area element in spherical coordinates

Another application is ergonomic design, where r is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out. The answer is no, because the volume element in spherical coordinates depends also on the actual position of the point. Can I tell police to wait and call a lawyer when served with a search warrant? In this case, $\psi^2(r,\theta,\phi)=A^2e^{-2r/a_0}$. ( This simplification can also be very useful when dealing with objects such as rotational matrices. :URn{\displaystyle \varphi :U\to \mathbb {R} ^{n}} 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. 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 $y$. The wave function of the ground state of a two dimensional harmonic oscillator is: $\psi(x,y)=A e^{-a(x^2+y^2)}$. The straightforward way to do this is just the Jacobian. The relationship between the cartesian and polar coordinates in two dimensions can be summarized as: \[\label{eq:coordinates_1} x=r\cos\theta\], \[\label{eq:coordinates_2} y=r\sin\theta\], \[\label{eq:coordinates_4} \tan \theta=y/x\]. The line element for an infinitesimal displacement from (r, , ) to (r + dr, + d, + d) is. The same situation arises in three dimensions when we solve the Schrdinger equation to obtain the expressions that describe the possible states of the electron in the hydrogen atom (i.e. The volume of the shaded region is, \[\label{eq:dv} dV=r^2\sin\theta\,d\theta\,d\phi\,dr\]. We'll find our tangent vectors via the usual parametrization which you gave, namely, These formulae assume that the two systems have the same origin, that the spherical reference plane is the Cartesian xy plane, that is inclination from the z direction, and that the azimuth angles are measured from the Cartesian x axis (so that the y axis has = +90). The polar angle may be called colatitude, zenith angle, normal angle, or inclination angle. In the cylindrical coordinate system, the location of a point in space is described using two distances (r and z) and an angle measure (). $$ rev2023.3.3.43278. 4: spherical coordinate area element = r2 Example Prove that the surface area of a sphere of radius R is 4 R2 by direct integration. E & F \\ , \[\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{0}^{\infty}\psi^*(r,\theta,\phi)\psi(r,\theta,\phi) \, r^2 \sin\theta \, dr d\theta d\phi=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{0}^{\infty}A^2e^{-2r/a_0}\,r^2\sin\theta\,dr d\theta d\phi=1 \nonumber\], \[\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{0}^{\infty}A^2e^{-2r/a_0}\,r^2\sin\theta\,dr d\theta d\phi=A^2\int\limits_{0}^{2\pi}d\phi\int\limits_{0}^{\pi}\sin\theta \;d\theta\int\limits_{0}^{\infty}e^{-2r/a_0}\,r^2\;dr \nonumber\]. The blue vertical line is longitude 0. According to the conventions of geographical coordinate systems, positions are measured by latitude, longitude, and height (altitude). Latitude is either geocentric latitude, measured at the Earth's center and designated variously by , q, , c, g or geodetic latitude, measured by the observer's local vertical, and commonly designated . Here is the picture. For a wave function expressed in cartesian coordinates, \[\int\limits_{all\;space} |\psi|^2\;dV=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\psi^*(x,y,z)\psi(x,y,z)\,dxdydz \nonumber\]. 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. The difference between the phonemes /p/ and /b/ in Japanese. for physics: radius r, inclination , azimuth ) can be obtained from its Cartesian coordinates (x, y, z) by the formulae. Chapter 1: Curvilinear Coordinates | Physics - University of Guelph (a) The area of [a slice of the spherical surface between two parallel planes (within the poles)] is proportional to its width. It is also possible to deal with ellipsoids in Cartesian coordinates by using a modified version of the spherical coordinates. $$ A spherical coordinate system is represented as follows: Here, represents the distance between point P and the origin. In cartesian coordinates, all space means $-\infty The angle $\theta$ runs from the North pole to South pole in radians. Spherical coordinates are somewhat more difficult to understand. Lets see how we can normalize orbitals using triple integrals in spherical coordinates. Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose. 