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Physics of Important Developments That Predestined Graphene
Published in Andre U. Sokolnikov, Graphene for Defense and Security, 2017
The two parameters together constitute a parameterization of the elastic moduli for homogeneous isotropic media, popular in mathematical literature, and are thus related to the other elastic moduli, for instance, the bulk modulus can be expressed as Κ=λ+(2/3)μ. Although the sheer modulus, µ must be positive, the Lame’s first parameter, x, can be negative, in principle; however, for most materials, it is also positive. Gabrial Leon Jean Baptiste Lame (1795 – 1870) was a French mathematician who contributed to the theory of partial differential equations. He made use of curvilinear coordinates along with the elasticity theory. The Lame parameters (Lame constants) are λ, the Lame’s first parameter and μ, the second parameter, also called the dynamic viscosity or sheer modulus. A metric tensor is a type of function that has input of tangent vectors v and w that produce a scalar (real number) g(v, w) that generalize the properties of the dot product of vectors in Euclidian space.
Tensors In General Coordinates
Published in C. Young Eutiquio, Vector and Tensor Analysis, 2017
The covariant and contravariant components of a tensor are related to each other through the metric tensor. Here we derive the relationship for tensors of the first order or vectors. Let us consider a vector A = ajej = ajej represented by its covariant as well as contravariant components in a coordinate system defined by the basis ei (1 ≤i≤3). Taking the dot product of the vector with the base vector ei, noting that ej· ek = δkj, we find
Celestial Mechanics and Astrodynamics
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
Physically, the metric tensor can be used to define the length of and the angle between tangent vectors on a manifold. Integration of the metric tensor allows us to calculate the length of curves on the manifold. In Euclidean space, the components of the metric tensor can be found by dot product between the tangent vectors. Riemann geometry is positive definite in the sense that gαα>0 $$ g_{{\alpha \alpha }}> 0 $$
Black hole entropy, the black hole information paradox, and time travel paradoxes from a new perspective
Published in Journal of Modern Optics, 2020
Our current view of nature is that it is composed entirely of quantum fields (although the proper description of quantum gravity is still undecided). All physics can be regarded as ultimately derived from an enormous path integral with the form where the action is now a functional of all fields over all spacetime. The gravitational field is described by a metric tensor . The other fields will be represented by . contains all physically distinct field configurations that are consistent with whatever boundary conditions are imposed.
Robust and efficient tool path generation for machining low-quality triangular mesh surfaces
Published in International Journal of Production Research, 2021
The last missing piece in converting geodesic computation to heat diffusion is to add the scallop metric to the above formulations. Loosely, a metric tensor like is a generalisation of the first fundamental form of the design surface, which equips the surface with new inner products on the tangent planes (Carmo 1976). Inner products are responsible for calculating magnitudes of vectors and the angle between two vectors. So to add to the above formulations, we simply need to re-express all the calculations about distances and angles involved in Equations (5)–(10) in terms of the scallop metric . To be specific, the length of a vector v is given by: The angle between two vectors is given by: An equivalent but more compact way for adding the scallop metric is to first decompose into and then apply the transformation, to each vector v in question. The inner product on v under the scallop metric is now equivalent to the usual inner product on , as justified by the identity: . Based on this transformation, all calculations in Equations (5)–(10) remain unchanged except that vectors are transformed using Equation (11). It should be noted that, as is positive definite, the decomposition is always possible.