Sample Midterm Problem 1

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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

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Part (a)

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Part (b)

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Part (c)

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Part (d)

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Part (e)

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Because   cannot be an even or odd permutation of  .

Part (f)

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The basis transformation rule for vectors is

 

where

 

Therefore,

 

Hence,

 

Thus,

 

Part (g)

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The basis transformation rule for second-order tensors is

 

Therefore,