Christoffel symbols

Christoffel symbols are a set of real numbers that describe the effects of parallel transportation over curved surfaces in mathematical space coordinates. They represent a measure of the curvature of the curvilinear coordinate axes and play an important role in tensor calculus. The calculation of Christoffel symbols can be geometrized by vanishing the covariant derivative of the mass tensor, as noticed by Gregorio Ricci-Curbastro.From: Illustrated Encyclopedia of Applied and Engineering Physics [2017], Formulation of 3D finite elements using curvilinear coordinates [2022], Environmental Fluid Mechanics [2001], Optimal controller applied to robotic systems using covariant control equations [2022]

Calculus on Manifolds

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Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018

Mattias Blennow

Based on this embedding, compute the following: The components gab of the induced metric on the torus resulting from the standard Euclidean metric in ℝ3.The Christoffel symbols of the Levi-Civita connection.The independent non-zero component Rθφθφ of the curvature tensor.The components of the Ricci tensor Rab and the Ricci scalar R. Verify that R is not constant and that it has a different sign on different points in the manifold.

Optimal controller applied to robotic systems using covariant control equations

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Published in International Journal of Control, 2022

Juan Antonio Rojas-Quintero, Juan Antonio Rojas-Estrada, Jorge Villalobos-Chin, Victor Santibañez, Eusebio Bugarin

As noticed by Gregorio Ricci-Curbastro, the formulae (3) and (4), allowing the calculations of the Christoffel symbols of the first and second kind, can be geometrised by vanishing the covariant derivative of the mass tensor (Bianchi, 1902; Grinfeld, 2013; Levi-Cività & Persico, 1925; Lovelock & Rund, 1975; Ricci-Curbastro & Levi-Cività, 1900). Therefore The same property holds for the inverse mass tensor, and

Formulation of 3D finite elements using curvilinear coordinates

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Published in Mechanics of Advanced Materials and Structures, 2022

Maria Cinefra

This quantity represents the Christoffel three-index symbol of the second kind. The Christoffel symbols are a measure of the curvature of the curvilinear coordinate axes, and play an important role in tensor calculus. From Eqs. (6) and (8), we obtain the derivative of the contravariant base vector with respect to as follows:

Christoffel symbols

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