Spherical Definition and 1000 Threads

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.

View More On Wikipedia.org
Recent contents Top users
  1. U

    Jacobian in spherical coordinates?

    Hi, Started to learn about Jacobians recently and found something I do not understand. Say there is a vector field F(r, phi, theta), and I want to find the flux across the surface of a sphere. eg: ∫∫F⋅dA Do I need to use the Jacobian if the function is already in spherical...
  2. R

    Coefficients of capacitance with spherical shell problem

    Homework Statement Metal sphere of radius R1 is surrounded by a concentric metal shell of inner radius R2 and outer radius R3. The dielectric is air. Calculate coefficients of capacitance for the given setup. Homework Equations Picture of the problem: The Attempt at a Solution...
  3. E

    Differentiation spherical coordinates

    Hi ! I'm trying to inverse a mass matrix so I need to do something like this \dfrac{q}{\partial \mathbf{r}} where \cos q = \dfrac{\mathbf{r}\cdot \hat{\mathbf{k}}}{r} However, when \mathbf{r} = \hat{\mathbf{k}} \text{ or } -\hat{\mathbf{k}} I have problems. ¿What can I do...
  4. M

    Looking for a part that can lock a spherical bearing.

    For context: I'm building a handle that needs to rotate in all directions. The handle will be placed inside the spherical bearing - and the bearing will allow it to rotate as it's required to. The handle will also need to be able to be locked in place at angles chosen by the user. Whatever...
  5. Philosophaie

    Formulating x^n Coordinate System for Non-Rectangular/Spherical Riemann Manifold

    I want to be able to formulate x^{n} coordinate system. x^{n} =(x^{1}, x^{2}, x^{3}, x^{4}) How do you do this when the Riemann Manifold is not rectangular or spherical? Also how do you differentiate with respect to "s" in that case. \frac{dx^n}{ds}
  6. A

    Convert unit vector from cartesian to spherical coordination

    i have a problem : A small loop antenna in free space and centered about the origin on the xy-plane is producing a (far-field) radiation electric field (in phasor notation) : http://postimg.org/image/63tm76h5l/ and their solution : http://postimg.org/image/6mdm6roh9/ i don't understand how...
  7. Y

    Divergence in spherical coordinates.

    I want to verify: \vec A=\hat R \frac{k}{R^2}\;\hbox{ where k is a constant.} \nabla\cdot\vec A=\frac{1}{R^2}\frac{\partial (R^2A_R)}{\partial R}+\frac{1}{R\sin\theta}\frac{\partial (A_{\theta}\sin\theta)}{\partial \theta}+\frac{1}{R\sin\theta}\frac{\partial A_{\phi}}{\partial \phi}...
  8. P

    Particle density in spherical geometry

    Homework Statement Neutrons are emitted uniformly from the inner surface of a thin spherical shell of radius R at a velocity V. They are emitted normal to the inner surface and fly radially across the volume of the sphere to be absorbed at diametrically opposed points. The neutrons are non...
  9. Ackbach

    MoI of a Sphere using Spherical Coordinates

    Homework Statement Calculate the moment of inertia of a uniformly distributed sphere about an axis through its center. Homework Equations I know that $$I= \frac{2}{5} M R^{2},$$ where ##M## is the mass and ##R## is the radius of the sphere. However, for some reason, when I do this...
  10. alemsalem

    Finding large order spherical harmonics

    is there an approximation for spherical harmonics for very large l and m in closed form?
  11. I

    How to deduct the gradient in spherical coordinates?

    http://en.wikipedia.org/wiki/Gradient#Cylindrical_and_spherical_coordinates which formula do we apply to get the gradient in spherical coordinates?
  12. E

    Finding voltage due to spherical charge

    Homework Statement A volume charge density of ρv=1/r^2 μC/m^3 exists in the region bounded by 1.0m<r<1.5m. Find the potential difference between point A(3.0,4.0,12.0) and point B(2.0,2.0,2.0)Homework Equations dQ=(ρv)dv dV=[dQ/(4∏ε0]*norm(R) where R is position vector of a point charge...
  13. G

    Electric field of a spherical cap

    Hi I am looking for the electric field caused by a uniformly charged spherical cap. Actually, I need only the potential inside the sphere. Is there anybody who knows how to do this. Frankly, I do not have a clue. Or could somebody at least give me integral, that I have to solve?
  14. O

    Why must be that for curl vector in spherical coordinate?

