Math 603-601 - Fall 2002

Homework

Assignment 9

  1. Problem 8, pg. 101, in Borisenko and Tarapov.

  2. Problem 9, pg. 101, in Borisenko and Tarapov.

  3. Let q1=r, q2 =θ, and q3=φ be the radius, co-latitude, and azimuthal angle in spherical coordinates. Find the following.
    1. The basis vectors: e1, e2, and e3, in terms of {i,j,k}.
    2. The reciprocal basis vectors: e1, e2, e3, in terms of {i,j,k}.
    3. The metric tensor g and the square of the arc length, ds2

  4. Repeat the previous exercise for parabolic cylinder coordinates q1, q2, and q3, which are defined by the equations below.
    x = ½((q1)2 - (q2)2)
    y = q1q2
    z = q3

  5. Let q1, q2, and q3 be a set of generalized coordinates and let x1, x2, and x3 be rectangular coordinates.
    1. Show that [dx1 dx2 dx3]T = K[dq1 dq2 dq3]T, where
      K = [ [e1]B [e2]B [e3]B], B = {i,j,k}
    2. Show that the metric tensor g = KTK, and use the properties of determinants to show that
      G := det(g) = (det(K))2
    3. Use part b above to show that the volume dV of the parallelepiped with edges e1dq1, e2dq2, e3dq3 is given by
      dV = G½dq1dq2dq3
Due Thursday, 21 November.