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Electrostatic Fields
Published in Martin J. N. Sibley, Introduction to Electromagnetism, 2021
if dϕ is expressed in radians. (This is a direct consequence of expressing dϕ in radians. The circumference of the disc is 2πr, and this encloses an angle of 360°, or 2π radians. So, the length of an arc that subtends an angle of 180°, or π radians, is πr. Thus, the length of an arc is equal to the product of the angle (in radians) and the radius of the arc.)
Calculus of pasta, sausages, and bagels: can their surface areas be derivatives of their volumes?
Published in International Journal of Mathematical Education in Science and Technology, 2020
If we look at a sphere of radius r, its volume, V, and surface area, A, are calculated by the well-known formulas and , which immediately offers the relationship linking them: , where both V and A are functions of the radius of a sphere. So more precisely and consequently, the differentiation of volume as a function of a radius leads to the relationship Similarly, the derivative of the area of a disk equals the circumference of the circle: If high-school students do observe the presence of any derivative connections such as in the first two examples, they typically regard them to be unusual and infrequent coincidences.
Harmonic interpolation of Hermite type based on Radon projections with constant distances
Published in Applicable Analysis, 2019
Let D be the open unit disk and the unit circle. For any given pair , we denote by the line segment of D, where the line passes through the point and is perpendicular to the vector (see Figure 1). The Radon projection of a real-valued function u defined on D is the line integral of u over . More precisely In the literature, it is also called an X-ray. It is known that a function in with compact support in D is uniquely determined if X-rays for all t and θ are given (see [1,2]). Hence reconstructing u from full or partial knowledge of is very useful, because it plays an important role in the computer tomography with many applications in medicine, radiology, geology, etc. In practice, since only finite data set is available, one wishes to approximate the function from a large collection of its X-rays. Therefore, it is natural to study the problem of determining a polynomial from its finite Radon projections. Such approach has been considered in [1,3–5].
Topological and hydrodynamic analyses of solar thermochemical reactors for aerodynamic-aided window protection
Published in Engineering Applications of Computational Fluid Mechanics, 2022
Bo Wang, Alireza Rahbari, Morteza Hangi, Xian Li, Chi-Hwa Wang, Wojciech Lipiński
Stagnation points in the vicinity of the window are crucial areas where the contamination takes place due to the lower fluid velocities in these regions. In order to protect the window, the number of stagnation points near the window should be minimized. From the perspective of differential topology, the solar reactor windows, either flat or spherical in geometry, have the same topology as a disk. A stagnation point in an actual flow field can be regarded as a singular point (a point where the vector field is not continuous) in an ideal vector field. Thus, the problem of minimizing the number of stagnation points near the window can be interpreted as the problem of minimizing the number of singular points in a 2D vector field on a disk.