    The correct one is 2nd, but why not first? Please guide , or tell me any link that relate to this derivation. Thanks
  15. N

    Obtaining spherical coordinates by rotations

    Hi Say I have a point on a unit sphere, given by the spherical coordinate $(r=1, \theta, \phi)$. Is this point equivalent to the point that one can obtain by $(x,y,z)=(1,0,0)$ around the $y$-axis by an angle $\pi/2-\theta$ and around the $z$-axis by the angle $\phi$? I'm not sure this is...
  16. A

    Gauss' Law for spherical shell vs Coulomb's law, regarding reativity

    Shalom We are used to hearing that Coulomb's law doesn't settle with the relativity principle that nothing moves faster than the speed of light, in the sense that it embeds 'Action in a Distance'. Meaning that if somthing changes in r1 at time t1, and we write the law for any t before...
  17. C

    Spherical tensor operators' commutation with lowering/raising operator

    I'm studying Shankar's book (2nd edition), and I came across his equation (15.3.11) about spherical tensor operators: [J_\pm, T_k^q]=\pm \hbar\sqrt{(k\mp q)(k\pm q+1)}T_k^{q\pm 1} I tried to derive this using his hint from Ex 15.3.2, but the result I got doesn't have the overall \pm sign on the...
  18. M

    Why only l=1 of spherical harmonics survives?

    Homework Statement The question is about page 198 of Jackson's Classical Electrodynamics. The magnetic scalar potential is set to be: Phi = ∫ (dΩ' cosθ'/ |x-x'|). Using the spherical harmonics expansion of 1/|x-x'|, the book claims that only l=1 survives. I...
  19. K

    The length of a path on a sphere (in spherical coordinates)

    So, I'm to show that in spherical coordinates, the length of a given path on a sphere of radius R is given by: L= R\int_{\theta_1}^{\theta_2} \sqrt{1+\sin^2(\theta) \phi'^2(\theta)}d\theta, where it is assumed \phi(\theta), and start coordinates are (\theta_1,\phi_1) and (\theta_2, \phi_2)...
  20. L

    How the planets get their spherical shape

    how the planets get in spherical shape if the they are formed by a big explosion ?
  21. Hepth

    Massive Vector Polarizations in Spherical Coordinates

    I can't seem to find one, but does anyone have a reference to the fourvector polarizations for a massive vector particle in spherical coordinates where a momentum is defined as p = \{E, |\vec{p}| \sin \theta \sin \phi, |\vec{p}|\sin \theta \cos \phi , |\vec{p}| \cos \theta\} theta goes...
  22. D

    Converting from cartesian to spherical boundaries

    If I had a sphere centred at the origin with x > 0, y > 0 and z > 0 Would the angle boundaries be: 0 < θ < pi/2 0 < α < pi/2 ?
  23. tom.stoer

    Spherical symmetric collapse of pressureless dust

    Is there an exact solution for the spherical symmetric collaps of pressureless dust? Can one see a Schwarzschild solution for r > Rdust with shrinking Rdust(t) ?
  24. Y

    Stuck on proof irreducible spherical tensor operator

    Hi everyone, I'm stuck on proving that a certain operator is an irreducible spherical tensor operator. These are tensor operators T^{k}_{q} with -k \leq q\leq k satisfying \mathscr{D} T^{k}_{q} \mathscr{D}^{\dagger} = \sum_{q'} \mathscr{D}^{k}_{q' q} T^{k}_{q'} where...
  25. I

    How to evaluate this nabla expression in spherical coordinates?

    I'm currently working out the Schrödinger equation for a proton in a constant magnetic field for a research project, and while computing the Hamiltonian I came across this expression: (\vec{A}\cdot\nabla)\Psi where \Psi is a scalar function of r, theta, and phi. How do you evaluate this...
  26. M

    How do spherical coordinates work for finding volume in a given region?

