystyle (r,\theta ,\varphi )} the orbitals of the atom). The radial distance r can be computed from the altitude by adding the radius of Earth, which is approximately 6,36011km (3,9527 miles). However, in polar coordinates, we see that the areas of the gray sections, which are both constructed by increasing $r$ by $dr$, and by increasing $\theta$ by $d\theta$, depend on the actual value of $r$. It is also convenient, in many contexts, to allow negative radial distances, with the convention that Is the God of a monotheism necessarily omnipotent? We are trying to integrate the area of a sphere with radius r in spherical coordinates. ) r $$\int_{-1 \leq z \leq 1, 0 \leq \phi \leq 2\pi} f(\phi,z) d\phi dz$$. }{a^{n+1}}, \nonumber\]. Why do academics stay as adjuncts for years rather than move around? The spherical coordinate system is also commonly used in 3D game development to rotate the camera around the player's position[4]. Calculating Infinitesimal Distance in Cylindrical and Spherical Coordinates Calculating $d\rr$in Curvilinear Coordinates Scalar Surface Elements Triple Integrals in Cylindrical and Spherical Coordinates Using $d\rr$on More General Paths Use What You Know 9Integration Scalar Line Integrals Vector Line Integrals