Optical implementation of integral transforms with Bessel function kernels

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Bessel Function of Order Zero01:20

Bessel Function of Order Zero

A common physical example of wave propagation with radial symmetry is the ripple formed when a stone is dropped into a still pond. The disturbance originates at a central point and travels outward as a circular wave. As the radius of the wavefront increases, the same initial energy is distributed along a progressively larger circumference. Consequently, the amplitude, or height, of the wave decreases with distance from the center. This decay behavior cannot be captured by simple sine or cosine...
Triple Integrals in Cylindrical Coordinates01:28

Triple Integrals in Cylindrical Coordinates

Cylindrical coordinates describe a point in three-dimensional space using three values: radial distance, angle, and height. The height gives the position above the xy-plane, the radial distance measures how far the point is from the z-axis, and the angle describes the point鈥檚 direction from the positive x-axis in the xy-plane. This system is especially useful for regions with circular symmetry because it matches the natural geometry of cylinders, disks, and circular tanks.To calculate volume,...
Integrals of Vector Functions01:23

Integrals of Vector Functions

Vector-valued functions provide a convenient framework for describing motion in space when both magnitude and direction are important. A drone鈥檚 velocity at any instant has a direction and a speed, and as the drone moves, both can change. A vector-valued function captures this behavior by assigning to each time a vector whose components are real-valued functions. Each component represents motion along a particular axis in space. Such functions can describe motion in either two-dimensional or...
Double Integrals in Polar Coordinates01:27

Double Integrals in Polar Coordinates

Double integrals provide an effective method for calculating areas and other physical quantities distributed across two-dimensional regions. In engineering and design applications, curved geometries often appear in structures such as ponds, reservoirs, and circular foundations. When these regions possess circular symmetry, polar coordinates offer a more natural and efficient description than Cartesian coordinates. This coordinate system simplifies the integration process by representing points...
Integration Applied to Polar Coordinates to Find Areas01:15

Integration Applied to Polar Coordinates to Find Areas

A rotating lawn sprinkler with an uneven spray pattern produces a variable reach as it distributes water in different directions. This directional variation in spray distance can be effectively described using polar coordinates, where the distance from the center is represented as a function of the angle of rotation. The path traced by the spray then forms a polar curve, which captures the irregularities in the sprinkler鈥檚 reach across the full rotation.To calculate the total area watered by...
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...