Some Coordinate Systems

1. Rectilinear or Euclidean: (x,y,z) Volume Element = dx dy dz

2. Cylindrical: (r, �, z) Relation to Rectilinear: x = r cos� , y = r sin� , z = z (For 2-d problems, use only r and �) Volume Element = r dr d� dz (Sometimes will see ' instead of r, and/or 1 instead of �)

3. Spherical: (r, �, 1) Relation to Rectili near: x = r sin� cos1 , y = r sin� sin1 ,

z = r cos� Volume Element = r2 sin� dr d� d1

4. Elliptical: (µ, �, 1) with baseline 2a With R = 2a, µ = (ra + rb)/R , � = (ra - rb)/R, 1 is rotation from the XZ-plane. Relation to Rectili near: x = a ( (µ2-1)(1-�2) )1/2 cos1 , y = a ( (µ2-1)(1-�2) )1/2 sin1 , z = aµ�

Volume Element = R3/8 (µ2 - �2) dµ d� d1

For 2-d problems, use only µ and �.

µ = constant describes an elli psoid� = constant describes a hyberboloid

These coordinate systems all have the property that at any point in space, the three directionvectors defined by keeping two of the coordinates fixed, and varying the third by an infinitesimalamount, are mutually orthogonal. This is a useful (but not at all necessary) property for acoordinate system.