   need help computing covariant derivative
  I understand essentially what the covariant derivative is and how it works, I'm just having a hard time solving for it.  I read the following example in a text boox.Given a vector field w = sinz*e1 + cosz*e2 + (x^2 + y^2)*e3 and a vector v = 3*e1 -2*e2 + e3,  find the coviariant derivative of w in the direction of v.Hence find ^vW, where the ^ is the upside down triangle.Anyway, the book said that the covariant derivative was cosz*e1 - sinz*e2 + (6x-4y)*e3.  I do not understand how you get this answer.  Can anyone help ?   
  This looks like just a plain ol' directional derivative to me, not a covariant derivative. The answer you're looking for is obtained as follows.grad w dot v = dw/dx * v.x + dw/dy * v.y + dw/dz * v.zdw/dx = 2x * e3dw/dy = 2y * e3dw/dz = cos z * e1 - sin z * e2dw/dx * v.x = 6x * e3dw/dy * v.y = -4y * e3dw/dz * v.z = cos z * e1 - sin z * e2grad w dot v = (6x - 4y) * e3 + cos z * e1 - sin z * e2;  Awsome, thanks.   