Search results for "Surface integral"
showing 6 items of 6 documents
Star-product approach to quantum field theory: The free scalar field
1990
The star-quantization of the free scalar field is developed by introducing an integral representation of the normal star-product. A formal connection between the Feynman path integral in the holomorphic representation and the star-exponential is established for the interacting scalar fields.
Calculations for a disk source and a general detector using a radiation vector potential
2008
A closed form expression for a radiation vector potential is derived for a generalized disk radiation source. By applying Stokes's theorem the surface integral for the radiation flux into a general detector is converted into a much simpler line integral of the vector potential around the edge of the detector. This line integral can be easily evaluated for general detector geometry and general location and angular orientation relative to the disk source. For a number of cases the line integral reduces to integrals of Bessel functions which give various generalizations of Ruby's formula. Explicit formulas and numerical results for the geometric efficiency are given for circular and elliptical…
Measuring the black hole spin direction in 3D Cartesian numerical relativity simulations
2015
We show that the so-called flat-space rotational Killing vector method for measuring the Cartesian components of a black hole spin can be derived from the surface integral of Weinberg's pseudotensor over the apparent horizon surface when using Gaussian normal coordinates in the integration. Moreover, the integration of the pseudotensor in this gauge yields the Komar angular momentum integral in a foliation adapted to the axisymmetry of the spacetime. As a result, the method does not explicitly depend on the evolved lapse $\ensuremath{\alpha}$ and shift ${\ensuremath{\beta}}^{i}$ on the respective time slice, as they are fixed to Gaussian normal coordinates while leaving the coordinate label…
The General Stokes’s Theorem
2012
Let ω be a differential form of degree k - 1 and class C 1 in a neighborhood of a compact regular k-surface with boundary M of class C 2. The general Stokes’s theorem gives a relationship between the integral of ω over the boundary of M and the integral of the exterior differential dω over M. It can be viewed as a generalization of Green’s theorem to higher dimensions, and it plays a role not unlike that of the fundamental theorem of calculus in an elementary course of analysis. Particular cases of the general Stokes’s theorem that are of great importance are the divergence theorem, which relates a triple integral with a surface integral and what we know as the classical Stokes’s theorem, w…
Indefinite integrals involving complete elliptic integrals of the third kind
2017
ABSTRACTA method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete …
The planar two-body problem for spheroids and disks
2021
We outline a new method suggested by Conway (2016) for solving the two-body problem for solid bodies of spheroidal or ellipsoidal shape. The method is based on integrating the gravitational potential of one body over the surface of the other body. When the gravitational potential can be analytically expressed (as for spheroids or ellipsoids), the gravitational force and mutual gravitational potential can be formulated as a surface integral instead of a volume integral, and solved numerically. If the two bodies are infinitely thin disks, the surface integral has an analytical solution. The method is exact as the force and mutual potential appear in closed-form expressions, and does not invol…