    Homework Statement Find VR_{z}^{2} = \int \!\!\! \int \!\!\! \int_{E} (x^{2} + y^{2})dV given a constant density lying above upper half of x^{2}+y^{2} = 3z^{2} and below x^{2}+y^{2}+z^{2} = 4z.Homework Equations The Attempt at a Solution Why does it say upper half of x^{2}+y^{2} = 3z^{2}? It's...
  27. E

    Absorption of light by spherical nanoparticle

    Hello Can anyone tell me how the absorption of a polystyrene nanoparticles scales as a function of its diameter. The particle is spherical and it is placed in vacuum. A reference to a paper I can read would be nice. At this point I only want to know how the absorption scales not...
  28. A

    Lift of a helium spherical balloon

    So I was pondering this question: On a conceptual level, how does a perfectly spherical helium balloon rise? I understand that the density of helium gas is lower than that of our atmospheric composition of gases, but that is not giving the full perspective for me. On a molecular level, I feel...
  29. D

    Time taken for magnetic field to permeate a spherical volume

    1. For a stationary plasma of electrical conductivity 1.00 x108 Ω-1 m-1 estimate the time taken for a magnetic field to permeate a spherical volume of 3.0 m radius. I have been looking at the question for some time now and I am struggling with where to begin. Any help would be much appreciated.
  30. D

    Forming Hydrogen wave functions with real spherical harmonics

    Hi, I'm a little confused about how to apply the real spherical harmonics when building a hydrogen wave function. I'm doing a computational project, so I want to work with a wave function which is strictly real, and I'm hoping I can do so by building the orbitals using the real spherical...
  31. E

    Triple Integration from Rectangular to Spherical Coordinates

    Homework Statement Convert the integral from rectangular coordinates to spherical coordinates 2 √(4-x^2) 4 ∫ ∫ ∫ x dz dy dx -2 -√(4-x^2) x^2+y^2 Homework Equations x=ρ sin∅ cosθ y=ρ sin∅ cosθ z=ρ cos∅ In case the above integrals cannot be understood: -2...
  32. U

    Solving Spherical Capacitors: Find C1 &amp; C2 in Parallel

    Homework Statement Homework Equations The Attempt at a Solution Working backwards I found that adding C1 (of radius a and b) and C2 (of radius b and c) in parallel gives the answer. Not sure why they can be modeled as capacitors in parallel though..
  33. Q

    Partial wave scattering cross section in spherical well

    Homework Statement Consider the spherical well such that V(r<a) = -V0 and V(r≥a) = 0. Calculate the l = 0 partial wave scattering cross section in the low energy limit for this potential. Homework Equations σ = \frac{4 \pi}{k^2} * \Sigma (2l+1)*sin^2(\delta_l) The Attempt at a...
  34. @

    Hollow spherical earthed conductor

    Homework Statement three concentric hollow conducting shells are there . inner most is given charge +q , outer most is given charge -q and middle one is earthed , then find charge appearing on all the surfaces ? Homework Equations v= k q / r , E=k q /r2 The Attempt at a Solution no...
  35. baby_1

    Convert Cartesian coordinates to spherical shape

    Hello how can Convert Cartesian coordinates to spherical with shape? for clear my question i explain a way to convert my coordinates in different spherical. for example i use this diagram to convert Cartesian coordinates to Cylindrical(with image to axises) for example: now how can i do...
  36. K

    Use spherical cord to compute area of a disk

    Homework Statement Use spherical cords to compute area of a disk, that's center at x,y=0, and z=4, having a radius of 3 Homework Equations I set up a triple ∫∫∫ (r^2* sin(theta)), running from phi =0 to 2pi, theta=0 to arcsin(3/5), r=5sin(theta) to 5. The Attempt at a Solution It doesn't...
  37. Saitama

    The Temperature Profile of a Spherical Cloud of Ideal Gas

    Homework Statement (see attachment 1) Homework Equations The Attempt at a Solution (see attachment 2) As the gas is ideal and there is no gravity, the pressure is same throughout the cloud. In the thin sphere shown, the mass of the gas is ##dm=dV \cdot \rho(r)##. Let ##\mu## be...
  38. S

