e one upper and one lower since they are dummy indices and hence the position

E one upper and one lower since they are dummy

This preview shows page 64 - 66 out of 171 pages.

, i.e. one upper and one lower, since they are dummy indices and hence the position of the index of the unbarred coordinate should be opposite to its position in the unbarred tensor, (ii) an upper index in the denominator is in lieu of a lower index in the numerator, and (iii) the order of the coordinates should match the order of the indices in the tensor: ¯ A ijkl = x n ¯ x i x p ¯ x j x q ¯ x k x r ¯ x l A npqr ¯ A ijkl = ¯ x i x n ¯ x j x p ¯ x k x q ¯ x l x r A npqr (78) ¯ A ij kl = ¯ x i x n ¯ x j x p x q ¯ x k x r ¯ x l A np qr We also replaced the “ $ ” sign in the final set of equations with the strict equality sign “=” as the equations now are complete. The covariant and contravariant types of a tensor are linked through the metric tensor , as will be detailed later in the book (refer to § 6). As indicated before, for orthonormal Cartesian systems there is no difference between covariant and contravariant tensors, and hence the indices can be upper or lower although it is common to use lower indices in this case. A tensor of m contravariant indices and n covariant indices may be called type ( m, n ) tensor. When one or both variance types are absent, zero is used to refer to the absent variance type in this notation. Accordingly, A k ij is a type ( 1 , 2 ) tensor, B ik is a type ( 2 , 0 ) tensor, C m is a type ( 0 , 1 ) tensor, and D ts pqr is a type ( 2 , 3 ) tensor. The vectors providing the basis set for a coordinate system are of covariant type when they are tangent to the coordinate axes, and they are of contravariant type when they are perpendicular to the local surfaces of constant coordinates. These two sets, like the
Image of page 64
2.6.1 Covariant and Contravariant Tensors 64 tensors themselves, are identical for orthonormal Cartesian systems. Formally, the covariant and contravariant basis vectors are given respectively by: E i = r ∂x i E i = x i (79) where r = x i e i is the position vector in Cartesian coordinates and x i is a general curvi- linear coordinate. As before, a superscript in the denominator of partial derivatives is equivalent to a subscript in the numerator . It should be remarked that in general the basis vectors (whether covariant or contravariant ) are not necessarily of unit length and/or mutually orthogonal although they may be so. [32] The two sets of covariant and contravariant basis vectors are reciprocal systems and hence they satisfy the following reciprocity relation : E i · E j = δ j i (80) where δ j i is the Kronecker delta (refer to § 4.1) which can be represented by the unity matrix (see § Special Matrices). The reciprocity of these two sets of basis vectors is illustrated schematically in Figure 14 for the case of a 2D space. A vector can be represented either by covariant components with contravariant coor- dinate basis vectors or by contravariant components with covariant coordinate basis vectors. For example, a vector A can be expressed as: A = A i E i or A = A i E i (81) where E i and E i are the contravariant and covariant basis vectors respectively. This is illustrated graphically in Figure 15 for a vector A in a 2D space. The use of the covariant
Image of page 65
Image of page 66

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