Grad 1.  (a)  Show that (x y c_x + y^2 c_y, -x^2 c_x - x y c_y)

              forms a vector.  The quantities c_x and c_y are
  
              components of a constant vector (c_x, c_y).

         (b)  Repeat for (x y c_x - x^2 c_y, y^2 c_x- x y c_y).


SP1.    Calculate the integral of the scalar product of the 
        position vector r with the surface area element dS
        over an arbitrary closed surface of total volume V.

SP2.    Given a vector A defined by its components:

        A_x = (x-a)/[(x-a)^2 + y^2] - x/[x^2 + y^2]

        A_y = y/[x^2 + y^2] - y/[(x-a)^2+ y^2]

        A_z = 0


        Find the curl of the vector A and verify the result
        at the origin (r = 0) using Stokes' Theorem.

SP3.    For parabolic coordinates  (lambda, mu, phi), defined by

        x = lambda * mu * cos(phi)

        y = lambda * mu * sin(phi)

        z = (lambda^2 - mu^2)/2, 

        calculate the scale factors and direction cosines in

        terms of both the parabolic coordinates and the 

        rectangular coordinates.

Grad 2.  Consider a scalar function psi(lambda, mu, phi) in

         the parabolic coordinates of problem SP3 which has

         the form  psi = sqrt[lambda^2 + mu^2] + tan[phi].

         Calculate the gradient of the scalar function psi.

SP4.     Verify that Eq. 2.16 reduces to 

         V = h_1 h_2 h_3 dq_1 dq_2 dq_3 in the case of

         3-d spherical coordinates, (q_1, q_2, q_3 = r, theta, phi).

Grad 3.  (5.5.1 in text)

Grad 4.  (6.6.5 in text)

Grad 5.  (8.3.6 in text)