    Griffiths example no. 5.11 w×r switch from cartesian to spherical

    I am pretty much satisfied with the example of a rotating shell example 5.11 pg 367 griffiths electrodynamics.on many ocassions he chooses cartesian coordinates before integration (see 5.10 too) , integrates and finds w×r along y direction .then he manipulates w×r, and writes it down in...
  39. P

    Electric field on the surface of uniformly charged spherical shell

    We can easily find from gauss's theorem(or otherwise) the field inside and outside a uniformly charged spherical shell.But i was wondering what would be the field on the surface of the shell.
  40. T

    Spherical ball rolling on a concave surface

    Homework Statement A spherical ball of mass m and radius r rolls without slipping on a rough concave surface of large radius R .It makes small oscillations about the lowest point.Find the time period. Ans : 2∏\sqrt\frac{7(R-r)}{5g} Homework Equations The Attempt at a Solution...
  41. K

    Net charge of a spherical capacitor

    Suppose we have a hollow spherical shell made of a conducting metal, inside a slightly larger hollow spherical shell made of the same conducting metal. The shells are separated by a layer of insulation, so that the assembly is basically a spherical, hollow capacitor. If I cause the inner shell...
  42. M

    E field above charged spherical surface

    the calculation of the E field a distance z above a spherical surface of charge gives rise to this integral which can be done by partial fractions...I don't see this integral in Stewart's table did I not input correctly in Mathematica? Integrate [(z-R*u)/(R^2+z^2-2*z*R*u)^(3/2),{u,-1,1}]...
  43. S

    Differences Between Conducting and Nonconducting Spherical Shells

    Homework Statement I am very confused about the differences between a conducting and nonconducting spherical shell. The biggest problem that I am having is the way that electric fields act both inside,outside, and within these shells. Any explanation would be much appreciated.
  44. N

    Determinant in Transformation from spherical to cartesian space

    Homework Statement Evaluate the appropriate determinant to show that the Jacobian of the transformation from Cartesian (this is a typo, they mean spherical) pψθ-space to Cartesian xyz-space is ρ2sin(ψ).Homework Equations The Attempt at a Solution Uhm, I am lost. I'm supposed to prove that when...
  45. G

    Solve Spherical Shell Gauss Problem with Differential Form Only

    Homework Statement A hollow spherical shell carries charge density \rho=\frac{k}{r^2} in the region a<=r<=b, where a is the inner radius and b is the outer radius. Find the electric field in the region a<r<b. I'm not allowed to use integral form of Gauss's law, must use differential...
  46. T

    Patch of a surface in spherical coordinates?

    Homework Statement I am currently trying to prove: S = ∫∫a2sinΦdΦdθ Here is my work (note that in my work I use dS instead of S, this is an accident): I end up with: S = ∫∫a*da2sinΦdΦdθ Where da is the infinitesimal thickness of the surface. Why am I getting the wrong answer?
  47. J

    Combination of spherical lenses

    Two thin lenses having focal lengths of +15 cm and -15 cm are positioned 60 cm apart. A bird stands 25 cm in front of of the converging lens. a. describe the image of the bird. Is it real or virtual? Upright or inverted? Magnified or reduced? b. If the bird is 10 cm in height, what is the...
  48. S

    Point Charge in an uncharged spherical conductor

    Homework Statement Consider a point charge q > 0 which is surrounded by a hollow metal sphere (uncharged) with inner radius R1 and outer radius R2. Use Gauss Law to determine the electric field E=E(r)er in the following regions: (i) 0 < r < R1 (ii) R1 < r < R2 (iii) r > R2 Homework...
  49. A

    Surface Area of a Sphere in Spherical Coordinates; Concentric Rings

    Hey, folks. I'm trying to derive the surface area of a sphere using only spherical coordinates—that is, starting from spherical coordinates and ending in spherical coordinates; I don't want to convert Cartesian coordinates to spherical ones or any such thing, I want to work geometrically...
  50. M

    Kinetic Energy in Spherical Coordinates

    Homework Statement Derive the expression for kinetic energy of a classical particle in spherical coordinates. Homework Equations I believe the answer I am supposed to reach is: T=\frac{1}{2} m (\dot{r}^2 + r^2\dot{\theta^2} + r^2\dot{\phi ^2}sin^2\theta) The Attempt at a Solution...
Back
Top