Week 2 Lecture 3 Curvilinear Co-ordinates.

In the last lecture we learnt about the orthogonal curvilinear co-ordinate systems mainly the two dimensional Plane Polar and the three dimensional Spherical Polar and Cylindrical Polar Co-ordinate sytems. The important issue is that the SYMMETRY of the system decides the use of a specific co-ordinate system.


These systems are called orthogonal as the unit vectors always are mutually orthogonal and forms a Vector Triad ( that is a x b =c and cyclic permutations like i, j, k). They are called curvilinear because the constant co-ordinate surfaces intersect in curves. It is very important to understand that the unit vectors for the curvilinear co-ordinates systems are not constant vectors unlike (i, j, k)
and they have different directions dependent upon their locations.

Plane Polar Coordinates: This is used whenever there is planar circular symmetry in the system and given as (r, θ) where r is the radial distance from the origin and θ is the angle measured anticlockwise from the +ve direction of the x-axis.
The unit vectors are always in the direction of the increasing value of the corresponding coordinate. The best discussion of this coordinate system is in "Mechanics" by Klepner and Kolenkow which is used for PHY 102 N course.

Cylindrical Polar:  ( ρ,φ , z) where ρ is the radial distance of the point from the axis of a right circular cylinder about the z-axis, φ is the angle between the (x,z) plane and the plane containing the z axis and the point and z is the distance along the z-axis. The co-ordinate system may be easily understood as a stack of plane polar co-ordinates system ( ρ , φ ) along the z-axis. The unit vectors are in the direction of the increasing coordinates and are orthogonal vector triad but their directions are location dependent. The line element, area element and the volume element can be calculated from this and are discussed in Griffiths. Note that the area vector element depends on the location.

Spherical Polar: ( r, θ , φ ) where r is the radial distance from the origin or radius of a sphere centered on the origin and passing through the point in question. θ is the polar angle between the z-axis and the radial line passing through the point ( note that constant θ is a cone about the z-axis with the half angle θ ) and φ is the azimuthal angle between the plane containing the point and the (x, z) plane. The unit vectors are again direction dependent although they are an orthogonal vector triad. Note that this system may be thought of as two plane polar coordinate systems
(i) in (r, θ) with θ being measured from the z-axis on the plane containing the point and the z axis.
(ii) in (r sin θ, φ ) on the (x, y) plane. r sin θ is the projection of r on the (x, y) plane. Using this it is easy to obtain all the line area and volume elements.


You may also check the following links for quick references ( not all of it is relevant for us)
Polar Co-ordinates
Spherical and Cylindrical Co-ordinates