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 $r$. , because this orbital is a real function, $\psi^*(r,\theta,\phi)\psi(r,\theta,\phi)=\psi^2(r,\theta,\phi)$. Would we just replace $dx\;dy\;dz$ by $dr\; d\theta\; d\phi$? The differential $dV$ is $dV=r^2\sin\theta\,d\theta\,d\phi\,dr$, so, \[\int\limits_{all\;space} |\psi|^2\;dV=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{0}^{\infty}\psi^*(r,\theta,\phi)\psi(r,\theta,\phi)\,r^2\sin\theta\,dr d\theta d\phi=1 \nonumber\]. {\displaystyle (r,\theta ,\varphi )} is mass. 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. Spherical coordinates to cartesian coordinates calculator The vector product $\times$ is the appropriate surrogate of that in the present circumstances, but in the simple case of a sphere it is pretty obvious that ${\rm d}\omega=r^2\sin\theta\,{\rm d}(\theta,\phi)$. Linear Algebra - Linear transformation question. It is now time to turn our attention to triple integrals in spherical coordinates. In spherical coordinates, all space means $0\leq r\leq \infty$, $0\leq \phi\leq 2\pi$ and $0\leq \theta\leq \pi$. In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. The value of should be greater than or equal to 0, i.e., 0. is used to describe the location of P. Let Q be the projection of point P on the xy plane. In cartesian coordinates, the differential volume element is simply $dV= dx\,dy\,dz$, regardless of the values of $x, y$ and $z$. $X(\phi,\theta) = (r \cos(\phi)\sin(\theta),r \sin(\phi)\sin(\theta),r \cos(\theta)),$ If it is necessary to define a unique set of spherical coordinates for each point, one must restrict their ranges. {\displaystyle (r,\theta ,\varphi )} X_{\theta} = (r\cos(\phi)\cos(\theta),r\sin(\phi)\cos(\theta),-r\sin(\theta)) While in cartesian coordinates $x$, $y$ (and $z$ in three-dimensions) can take values from $-\infty$ to $\infty$, in polar coordinates $r$ is a positive value (consistent with a distance), and $\theta$ can take values in the range $[0,2\pi]$. The function $\psi(x,y)=A e^{-a(x^2+y^2)}$ can be expressed in polar coordinates as: $\psi(r,\theta)=A e^{-ar^2}$, \[\int\limits_{all\;space} |\psi|^2\;dA=\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi} A^2 e^{-2ar^2}r\;d\theta dr=1 \nonumber\]. For example, in example [c2v:c2vex1], we were required to integrate the function ${\left | \psi (x,y,z) \right |}^2$ over all space, and without thinking too much we used the volume element $dx\;dy\;dz$ (see page ). flux of $\langle x,y,z^2\rangle$ across unit sphere, Calculate the area of a pixel on a sphere, Derivation of $\frac{\cos(\theta)dA}{r^2} = d\omega$. Now this is the general setup. Vectors are often denoted in bold face (e.g. 180 The relationship between the cartesian coordinates and the spherical coordinates can be summarized as: \[\label{eq:coordinates_5} x=r\sin\theta\cos\phi\], \[\label{eq:coordinates_6} y=r\sin\theta\sin\phi\], \[\label{eq:coordinates_7} z=r\cos\theta\]. gives the radial distance, azimuthal angle, and polar angle, switching the meanings of and . In this case, $n=2$ and $a=2/a_0$, so: \[\int\limits_{0}^{\infty}e^{-2r/a_0}\,r^2\;dr=\dfrac{2! $$\int_{0}^{ \pi }\int_{0}^{2 \pi } r \, d\theta * r \, d \phi = 2 \pi^2 r^2$$. gives the radial distance, polar angle, and azimuthal angle. Surface integral - Wikipedia the orbitals of the atom). It is now time to turn our attention to triple integrals in spherical coordinates. ) Area element of a surface[edit] A simple example of a volume element can be explored by considering a two-dimensional surface embedded in n-dimensional Euclidean space. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. It is because rectangles that we integrate look like ordinary rectangles only at equator! Find $A$. The spherical system uses r, the distance measured from the origin; , the angle measured from the + z axis toward the z = 0 plane; and , the angle measured in a plane of constant z, identical to in the cylindrical system. the orbitals of the atom). x >= 0. What Is the Difference Between 'Man' And 'Son of Man' in Num 23:19? 