trying to understand contravariance

Chestermiller

Here is another example for you to consider.

Assume that you have a flat compact disc (CD), and that the CD lies within the horizontal x-y plane of a rectangular cartesian coordinate system x-y-z. The axis of the CD coincides with the z-axis of the cartesian coordinate system, and the CD is rotating as a rigid body about its axis (relative to the x-y-z coordinate system) with a constant angular velocity of ω. Each material particle of the CD travels in a perfect circle about the z-axis. Assume that there is also a cylindrical polar coordinate system (r-θ-z) present, that coincides with the x-y-z coordinate system.

As reckoned from the r-θ-z coordinate system, the velocities of the various material particles comprising the CD are given by:

V = (ωr) i_θ = ω (r i_θ) = ω e_θ

where e_θ is the coordinate basis vector in the θ direction:

e_θ = r i_θ

According to the equations above, even though the velocity vector V varies with r and θ, its contravariant component in the r-direction is zero, and its contravariant component in the θ-direction is a constant, and equal to ω:

V^r = 0

V^θ = ω

Note that the velocity V is a vector tangent to the circles (trajectories) around which the material particles are traveling.

Now use your transformation formula to show that, in cartesian coordinates, the contravariant components of the velocity V are given by:

V^x = -ωy

V^y = +ωx

so that

V = -ωy i_x + ωx i_y

cin-bura

all right, i think i got it now
thank you everyone!!