Using this notation, r (t) takes the form, r ( t ) = r r ^ ^ + z ( t ) z ^ ^ . In general, the trajectory r (t) is not constrained to lie on a circular cylinder, so the radius R varies with time and the trajectory of the particle in cylindrical-polar coordinates becomes: r ( t ) = r ( t ) r ^ ^ + z ( t ) z ^ ^ . Where r, θ, and z might be continuously differentiable functions of time and the function notation is dropped for simplicity. The velocity vector v P is the time derivative of the trajectory r (t), which yields: v P = d d t ( r r ^ ^ + z z ^ ^ ) = v r ^ ^ + r ω ω θ θ ^ ^ + v z z ^ ^ = v ( r ^ ^ + θ θ ^ ^ ) + v z z ^ ^ .