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Volumes of Solids of Revolution01:29

Volumes of Solids of Revolution

Volumes of irregularly shaped objects can be systematically determined using the concept of solids of revolution. This approach begins with a region defined by a curve in a two-dimensional plane. When this region is rotated about a fixed line, known as the axis of revolution, it generates a three-dimensional object with rotational symmetry. Such objects frequently arise in mathematical modeling, physics, and engineering applications.When the region being rotated lies directly against the axis...
Gauss's Law: Spherical Symmetry01:26

Gauss's Law: Spherical Symmetry

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 蟻0, 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 蟻1 and the bottom half has a uniform...
Gauss's Law: Cylindrical Symmetry01:20

Gauss's Law: Cylindrical Symmetry

A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
Rotation of Asymmetric Top01:11

Rotation of Asymmetric Top

By definition, a spherically symmetric body has the same moment of inertia about any axis passing through its center of mass. This situation changes if there is no spherical symmetry. Since most rigid bodies are not spherically symmetric, these require special treatment.
The relationship between the angular momentum of any rigid body and its angular velocity, both of which are vectors, involves the moment of inertia. The moment of inertia is a scalar quantity only for spherically symmetric...
Sampling Theorem01:15

Sampling Theorem

In signal processing, the analysis of continuous-time signals, denoted as x(t), often involves sampling techniques to convert these signals into discrete-time signals. This process is essential for digital representation and manipulation. A critical component in sampling is the train of impulses, characterized by the sampling interval and the sampling frequency. The relationship between these parameters and the original signal's properties dictates the success of the sampling process.
Triple Integrals in Spherical Coordinates01:27

Triple Integrals in Spherical Coordinates

Triple integrals in spherical coordinates provide an efficient method for evaluating volumes over regions with central symmetry, such as spheres. Instead of describing points by rectangular coordinates, spherical coordinates use three variables: 饾湆, 饾渻, and 饾湋. Here, 饾湆 is the distance from the origin, 饾渻 is the angle in the xy-plane measured from the positive x-axis, and 饾湋 is the angle measured downward from the positive z-axis.To derive the volume of a sphere, the solid region can be divided...

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Related Experiment Video

Updated: Jun 20, 2026

Methods for Measuring the Orientation and Rotation Rate of 3D-printed Particles in Turbulence
12:34

Methods for Measuring the Orientation and Rotation Rate of 3D-printed Particles in Turbulence

Published on: June 24, 2016

Sampling theorem for rotationally symmetric systems based on Dini expansion.

S Szapiel

    Optics Letters
    |August 28, 2009
    PubMed
    Summary

    A novel sampling theorem formulation for circular apertures uses Dini series, simplifying point spread function evaluation. This method efficiently handles even thin-ring apertures, improving computational speed.

    Area of Science:

    • Optics
    • Mathematical Physics

    Background:

    • The sampling theorem is crucial for reconstructing signals from discrete samples.
    • Traditional methods for circular apertures often rely on Fourier-Bessel analysis, which can be computationally intensive.

    Purpose of the Study:

    • To propose a new formulation of the sampling theorem for circular apertures.
    • To develop a computationally efficient method for evaluating point spread functions.

    Main Methods:

    • A Dini series approach is employed as an alternative to the Fourier-Bessel method.
    • The formulation is applied to circular apertures, including thin-ring cases.

    Main Results:

    • The Dini series approach provides a convenient formulation for the sampling theorem.

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    Micro/Nano-scale Strain Distribution Measurement from Sampling Moiré Fringes
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    Micro/Nano-scale Strain Distribution Measurement from Sampling Moiré Fringes

    Published on: May 23, 2017

    Related Experiment Videos

    Last Updated: Jun 20, 2026

    Methods for Measuring the Orientation and Rotation Rate of 3D-printed Particles in Turbulence
    12:34

    Methods for Measuring the Orientation and Rotation Rate of 3D-printed Particles in Turbulence

    Published on: June 24, 2016

    Micro/Nano-scale Strain Distribution Measurement from Sampling Moiré Fringes
    06:56

    Micro/Nano-scale Strain Distribution Measurement from Sampling Moiré Fringes

    Published on: May 23, 2017

  • Fast evaluation of point spread functions is achieved, even for complex aperture geometries like thin rings.

    • Conclusions:

    • The proposed Dini series formulation offers a more efficient alternative for sampling theorem applications with circular apertures.
    • This method simplifies the analysis and computation of optical system performance metrics.