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

256 are the covariant and contravariant forms of a

This preview shows page 80 - 83 out of 171 pages.

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.
Image of page 80
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.
Image of page 81
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.
Image of page 82
Image of page 83

You've reached the end of your free preview.

Want to read all 171 pages?

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

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes
A+ icon
Ask Expert Tutors