275 Do we have symmetricanti symmetric scalars or vectors If not why 276 Is it

275 do we have symmetricanti symmetric scalars or

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2.75 Do we have symmetric/anti-symmetric scalars or vectors? If not, why? 2.76 Is it the case that any tensor of rank > 1 should be either symmetric or anti- symmetric? 2.77 Give an example, writing all the components in numbers or symbols, of a symmetric tensor of rank-2 in a 3D space. Do the same for an anti-symmetric tensor of the same rank. 2.78 Give, if possible, an example of a rank-2 tensor which is neither symmetric nor anti- symmetric assuming a 4D space. 2.79 Using the Index in the back of the book, gather all the terms related to the symmetric and anti-symmetric tensors including the symbols used in their notations. 2.80 Is it true that any rank-2 tensor can be decomposed into a symmetric part and an anti-symmetric part? If so, write down the mathematical expressions representing these parts in terms of the original tensor. Is this also true for a general rank- n tensor? 2.81 What is the meaning of the round and square brackets which are used to contain indices in the indexed symbol of a tensor (e.g. A ( ij ) and B [ km ] n )? 2.82 Can the indices of symmetry/anti-symmetry be of different variance type? 2.83 Is it possible that a rank- n ( n > 2 ) tensor is symmetric/anti-symmetric with respect to some, but not all, of its indices? If so, give an example of a rank-3 tensor which is symmetric or anti-symmetric with respect to only two of its indices.
2.7 Exercises 82 2.84 For a rank-3 covariant tensor A ijk , how many possibilities of symmetry and anti- symmetry do we have? Consider in your answer total, as well as partial, symmetry and anti-symmetry. Is there another possibility (i.e. the tensor in neither symmetric nor anti-symmetric with respect to any pair of its indices)? 2.85 Can a tensor be symmetric with respect to some combinations of its indices and anti- symmetric with respect to other combinations? If so, can you give a simple example of such a tensor? 2.86 Repeat the previous exercise considering the additional possibility that the tensor is neither symmetric nor anti-symmetric with respect to another set of indices, i.e. it is symmetric, anti-symmetric and neither with respect to different sets of indices. [41] 2.87 A is a rank-3 totally symmetric tensor and B is a rank-3 totally anti-symmetric tensor. Write all the mathematical conditions that these tensors satisfy. 2.88 Justify the following statement: “For a totally anti-symmetric tensor, non-zero entries can occur only when all the indices are different”. Use mathematical, as well as descriptive, language in your answer. 2.89 For a totally anti-symmetric tensor B ijk in a 3D space, write all the elements of this tensor which are identically zero. Consider the possibility that it may be easier to find first the elements which are not identically zero, then exclude the rest. [42] [41] The best way to tackle this sort of exercises is to build a table or array of appropriate dimensions where the indexed components in symbolic or numeric formats are considered and a trial and error approach is used to investigate the possibility of creating such a tensor.

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

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