SET 2

 

  1. Problem 1
    1. Write down the Laplace equation in spherical coordinates.
    2. Use separation of variables to find the equation that will generate the angular dependence. (You may assume that there is no f-dependence in order to simplify the mathematics.)
    3. Show that this is equivalent to Legendre’s equation by letting .

Legendre eq.       

 

  1. Consider two vectors in a plane  .
    1. Transform these vectors using the matrix formulation (show work) by rotating them in the plane through and angle  to find the new vectors .
    2. Show that the dot product doesn’t change

 

  1. Consider two vectors in 4-space 
    1. Use a Lorentz transformation to find the new vectors  in the boosted coordinate system. Show your work using matrix mathematics (column vectors, row vectors and 4x4 matrices)
    2. Show that the inner product

 

  1. In class we developed a matrix operator

Which could be used to produce a rotation through the angle  for the case of a two component object. How would I extend this so as to rotate a 3-d vector.