256 Are the covariant and contravariant forms of a specific tensor A represent

# 256 are the covariant and contravariant forms of a

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2.56 Are the covariant and contravariant forms of a specific tensor A represent the same mathematical object? If so, in what sense they are equal from the perspective of different coordinate systems? 2.57 Correct, if necessary, the following statement: “A tensor of any rank ( 1 ) can be rep- resented covariantly using contravariant basis tensors of that rank, or contravariantly using contravariant basis tensors, or in a mixed form using a mixed basis of the same type”. 2.58 Make corrections, if needed, to the following equations assuming a general curvilinear coordinate system where, in each case, all the possible ways of correction should be considered: B = B i E i M = M ij E i D = D i E i E j C = C i E j F = F n E n T = T rs E s E r 2.59 What is the technical term used to label the following objects: E i E j , E i E j , E i E j and E i E j ? What they mean? 2.60 What sort of tensor components that the objects in the previous question should be associated with? 2.61 What is the difference between true and pseudo vectors? Which of these is called axial and which is called polar? 2.62 Make a sketch demonstrating the behavior of true and pseudo vectors.
2.7 Exercises 80 2.63 Is the following statement correct? “The terms of tensor expressions and equations should be uniform in their true and pseudo type”. Explain why. 2.64 There are four possibilities for the direct product of two tensors of true and pseudo types. Discuss all these possibilities with respect to the type of the tensor produced by this operation and if it is true or pseudo. Also discuss in detail the cross product and curl operations from this perspective. 2.65 Give examples for the true and pseudo types of scalars, vectors and rank-2 tensors. 2.66 Explain, in words and equations, the meaning of absolute and relative tensors. Do these intersect in some cases with true and pseudo tensors (at least according to some conventions)? 2.67 What “Jacobian” and “weight” mean in the context of absolute and relative tensors? 2.68 Someone stated: “ A is a tensor of type ( 2 , 4 , - 1 )”. What these three numbers refer to? 2.69 What is the type of the tensor in the previous exercise from the perspectives of lower and upper indices and absolute and relative tensors? What is the rank of this tensor? 2.70 What is the weight of a tensor A produced from multiplying a tensor of weight - 1 by a tensor of weight 2? Is A relative or absolute? Is it true or not? 2.71 Define isotropic and anisotropic tensors and give examples for each using tensors of different ranks. 2.72 What is the state of the inner and outer products of two isotropic tensors? 2.73 Why if a tensor equation is valid in a particular coordinate system it should also be valid in all other coordinate systems under admissible coordinate transformations? Use the isotropy of the zero tensor in your explanation.
2.7 Exercises 81 2.74 Define “symmetric” and “anti-symmetric” tensors and write the mathematical condi- tion that applies to each assuming a rank-2 tensor.

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

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