Line integral

Summary

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as W = \mathbf{F} \cdot \mathbf{s}, have natural continuous analogues in terms of line integrals, in this case W =

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This paper presents a novel method for solving partial differential equations on three-dimensional CAD geometries by means of immersed isogeometric discretizations that do not require quadrature schemes. It relies on a newly developed technique for the evaluation of polynomial integrals over spline boundary representations that is exclusively based on analytical computations. First, through a consistent polynomial approximation step, the finite element operators of the Galerkin method are transformed into integrals involving only polynomial integrands. Then, by successive applications of the divergence theorem, those integrals over B-Reps are transformed into the first surface and then line integrals with polynomials integrands. Eventually, these line integrals are evaluated analytically with machine precision accuracy. The performance of the proposed method is demonstrated by means of numerical experiments in the context of 2D and 3D elliptic problems, retrieving optimal error convergence order in all cases. Finally, the methodology is illustrated for 3D CAD models with an industrial level of complexity.

SPRINGER2022

We introduce the analog of Kramers-Kronig dispersion relations for correlators of four scalar operators in an arbitrary conformal field theory. The correlator is expressed as an integral over its "absorptive part", defined as a double discontinuity, times a theory-independent kernel which we compute explicitly. The kernel is found by resumming the data obtained by the Lorentzian inversion formula. For scalars of equal scaling dimensions, it is a remarkably simple function (elliptic integral function) of two pairs of cross-ratios. We perform various checks of the dispersion relation (generalized free fields, holographic theories at tree-level, 3D Ising model), and get perfect matching. Finally, we derive an integral relation that relates the "inverted" conformal block with the ordinary conformal block.

2020

We consider the general problem of describing the dynamics of subnetworks of larger biochemical reaction networks, e.g., protein interaction networks involving complex formation and dissociation reactions. We propose the use of model reduction strategies to understand the "extrinsic" sources of stochasticity arising from the rest of the network. Our approaches are based on subnetwork dynamical equations derived by projection methods and path integrals. The results provide a principled derivation of different components of the extrinsic noise that is observed experimentally in cellular biochemical reactions, over and above the intrinsic noise from the stochasticity of biochemical events in the subnetwork. We explore several intermediate approximations to assess systematically the relative importance of different extrinsic noise components, including initial transients, long-time plateaus, temporal correlations, multiplicative noise terms, and nonlinear noise propagation. The best approximations achieve excellent accuracy in quantitative tests on a simple protein network and on the epidermal growth factor receptor signaling network. Published under license by AIP Publishing.

AMER INST PHYSICS2020