212 Describe direct and inverse coordinate transformations between spaces and

212 describe direct and inverse coordinate

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2.12 Describe direct and inverse coordinate transformations between spaces and write the generic equations for these transformations.
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2.7 Exercises 75 2.13 What are proper and improper transformations? Draw a simple sketch to demonstrate them. 2.14 Define the following terms related to permutation of indices: permutation, even, odd, parity, and transposition. 2.15 Find all the permutations of the following four letters assuming no repetition: ( i, j, k, l ). 2.16 Give three even permutations and three odd permutations of the symbols ( α, β, γ, δ ) in the stated order. 2.17 Discuss all the similarities and differences between free and dummy indices. 2.18 What is the maximum number of repetitive indices that can occur in each term of a legitimate tensor expression? 2.19 How many components are represented by each one of the following assuming a 4D space? A jk i f + g C mn - D nm 5 D k + 4 A k = B k 2.20 What is the “summation convention”? To what type of indices this convention applies? 2.21 Is it always the case that the summation convention applies when an index is repeated? If not, what precaution should be taken to avoid ambiguity and confusion? 2.22 In which cases a pair of dummy indices should be of different variance type (i.e. one upper and one lower)? In what type of coordinate systems these repeated indices can be of the same variance type and why? 2.23 What are the rules that the free indices should obey when they occur in the terms of tensor expressions and equalities?
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2.7 Exercises 76 2.24 What is illegitimate about the following tensor expressions and equalities considering in your answer all the possible violations? A ij + B k ij C n - D n = B m A i j = A j i A j = f 2.25 Which of the following tensor expressions and equalities is legitimate and which is illegitimate? B i + C ij j A i - B k i C m + D m = B m mm B i k = A i k State in each illegitimate case all the reasons for illegitimacy. 2.26 Which is right and which is wrong of the following tensor equalities? n A n = A n n [ B ] k + [ D ] k = [ B + D ] k ab = ba A ij M kl = M kl A ji Explain in each case why the equality is right or wrong. 2.27 Choose from the Index six entries related to the general rules that apply to the math- ematical expressions and equalities in tensor calculus. 2.28 Give at least two examples of tensors used in mathematics, science and engineering for each one of the following ranks: 0 , 1 , 2 and 3. 2.29 State the special names given to the rank-0 and rank-1 tensors. 2.30 What is the difference, if any, between rank-2 tensors and matrices? 2.31 Is the following statement correct? If not, re-write it correctly: “all rank-0 tensors are vectors and vice versa, and all rank-1 tensors are scalars and vice versa”.
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2.7 Exercises 77 2.32 Give clear and detailed definitions of scalars and vectors and compare them. What is common and what is different between the two? 2.33 Make a simple sketch of the nine unit dyads associated with the double directions of rank-2 tensors in a 3D space.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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