Gotowa bibliografia na temat „Intrinsic field tensors”

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as wel

Stress is a tensor quantity that describes the mechanical force density (force per unit area) on the complete surface of a domain inside a material body. A stress exists wherever one part of a body exerts a force on neighboring parts. Its orientation is not tied to any particular directions that are intrinsic to the material like, say, crystallographic axes. It is thus distinct from the matter tensors that were discussed in the preceding chapter, all of which have definitive orientations within a crystal or other anisotropic material; it is called a field tensor (and so is strain). The definition of stress depends on the concept of a continuum. Let f(xi) be a single-valued function defined for every point xi in a region. This function is said to be continuous at the point xi if the following holds for all paths of approach of xi to °xi:. . . f(xi) → f(°xi) as xi → °xi (4.1)· . . . Equivalently, for any number ∊, no matter how small, there exists a neighborhood of nonzero radius around the point xi in which: . . . . 〈f(xi) − f(°xi)〉2 < ∊,­ (4.2) . . . for all points xi in that neighborhood. A continuum is an idealized material whose physical attributes are continuous functions of position. Thus neighboring points remain neighbors, and a continuum cannot have gaps or jumps (discontinuities) in its properties. Surfaces bounding gaps or defining discontinuities must be specially treated in continuum mechanics. Examples are surfaces between two fluids of differing density or viscosity, or between solids with different thermal conductivity or elastic properties. Real materials are never continua; they are discontinuous at the atomic scale, and often at larger scales as well. The notion of a continuum is, therefore, only a macroscopic approximation, but it allows useful mathematical approaches to the treatment of real phenomena.

The intrinsic curvature of a metric space is captured by the Riemann curvature tensor, which can be contracted to the Ricci tensor and the Ricci scalar. Einstein took these curvature quantities and constructed the Einstein field equations that relate the curvature of space-time to energy and mass density. For an isotropic density, a solution to the field equations is the Schwarzschild metric, which contains mass terms that modify both the temporal and the spatial components of the invariant element. Consequences of the Schwarzschild metric include gravitational time dilation, length contraction, and redshifts. Trajectories in curved space-time are expressed as geodesics through the Schwarzschild metric space. Solutions to the geodesic equation lead to the precession of the perihelion of Mercury and to the deflection of light by the Sun.

The real and perceived conflicts among economic growth and ecological protection define one of the central tensions in environmental policy. The premise of the concept of a green economy is that it is possible at some level to transform this assumed, traditional zero-sum into a positive relationship. Although its intellectual origins may be traced to the fields of ecological economics, business greening, and ecological modernization, the green economy concept gained particular visibility in the wake of the 2009 financial crisis. This chapter argues that the green economy concept may be influential both in policy framing and as a framework for guiding and carrying out economic and political change. Three modifications in treatments of the concept would make it more acceptable and relevant for future policy debates: rethinking the purpose of growth as well as how it is achieved; shedding more light on economic inequality and its ecological consequences; and emphasizing the intrinsic as well as the utilitarian value of ecological assets and services.

Non-contact geomagnetic anomaly detection, as one of passive non-destructive testing (NDT) techniques, can be used to locate pipeline defects, while its accuracy is affected by random noise and detection orientation. In order to extract effective geomagnetic anomaly signals of pipeline defects, a method based on empirical mode decomposition (EMD) and magnetic gradient tensor was studied. In order to filter random noise, EMD was performed to self-adaptively decompose magnetic field signals into a series of intrinsic mode functions (IMFs), and then Hurst exponent was implemented to exclude false

In contrast to the classical local and nonlocal theories, the peridynamic equation of motion introduced by Silling (J. Mech. Phys. Solids 2000; 48: 175–209) is free of any spatial derivatives of displacement. The new general integral equations (GIE) connecting the displacement fields in the point being considered and the surrounding points of random structure composite materials (CMs) is proposed. For statistically homogeneous thermoperistatic media subjected to homogeneous volumetric boundary loading, one proved that the effective behaviour of this media is governing by conventional effective