32.4: Spherical Coordinates - Chemistry LibreTexts Relevant Equations: F & G \end{array} \right), 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! ), geometric operations to represent elements in different Explain math questions One plus one is two. 10.2: Area and Volume Elements - Chemistry LibreTexts The Cartesian unit vectors are thus related to the spherical unit vectors by: The general form of the formula to prove the differential line element, is[5]. $$. Be able to integrate functions expressed in polar or spherical coordinates. . When solving the Schrdinger equation for the hydrogen atom, we obtain $\psi_{1s}=Ae^{-r/a_0}$, where $A$ is an arbitrary constant that needs to be determined by normalization. The volume element is spherical coordinates is: We also knew that all space meant $-\infty\leq x\leq \infty$, $-\infty\leq y\leq \infty$ and $-\infty\leq z\leq \infty$, and therefore we wrote: \[\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }{\left | \psi (x,y,z) \right |}^2\; dx \;dy \;dz=1 \nonumber\]. In any coordinate system it is useful to define a differential area and a differential volume element. $$ Two important partial differential equations that arise in many physical problems, Laplace's equation and the Helmholtz equation, allow a separation of variables in spherical coordinates. The spherical coordinates of a point P are then defined as follows: The sign of the azimuth is determined by choosing what is a positive sense of turning about the zenith. This will make more sense in a minute. If you preorder a special airline meal (e.g. Legal. In cartesian coordinates, all space means $-\inftyQ1P Find ds2 in spherical coordin [FREE SOLUTION] | StudySmarter Spherical charge distribution 2013 - Purdue University What happens when we drop this sine adjustment for the latitude? The wave function of the ground state of a two dimensional harmonic oscillator is: \(\psi(x,y)=A e^{-a(x^2+y^2)}$. 16.4: Spherical Coordinates - Chemistry LibreTexts dA = | X_u \times X_v | du dv = \sqrt{|X_u|^2 |X_v|^2 - (X_u \cdot X_v)^2} du dv = \sqrt{EG - F^2} du dv. where $B$ is the parameter domain corresponding to the exact piece $S$ of surface. For the polar angle , the range [0, 180] for inclination is equivalent to [90, +90] for elevation. Cylindrical Coordinates: When there's symmetry about an axis, it's convenient to . 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. Their total length along a longitude will be $r \, \pi$ and total length along the equator latitude will be $r \, 2\pi$. In space, a point is represented by three signed numbers, usually written as $(x,y,z)$ (Figure $\PageIndex{1}$, right). From (a) and (b) it follows that an element of area on the unit sphere centered at the origin in 3-space is just dphi dz. where we used the fact that $|\psi|^2=\psi^* \psi$. (g_{i j}) = \left(\begin{array}{cc} the spherical coordinates. We will see that $p$ and $d$ orbitals depend on the angles as well. where we used the fact that $|\psi|^2=\psi^* \psi$. , ) to use other coordinate systems. The answer is no, because the volume element in spherical coordinates depends also on the actual position of the point. The lowest energy state, which in chemistry we call the 1s orbital, turns out to be: This particular orbital depends on $r$ only, which should not surprise a chemist given that the electron density in all $s$-orbitals is spherically symmetric. Spherical coordinates are useful in analyzing systems that are symmetrical about a point. $$I(S)=\int_B \rho\bigl({\bf x}(u,v)\bigr)\ {\rm d}\omega = \int_B \rho\bigl({\bf x}(u,v)\bigr)\ |{\bf x}_u(u,v)\times{\bf x}_v(u,v)|\ {\rm d}(u,v)\ ,$$ (25.4.7) z = r cos . Mutually exclusive execution using std::atomic? We already introduced the Schrdinger equation, and even solved it for a simple system in Section 5.4. , Regardless of the orbital, and the coordinate system, the normalization condition states that: \[\int\limits_{all\;space} |\psi|^2\;dV=1 \nonumber\]. In geography, the latitude is the elevation. $$\int_{-1 \leq z \leq 1, 0 \leq \phi \leq 2\pi} f(\phi,z) d\phi dz$$. for any r, , and . The same value is of course obtained by integrating in cartesian coordinates. $$dA=h_1h_2=r^2\sin(\theta)$$. Then the integral of a function f (phi,z) over the spherical surface is just $$\int_ {-1 \leq z \leq 1, 0 \leq \phi \leq 2\pi} f (\phi,z) d\phi dz$$. In the case of a constant or else = /2, this reduces to vector calculus in polar coordinates. the area element and the volume element The Jacobian is The position vector is Spherical Coordinates -- from MathWorld Page 2 of 11 . r) without the arrow on top, so be careful not to confuse it with $r$, which is a scalar. When using spherical coordinates, it is important that you see how these two angles are defined so you can identify which is which. { "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.

area element in spherical coordinates 2023