- Curvilinear Coordinates, Vectors, Operators, and Differential Relations

Parabolic partial differential equations used in chemical engineering for the description of mass transport and heat transfer and analogous relationship derived in stochastic processes theory are given. A standard transformation procedure is applied, allowing these relations to be generally written in curvilinear coordinates and particular expressions for cylindrical and spherical coordinates to be derived. The relation between the probability density for the position of a discernible particle and the concentration of a set of such particles is discussed.

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The Bateman Functions Revisited after 90 Years—A Survey of Old and New Results

Mathematics ◽

10.3390/math9111273 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1273

Author(s):

Alexander Apelblat ◽

Armando Consiglio ◽

Francesco Mainardi

Keyword(s):

Hypergeometric Function ◽

Laplace Transforms ◽

Confluent Hypergeometric Function ◽

Integral Function ◽

Basic Properties ◽

Differential Relations

The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman function is only mentioned as a particular case of the confluent hypergeometric function. In order to revive our knowledge on these functions, their basic properties (recurrence functional and differential relations, series, integrals and the Laplace transforms) are presented. Some new results are also included. Special attention is directed to the Bateman and Havelock functions with integer orders, to generalizations of these functions and to the Bateman-integral function known in the literature.

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Does general relativity highlight necessary connections in nature?

Synthese ◽

10.1007/s11229-020-03009-z ◽

2021 ◽

Author(s):

Antonio Vassallo

Keyword(s):

General Relativity ◽

Laws Of Nature ◽

Coupling Mechanism ◽

Einstein Field Equations ◽

Field Equations ◽

Bianchi Identities ◽

Physical Role ◽

General Relativistic ◽

Differential Relations ◽

Necessary Connections

AbstractThe dynamics of general relativity is encoded in a set of ten differential equations, the so-called Einstein field equations. It is usually believed that Einstein’s equations represent a physical law describing the coupling of spacetime with material fields. However, just six of these equations actually describe the coupling mechanism: the remaining four represent a set of differential relations known as Bianchi identities. The paper discusses the physical role that the Bianchi identities play in general relativity, and investigates whether these identities—qua part of a physical law—highlight some kind of a posteriori necessity in a Kripkean sense. The inquiry shows that general relativistic physics has an interesting bearing on the debate about the metaphysics of the laws of nature.

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The Differential Relations Between ADHD and Reading Comprehension: A Quantile Regression and Quantile Genetic Approach

Behavior Genetics ◽

10.1007/s10519-021-10077-5 ◽

2021 ◽

Author(s):

Jeffrey A. Shero ◽

Jessica A. R. Logan ◽

Stephen A. Petrill ◽

Erik Willcutt ◽

Sara A. Hart

Keyword(s):

Reading Comprehension ◽

Quantile Regression ◽

Genetic Approach ◽

Differential Relations

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XVI.—On Triple Systems of Surfaces and Non-Orthogonal Curvilinear Coordinates

Proceedings of the Royal Society of Edinburgh ◽

10.1017/s0370164600021982 ◽

1927 ◽

Vol 46 ◽

pp. 194-205 ◽

Cited By ~ 3

Author(s):

C. E. Weatherburn

Keyword(s):

Curvilinear Coordinates ◽

Triple Systems ◽

Orthogonal Curvilinear Coordinates ◽

Orthogonal Systems ◽

Considerable Detail

The properties of “triply orthogonal” systems of surfaces have been examined by various writers and in considerable detail; but those of triple systems generally have not hitherto received the same attention. It is the purpose of this paper to discuss non-orthogonal systems, and to investigate formulæ in terms of the “oblique” curvilinear coordinates u, v, w which such a system determines.

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A New Nonlinear Higher-Order Shear Deformation Theory for Nonlinear Vibrations of Laminated Shells

Volume 8: Dynamic Systems and Control, Parts A and B ◽

10.1115/imece2010-39483 ◽

2010 ◽

Author(s):

M. Amabili ◽

J. N. Reddy

Keyword(s):

Shear Deformation ◽

Deformation Theory ◽

Nonlinear Vibrations ◽

Circular Cylindrical Shell ◽

Shear Deformation Theory ◽

Higher Order ◽

Curvilinear Coordinates ◽

Geometrically Nonlinear ◽

Geometric Imperfections ◽

Laminated Shells

A consistent higher-order shear deformation nonlinear theory is developed for shells of generic shape; taking geometric imperfections into account. The geometrically nonlinear strain-displacement relationships are derived retaining full nonlinear terms in the in-plane displacements; they are presented in curvilinear coordinates in a formulation ready to be implemented. Then, large-amplitude forced vibrations of a simply supported, laminated circular cylindrical shell are studied (i) by using the developed theory, and (ii) keeping only nonlinear terms of the von Ka´rma´n type. Results show that inaccurate results are obtained by keeping only nonlinear terms of the von Ka´rma´n type for vibration amplitudes of about two times the shell thickness for the studied case.

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