What is Spherical: Definition and 1000 Discussions

A sphere (from Greek σφαῖρα—sphaira, "globe, ball") is a geometrical object in three-dimensional space that is the surface of a ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").
Like a circle in a two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point in a three-dimensional space. This distance r is the radius of the ball, which is made up from all points with a distance less than (or, for a closed ball, less than or equal to) r from the given point, which is the center of the mathematical ball. These are also referred to as the radius and center of the sphere, respectively. The longest straight line segment through the ball, connecting two points of the sphere, passes through the center and its length is thus twice the radius; it is a diameter of both the sphere and its ball.
While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics the above distinction is made between a sphere, which is a two-dimensional closed surface embedded in a three-dimensional Euclidean space, and a ball, which is a three-dimensional shape that includes the sphere and everything inside the sphere (a closed ball), or, more often, just the points inside, but not on the sphere (an open ball). The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. This is analogous to the situation in the plane, where the terms "circle" and "disk" can also be confounded.

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  1. S

    Solving Spherical Harmonics Homework

    Homework Statement The spherical harmonic, Ym,l(θ,φ) is given by: Y2,3(θ,φ) = √((105/32π))*sin2θcosθe2iφ 1) Use the ladder operator, L+ = +ħeiφ(∂/∂θ+icotθ∂/∂φ) to evaluate L+Y2,3(θ,φ) 2) Use the result in 1) to calculate Y3,3(θ,φ) Homework Equations L+Ym,l(θ,φ)=Am,lYm+1,l(θ,φ)...
  2. S

    B Are the profiles for reflecting/refracting spherical?

    Are the profiles of the reflecting/refracting surfaces spherical for the bleeding-edge astronomical instruments? I realize that because of the paraxial approximation, a "small angle" for a "ray" of light on spherical reflecting & refracting surfaces allows for a clean focus to take place...
  3. Robin04

    Divergence of a vector field in a spherical polar coordinate system

    Homework Statement I have to calculate the partial derivative of an arctan function. I have started to calculate it but I wonder if there is any simpler form, because if the simplest solution is this complex then it would make my further calculation pretty painful... Homework Equations $$\beta...
  4. karush

    MHB Evaluate the spherical coordinate integrals

    $\textsf{Evaluate the spherical coordinate integrals}$ \begin{align*}\displaystyle DV_{22}&=\int_{0}^{2\pi}\int_{0}^{\pi/4}\int_{0}^{2} \, (\rho \cos \phi) \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta \\ %&=\color{red}{abc} \end{align*} so then next ...
  5. H

    A Applying boundary conditions on an almost spherical body

    I am solving the Laplace equation in 3D: \nabla^{2}V=0 I am considering azumuthal symmetry, so using the usual co-ordinates V=V(r,\theta). Now suppose I have two boundary conditions for [V, which are: V(R(t)+\varepsilon f(t,\theta),\theta)=1,\quad V\rightarrow 0\quad\textrm{as}\quad...
  6. A

    MHB How to find volume of cone+hemisphere on the cone using spherical coordinate

    i have a problem about find volume of hemisphere I do not know the true extent of radius r (0 to ?) i think... cone ( 0 < r < R cosec(\theta) ) hemisphere (0 < r < R)
  7. Likith D

    Electric field experienced by a uniformly charged spherical shell due to itself

    The electric field experienced by the points on the surface of the shell is put out as KQ/R^2 where Q is charge on shell and R is radius of shell... But the gaussian surface corresponding to the case intersects the sphere, which means there are non-infinitesimal charge quantity sitting on the...
  8. W

    Electrostatics: "Compressing" Charged Spherical Shell

    Homework Statement How much work is required to squeeze a uniformly charged spherical shell from a radius of ##r## to a radius of ##r−dr##, if (a) the total charge q is a constant, (b) the sphere is kept at a constant potential, e.g. grounded. (c) Are the answers the same or different...
  9. R

    Electric field at the surface of a spherical shell?

    What is the value of electric field at the surface of a spherical shell?
  10. K

    MHB Calculating Coordinates of Spherical Triangles

    Hello, I'm not a student, I'm just trying to figure out how to calculate coordinates on a globe, and I would like to ask for some help. Let's say I have POINT A on the globe with the following coordinates: POINT A Latitude 45° 27' 50.95" N Longitude 9° 11' 23.98" E Also I have POINT B which...
  11. N

    About a concave mirror with a large focal length (1000mm)

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  12. Wrichik Basu

    Intro Math Books on Spherical Trigonometry

    What are some good books to learn spherical trigonometry from basics to the advanced level?
  13. Dealingwithphysics

    Electrodynamics, Potentials, spherical uncharged shells

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  14. Arman777

    B Spherical Geometry (Two dimension ) Defining a metric

    I am trying to understand how to define a metric for a positively curved two-dimensional space.I am reading a book and in there it says, On the surface of a sphere, we can set up a polar coordinate system by picking a pair of antipodal points to be the “north pole” and “south pole” and by...
  15. Ethan Singer

    Why aren't less massive objects also spherical like planets and moons?

    given that gravity pulls things together into spheres, how much mass is needed to do so? Smaller objects such as asteroids, meteors, and that bench in the park don't just turn into spheres, because they lack enough gravitational force to dominate the shape. So at approximately how massive do...
  16. K

    Is it correct that spherical buoy doesn't rotate?(Buoyancy)

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  17. M

    A Unraveling the Confusion: Mistakes in Solving PDEs in Spherical Coordinates?

