Rotational symmetry | Partial differential equations | Harmonic analysis | Fourier analysis | Special hypergeometric functions
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Spherical harmonics originate from solving Laplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are called harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree in that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence from the above-mentioned polynomial of degree ; the remaining factor can be regarded as a function of the spherical angular coordinates and only, or equivalently of the orientational unit vector specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition. Notice, however, that spherical harmonics are not functions on the sphere which are harmonic with respect to the Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the Maximum principle. Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). A specific set of spherical harmonics, denoted or , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above. Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configurations, gravitational fields, geoids, the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation. In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and modelling of 3D shapes. (Wikipedia).
We calculate the functional form of some example spherical harmonics, and discuss their angular dependence.
From playlist Quantum Mechanics Uploads
Definition of spherical coordinates | Lecture 33 | Vector Calculus for Engineers
We define the relationship between Cartesian coordinates and spherical coordinates; the position vector in spherical coordinates; the volume element in spherical coordinates; the unit vectors; and how to differentiate the spherical coordinate unit vectors. Join me on Coursera: https://www
From playlist Vector Calculus for Engineers
Introduction to Spherical Coordinates
Introduction to Spherical Coordinates This is a full introduction to the spherical coordinate system. The definition is given and then the formulas for converting rectangular to spherical and spherical to rectangular. We also look at some of the key graphs in spherical coordinates. Final
From playlist Calculus 3
Introduction to Cylindrical Coordinates
Introduction to Cylindrical Coordinates Definition of a cylindrical coordinate and all of the formulas used to convert from cylindrical to rectangular and from rectangular to cylindrical. Examples are also given.
From playlist Calculus 3
Classical spherical trigonometry | Universal Hyperbolic Geometry 36 | NJ Wildberger
This video presents a summary of classical spherical trigonometry. First we define spherical distance between two points on a sphere, then the angle between two lines on a sphere (i.e. great circles). After a quick reminder of the circular functions cos,sin and tan, we present the main la
From playlist Universal Hyperbolic Geometry
Spherical Harmonics and Angular Momentum
We make a connection between spherical harmonics and eigenfunctions of angular momentum.
From playlist Quantum Mechanics Uploads
Converting Between Spherical and Rectangular Equations
This video provides example of how to convert between rectangular equation and spherical equations and vice versa. http://mathispower4u.com
From playlist Quadric, Surfaces, Cylindrical Coordinates and Spherical Coordinates
Introduction to Spherical Coordinates
This video defines spherical coordinates and explains how to convert between spherical and rectangular coordinates. http://mathispower4u.yolasite.com/
From playlist Quadric, Surfaces, Cylindrical Coordinates and Spherical Coordinates
Spherical Tensor Operators | Wigner D-Matrices | Clebsch–Gordan & Wigner–Eckart
In this video, we will explain spherical tensor operators. They are defined like this: A spherical tensor operator T^(k)_q with rank k is a collection of 2k+1 operators that are numbered by the index q, which transform under rotations in the same way as spherical harmonics do. They are als
From playlist Quantum Mechanics, Quantum Field Theory
Converting Between Cylindrical and Rectangular Equations
This video explains how to convert between cylindrical and rectangular equations. http://mathispower4u.yolasite.com/
From playlist Quadric, Surfaces, Cylindrical Coordinates and Spherical Coordinates
[Lesson 9] QED Prerequisites - Mind Map of Angular Momentum Part I
This is the start of a high level review of the Quantum Theory of Angular Momentum. It is intended to point you into directions for deeper review. This material will lead us to our detailed review of non-relativistic scattering theory. Please consider supporting this channel on Patreon: h
From playlist QED- Prerequisite Topics
Visualizing (Earth) Gravity Potential Models and Perturbed Kepler Orbits
Gottlob Gienger To learn more about the Wolfram Technologies, visit http://www.wolfram.com The European Wolfram Technology Conference featured both introductory and expert sessions on all major technologies and many applications made possible with Wolfram technology. Learn to achieve in
From playlist European Wolfram Technology Conference 2015
Black hole perturbation theory (Lecture 3) by Emanuele Berti
DATES Monday 25 Jul, 2016 - Friday 05 Aug, 2016 VENUE Madhava Lecture Hall, ICTS Bangalore APPLY Over the last three years ICTS has been organizing successful summer/winter schools on various topics of gravitational-wave (GW) physics and astronomy. Each school from this series aimed at foc
From playlist Summer School on Gravitational-Wave Astronomy
Vlad Yaskin: A solution to the 5th and 8th Busemann-Petty problems near the Euclidean ball
We show that the 5th and the 8th Busemann-Petty problems have positive solutions for bodies that are sufficiently close to the Euclidean ball in the Banach-Mazur distance. Joint work with M. Angeles Alfonseca, Fedor Nazarov, and Dmitry Ryabogin.
From playlist Workshop: High dimensional measures: geometric and probabilistic aspects
Benjamin Stamm: An embedded corrector problem for stochastic homogenization
The lecture was held within the framework of the Hausdorff Trimester Program Multiscale Problems: Workshop on Numerical Inverse and Stochastic Homogenization. (14.02.2017) A very efficient algorithm has recently been introduced in [1] in order to approximate the solution of implicit solva
From playlist HIM Lectures: Trimester Program "Multiscale Problems"
Tess Smidt: "Euclidean Neural Networks for Emulating Ab Initio Calculations and Generating Atomi..."
Machine Learning for Physics and the Physics of Learning 2019 Workshop I: From Passive to Active: Generative and Reinforcement Learning with Physics "Euclidean Neural Networks* for Emulating Ab Initio Calculations and Generating Atomic Geometries *also called Tensor Field Networks and 3D
From playlist Machine Learning for Physics and the Physics of Learning 2019
Tess Smidt - Learning how to break symmetry with symmetry-preserving neural networks - IPAM at UCLA
Recorded 26 January 2023. Tess Smidt of the Massachusetts Institute of Technology presents "Symmetry’s made to be broken: Learning how to break symmetry with symmetry-preserving neural networks" at IPAM's Learning and Emergence in Molecular Systems Workshop. Abstract: Symmetry-preserving (
From playlist 2023 Learning and Emergence in Molecular Systems
Condon-Shortley Phase Convention | Clebsch-Gordan Gets Real
In this video, we will explain the Condon-Shortley phase convention. This is a widely used set of two statements, which together lead to the conclusion that all Clebsch-Gordan coefficients are purely real numbers. It's named after the US physicists Edward Condon and George Shortley, who in
From playlist Quantum Mechanics, Quantum Field Theory
Introduction to Spherical Harmonics
Using separation of variables in spherical coordinates, we arrive at spherical harmonics.
From playlist Quantum Mechanics Uploads