Optical applications of von Neumann's alternating-projection theorem

The parallel-axis theorem provides a convenient and quick method of finding the moment of inertia of an object about an axis parallel to the axis passing through its center of mass. Consider a thin rod as an example. There is a striking similarity between the process of finding the moment of inertia of a thin rod about an axis through its middle, where the center of mass lies, and about an axis through its end using the conventional method. In the conventional method, the concept of linear mass...

The moment of inertia is a fundamental concept in mechanical engineering that plays a significant role in designing rotationally symmetric objects such as flywheels, gears, and other mechanical systems. In this context, we will discuss the moment of inertia of a flywheel rotating about its centroidal axis and how it relates to the moment of inertia about an axis parallel to it.
For a flywheel approximated as a solid disc, consider an infinitesimal differential element with an arbitrary distance...

Stokes’ Theorem provides a fundamental connection between the circulation of a vector field along a closed boundary and the cumulative rotational behavior across the surface it encloses. For a smooth three-dimensional surface with an oriented boundary curve, this theorem offers a unified way to relate motion along the edge to local rotational effects distributed over the surface.Mathematical FormulationThe theorem states that the circulation of a vector field along a closed curve is equal to...

The perpendicular-axis theorem states that the moment of inertia of a planar object about an axis perpendicular to its plane is equal to the sum of the moments of inertia about two mutually perpendicular concurrent axes lying in the plane of the body.
Consider a circular disc of mass M and radius R lying along an x-y plane. The origin lies at the center of the disc, and the z-axis is perpendicular to the disc's plane. All three axes coincide at the disc's center. The moment of inertia of this...

Learning to draw Fischer projections of molecules and understanding their relevance plays a crucial role in the visual depiction of organic molecules. A Fischer projection is a two-dimensional projection on a planar surface to simplify the three-dimensional wedge–dash representation of molecules. This is especially helpful in the case of molecules with multiple chiral centers that can be difficult to draw. Here, all the bonds of interest are represented as horizontal or vertical lines. While...

In vector calculus, flux measures the total flow of a vector field through a surface. For a closed surface in three-dimensional space, this means measuring how much of the field passes outward through every point on the boundary. Directly calculating this flux can be difficult when the surface has a complicated or irregular shape. The Divergence Theorem provides a powerful alternative by relating surface flux to behavior inside the enclosed region.The Divergence Theorem states that the outward...