715 What is the covariant derivative of the covariant and contravariant forms

# 715 what is the covariant derivative of the covariant

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7.15 What is the covariant derivative of the covariant and contravariant forms of the metric tensor for an arbitrary type of coordinate system? How is this related to the fact that the covariant derivative operator bypasses the index-shifting operator? 7.16 Which rules of ordinary differentiation apply equally to covariant differentiation and which do not? Make mathematical statements about all these rules with sufficient explanation of the symbols and operations involved. 7.17 Make corrections, where necessary, in the following equations explaining in each case why the equation should or should not be amended:
7.1 Exercises 161 ( C ± D ) ; i = ; i C ± ; i D ; i ( AB ) = B ( ; i A ) + A ; i B ( g ij A j ) ; m = g ij ( A j ) ; m ; i ; j = ; j ; i i j = j i 7.18 How do you define the second and higher order covariant derivatives of tensors? Do these derivatives follow the same rules as the ordinary partial derivatives of the same order in the case of different differentiation indices? 7.19 From the Index, find all the terms that refer to symbols used in the notation of ordinary partial derivatives and covariant derivatives.
References G.B. Arfken; H.J. Weber; F.E. Harris. Mathematical Methods for Physicists A Compre- hensive Guide . Elsevier Academic Press, seventh edition, 2013. R.B. Bird; R.C. Armstrong; O. Hassager. Dynamics of Polymeric Liquids , volume 1. John Wiley & Sons, second edition, 1987. R.B. Bird; W.E. Stewart; E.N. Lightfoot. Transport Phenomena . John Wiley & Sons, second edition, 2002. M.L. Boas. Mathematical Methods in the Physical Sciences . John Wiley & Sons Inc., third edition, 2006. C.F. Chan Man Fong; D. De Kee; P.N. Kaloni. Advanced Mathematics for Engineering and Science . World Scientific Publishing Co. Pte. Ltd., first edition, 2003. T.L. Chow. Mathematical Methods for Physicists: A concise introduction . Cambridge University Press, first edition, 2003. J.H. Heinbockel. Introduction to Tensor Calculus and Continuum Mechanics . 1996. D.C. Kay. Schaum’s Outline of Theory and Problems of Tensor Calculus . McGraw-Hill, first edition, 1988. K.F. Riley; M.P. Hobson; S.J. Bence. Mathematical Methods for Physics and Engineering . Cambridge University Press, third edition, 2006. D. Zwillinger, editor. CRC Standard Mathematical Tables and Formulae . CRC Press, 32nd edition, 2012. 162
Index Absolute differential calculus, 9 tensor, 68, 69, 80 Active transformation, 52 Addition/Subtraction of tensors, 69, 83, 84, 92, 94 Adjoint matrix, 39 Adjugate matrix, 39 Algebraic sum, 50, 86, 88, 158 Alternating tensor, 96 Anisotropic tensor, 70, 80 Anti- commutative, 26, 28 diagonal, 35 symmetric tensor, 59, 70–73, 81, 82, 96 symmetry of tensor, 73, 81, 114 Arithmetic, 1 Associate metric tensor, 151, 154 Associative, 29–32, 37, 42, 43, 46, 84, 85, 92 Axial tensor, 65 vector, 68, 79 Axis of coordinates, 16, 19, 51, 52, 59, 63, 66, 70, 101 Bar notation, 13, 23 Barred symbol, 51, 60–63, 69 Basis set, 26, 63, 103, 105 vector, 12, 16–22, 26, 41, 61, 156, 160 Bianchi, 10 identities, 10 Bound index, 50 Calculus, 1 Cartesian coordinate system, 7, 12, 16, 17, 41, 121, 152, 154, 155 Christoffel, 9 symbol, 9, 155–157, 159 Circulation of field, 31 Cofactor, 38, 39, 45 Column of matrix, 28, 35–38, 108, 114, 117 Comma notation, 13, 160 Commutative, 15, 30–32, 37, 43, 46, 84–86, 88, 92, 93 Component notation, 49 of metric tensor, 150 of tensor, 12, 13, 47–49, 53, 56, 64, 70, 75, 81, 83–85, 96, 97, 104, 114 of vector, 13, 17, 23, 28, 31, 66, 123, 133

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• Summer '20
• Rajendra Paramanik
• Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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