Learning Materials
Features
Discover
Chapter 10: Q12P (page 534)
Show that in a general coordinate system with variables x1, x2, x3, the contravariant basis vectors are given by
Hint:Write the gradient in terms of its covariant components and the basis
vectors to getand let .
The equation is proved.
Over 30 million students worldwide already upgrade their learning with 91影视!
All the tools & learning materials you need for study success - in one app.
Get started for free
As in (4.3) and (4.4), find the y and z components of (4.2) and the
other 6 components of the inertia tensor. Write the corresponding components
of the inertia tensor for a set of masses or an extended body as in (4.5).
Verify Hints: In Figure , consider the projection of the slanted face of area onto the three unprimed coordinate planes. In each case, show that the projection angle is equal to an angle between the axis and one of the unprimed axes. Find the cosine of the angle from the matrix A in .
Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions.In cylindrical coordinates .
Following what we did in equations (2.14) to (2.17), show that the direct product of a vector and a -rank tensor is a -rank tensor. Also show that the direct product of two -rank tensors is a -rank tensor. Generalize this to show that the direct product of two tensors of ranks m and n is a tensor of rank m + n .
Show that the nine quantities (which are the Cartesian components of where V is a vector) satisfy the transformation equations for a Cartesian -rank tensor. Show that they do not satisfy the general tensor transformation equations as in . Hint: Differentiate orpartially with respect to, say,. You should get the expected terms [as in ] plus some extra terms; these extraneous terms show that is not a tensor under general transformations. Comment: It is possible to express the components of correctly in general coordinate systems by taking into account the variation of the basis vectors in length and direction.
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine