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Related Concept Videos

Spherical Coordinates01:23

Spherical Coordinates

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Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe the surface of a sphere, Cartesian coordinates require all three coordinates. On the other hand, the spherical coordinate system requires only one parameter: the sphere's radius. As a result, the complicated mathematical calculations become simple. Spherical coordinates are used in science and engineering applications like electric and...
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A toroid is a closely wound donut-shaped coil constructed using a single  conducting wire. In general, it is assumed that a toriod consists of  multiple circular loops perpendicular to its axis.
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Gauss's Law: Spherical Symmetry01:26

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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a...
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In uniform circular motion, the particle executing circular motion has a constant speed, and the circle is at a fixed radius. However, not all circular motion occurs at a constant speed. A particle can travel in a circle and speed up or slow down, showing an acceleration in the direction of motion. In that case, the motion is called non-uniform circular motion, and an additional acceleration is introduced, which is in the direction tangential to the circle. 
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Rotational Motion about a Fixed Axis01:26

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A rigid body's rotation around a fixed axis makes every point within it trace a circular path around a specific line or point. The term given to this type of spinning is defined by the angular position, symbolized by the angle θ. This angle is gauged from a static reference line to the revolving object. From this angular position, any variation is referred to as angular displacement, denoted by dθ. The extent of this displacement can be calculated in degrees, radians, or...
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Uniform circular motion is a specific type of motion in which an object travels in a circle with a constant speed. For example, any point on a propeller spinning at a constant rate is undergoing uniform circular motion. The second, minute, and hour hands of a watch also undergo uniform circular motion. It is hard to believe that points on these rotating objects are actually accelerating, even though the rotation rate is constant. To understand this, we must analyze the motion in terms of...
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Applying Permanent, Robust Stenciled Patterns of Fine Particles to Elastomeric Surfaces
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Turing patterns on a sphere.

C Varea1, J L Aragón, R A Barrio

  • 1Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México, D.F., Mexico.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|April 24, 2002
PubMed
Summary
This summary is machine-generated.

Pattern formation on spheres is explored using Turing equations. Spherical geometry restricts pattern shapes, impacting biological processes influenced by curvature.

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Area of Science:

  • Mathematical Biology
  • Chemical Kinetics
  • Differential Equations

Background:

  • Turing equations model pattern formation via reaction-diffusion mechanisms.
  • Spherical geometry is prevalent in biological systems, influencing developmental processes.

Purpose of the Study:

  • Investigate pattern formation on spherical surfaces using Turing equations.
  • Analyze how surface curvature affects pattern characteristics.

Main Methods:

  • Numerical simulations of a generic reaction-diffusion model.
  • Exploration of patterns under varied parameter conditions.

Main Results:

  • Curvature of closed surfaces, like spheres, imposes geometric constraints on pattern formation.
  • Identified specific pattern morphologies influenced by spherical geometry.

Conclusions:

  • Spherical geometry is a critical factor in pattern formation.
  • Findings are relevant to understanding curvature-driven processes in biology, biochemistry, and embryology.