Sample Midterm Problem 1 edit

Given:

The vectors  ,  , and   are given, with respect to an orthonormal basis  , by

 

Find:

  • (a) Evaluate  .
  • (b) Evaluate  . Is   a tensor? If not, why not? If yes, what is the order of the tensor?
  • (c) Name and define   and  .
  • (d) Evaluate  .
  • (e) Show that  .
  • (f) Rotate the basis   by 30 degrees in the counterclockwise direction around   to obtain a new basis  . Find the components of the vector   in the new basis  .
  • (g) Find the component   of   in the new basis  .

Solution edit

Part (a) edit

 
 

Part (b) edit

 
 
 

Part (c) edit

 
 
 
 

Part (d) edit

 
 

Part (e) edit

 

Because   cannot be an even or odd permutation of  .

Part (f) edit

The basis transformation rule for vectors is

 

where

 

Therefore,

 

Hence,

 

Thus,

 

Part (g) edit

The basis transformation rule for second-order tensors is

 

Therefore,