    Given the PDE $$f_t=\frac{1}{r^2}\partial_r(r^2 f_r),\\ f(t=0)=0\\ f_r(r=0)=0\\ f(r=1)=1.$$ We let ##R(r)## be the basis function, and is determined by separation of variables: ##f = R(r)T(t)##, which reduces the PDE in ##R## to satisfy $$\frac{1}{r^2 R}d_r(r^2R'(r)) = -\lambda^2:\lambda^2 \in...
  18. Physics Dad

    Find the total electric charge in a spherical shell

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  19. P

    Fluid Dynamics -- a spherical particle immersed in water

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  20. L

    How to calculate the dipole moment of the spherical shell?

    Homework Statement A spherical shell of radius R has a surface charge distribution σ = k sinφ. Calculate the dipole moment of the spherical shell. Homework Equations P[/B]' = ∫r' σ(r') da' The Attempt at a Solution So I believe my dipole will be directed along the y axis, as the function...
  21. G

    Spherical conductor shell problem

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  22. M

    Spherical Boundary condition problem

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  23. amjad-sh

    A Spherical Harmonics: A Primer on Barton's Relations & Addition Theorem

    Hello. I was recently reading Barton's book. I reached the part where he proved that in spherical polar coordinates ##δ(\vec r - \vec r')=1/r^2δ(r-r')δ(cosθ-cosθ')δ(φ-φ')## ##=1/r^2δ(r-r')δ(\Omega -\Omega')## Then he said that the most fruitful presentation of ##δ(\Omega-\Omega')## stems from...
  24. amjad-sh

    Dirac-delta function in spherical polar coordinates

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  25. D

    Magnitude of the Electric Field at the Following Distances

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  26. Philosophaie

    I Convert from rectangular to Spherical Coordinates

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  27. karush

    MHB 15.5.63 - Rewrite triple integral in spherical coordinates

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  28. B

    Gaussian beam spherical mirror reflection question

    Homework Statement Gaussian beam of radius R_i and beam width w_i, The beam is reflected off a mirror with a radius of curvature R = R_i and the reflectivity of this mirror is given as rho(r) = rho_0*exp(-r^2/a^2), where r is the radial distance from the center of the mirror and a is a...
  29. T

    Electric field inside non-conducting spherical shell

    Homework Statement Figure 23-30 shows two nonconducting spherical shells fixed in place. Shell 1 has uniform surface charge density +6.0 µC/m2 on its outer surface and radius 3.0 cm. Shell 2 has uniform surface charge density -3.8 µC/m2 on its outer surface and radius 2.0 cm. The shell centers...
  30. T

    Rotating a spherical conductor in an Electric field

    Hi, I recently came across the familiar image of a metal sphere in an electric field: https://i.stack.imgur.com/x58Ia.jpg I noted how the free-charges on the surface of the sphere align with the electric field lines as opposite charges are attracted. Then I wondered, 'what if the sphere was...
  31. RJLiberator

    Volume Charge Density in Spherical and Cylindrical Coordinat

    Homework Statement Consider a ring of radius R placed on the xy-plane with its center at the origin. A total charge of Q is uniformly distributed on the ring. a) Express the volume charge density of this configuration ρ(s,Φ,z) in cylindrical coordinates. b) Express the volume charge density of...
  32. E

    Spherical magnet dropped through aluminum pole rotates?

    Lenz's law shows that dropping a magnet through an aluminum pole will cause an induced current that slows down its fall drastically. I found a website that talks about this a little: https://www.lhup.edu/~dsimanek/TTT-slowfall/slowfall.htm It has the following question: I don't see any...
  33. JTC

    A Understanding Metric Tensor Calculations for Different Coordinate Systems

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  34. davidge

    I Spherical Harmonics from operator analysis

    I found an interesting thing when trying to derive the spherical harmonics of QM by doing what I describe below. I would like to know whether this can be considered a valid derivation or it was just a coincidence getting the correct result at the end. Starting making a Fundamental Assumption...
  35. davidge

    I Spherical Harmonics: Arriving at Equation

    How does one arrive at the equation $$\bigg( (1-z^2) \frac{d^2}{dz^2} - 2z \frac{d}{dz} + l(l+1) - \frac{m^2}{1-z^2} \bigg) P(z) = 0$$ Solving this equation for ##P(z)## is one step in deriving the spherical harmonics "##Y^{m}{}_{l}(\theta, \phi)##". The problem is that the book I'm following...
  36. U

    Rectangular to Spherical Coordinate conversion....

    Homework Statement Convert from rectangular to spherical coordinates. (-(sqrt3)/2 , 3/2 , 1) Homework Equations We know the given equations are ρ = sqrt(x^2 + y^2 + z^2) tan theta = y/x cos φ = z / ρ The Attempt at a Solution My answer was (2, -pi/3, pi/3) It should be a simple plug and go...
  37. akkex

    MHB Change from cartesian coordinates to cylindrical and spherical

    Hello, I have 6 equations in Cartesian coordinates a) change to cylindrical coordinates b) change to spherical coordinate This book show me the answers but i don't find it If anyone can help me i will appreciate so much! Thanks for your time1) z = 2...
  38. LyleJr

    I Derivation of the Laplacian in Spherical Coordinates

    Hi all, Sorry if this is the wrong section to post this. For some time, I have wanted to derive the Laplacian in spherical coordinates for myself using what some people call the "brute force" method. I knew it would take several sheets of paper and could quickly become disorganized, so I...
  39. Pushoam

    Dirac delta function in spherical cordinates

    Homework Statement Calculate ##\int_{r=0}^\inf δ_r (r -r_0)\,dr## Homework Equations ##\int_V \delta^3(\vec{r} - \vec{r}') d\tau = 1## The Attempt at a Solution $$\int_V \delta^3(\vec{r} - \vec{r}') d\tau = \int_V \frac {1}{r^2 sinθ}\delta_r(r-r_0) \delta_θ (θ-θ_0) \delta_Φ (Φ-Φ_0) r^2...
  40. AdrianMachin

    Electric potential between two concentric spherical shells

    Homework Statement (The complete problem statement and solution are inside the attached picture) Two isolated, concentric, conducting spherical shells have radii ##R_1=0.500 m## and ##R_2=1.00 m##, uniform charges ##q_1=2.00 mC## and ##q_2=1.00 mC##, and negligible thicknesses. What is the...
  41. H

    A Heat Diffusion in 3D: Almost Spherical Flow

    Suppose I am considering the diffusion of heat in three dimensions: \frac{\partial T}{\partial t}=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial T}{\partial r}\right)+\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial...
  42. J

    Average of cos(x)^2 in spherical distribution

    << Mentor Note -- Thread moved from the technical forums so no Homework Help Template is shown >> I'm particle physics at the moment and I don't get why the average value of cosˆ2 is 1/3. The section : My solution is : < cos^{2}(\alpha)> = \frac{1}{\pi - 0} \int_{0}^{\pi }...
  43. C

    Improper integral with spherical coordinates

    Homework Statement I have a question. I have a function f(x,y,z) which is a continuous positive function in D = {(x,y,z); x^2 + y^2 +z^2<=1}. And let r = sqrt(x^2 + y^2 + z^2). I have to check whether the following jntegral is convergent. x^2y^2z^2/r^(17/2) * f(x,y,z)dV. Homework Equations...
  44. R

    Electric field in a cavity in a spherical charge density

    Homework Statement A uniform spherical charge density of radius R is centred at origin O. A spherical cavity of radius r and centre P is made. OP = D = R-r. If the electric field inside the cavity at position r is E(r), the correct statement is: 1)E is uniform, its magnitude is independent of r...
  45. chandrahas

    Difference between a tokamak and a spherical tokamak

    The title question is quite self-explanatory. Despite the fact that Spherical tokamaks are more spherical in shape, what else differentiates the ST from the conventional tokamak. I've heard that ST's use reverse field configurations from a website but I am skeptical about this since the rest...
  46. asteeves_

    Optics - spherical and plane mirror

    Homework Statement A convex spherical mirror with a focal length of magnitude 24.0 cm is placed 22.0 cm to the left of a plane mirror. An object 0.300 cm tall is placed midway between the surface of the plane mirror and the vertex of the spherical mirror. The spherical mirror forms multiple...
  47. A

    I Understanding the Form of the Y(2,0) Spherical Harmonic

    I am basically just rewriting a question that was posted on other forums. While watching videos of a MIT lecture on the eigenstates of angular momentum (video: '16. Eigenstates of the Angular Momentum II' by MIT OpenCourseWare) the professor visualized different spherical harmonics for low...
  48. E

    In which direction a spherical particle moves under torque?

    Consider a small rigid spherical particle of radius a, fully immersed in a viscous incompressible Newtonian fluid of shear viscosity η above a hard-wall with stick (no-slip) boundary conditions, located at the plane z = 0. A constant positive (external) torque Tx is applied on the particle. My...
  49. I

    Placing object between 2 spherical mirrors

    Homework Statement A concave mirror (f1 = 13.6 cm) and a convex mirror (f2 = −7.00 cm) are facing each other and are separated by a distance of 35.8 cm. An object is placed between the mirrors and is 17.9 cm from each mirror. Consider the light from the object that reflects first from the...
  50. Arman777

    Spherical Shell-Potential Energy, Energy density

    Homework Statement Homework EquationsThe Attempt at a Solution I fount these Part(a)its ##E=\frac {ρa^3} {3ε_0}## and ##υ=\frac 1 2ε_0E^2## Part (b) ##dU=4πr^2drυ## Part (c) ##U=\int_0^a 4πr^2udr## but it gives me ##U=\frac {-Q^2} {8πε_0a}## This"-" bothers me.
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