In multivariable calculus, motion in the plane can be described using a moving orthonormal frame of unit vectors, u_r and u_θ, defined relative to polar coordinates rather than the fixed Cartesian frame I and J. Since u_r (in the direction of the radius vector) and u_θ (its 90° rotation) vary with θ, differentiating the position vector R = r·u_r with respect to time via the product and chain rules yields velocity and acceleration expressed in u_r/u_θ components — independent of any physical reasoning. This formulation belongs to vector calculus / kinematics of planar motion, paralleling Cartesian (I, J) and tangential-normal (T, N) decompositions of velocity and acceleration, and is particularly significant for analyzing central force fields, where the acceleration's u_θ-component vanishes.
In multivariable calculus, motion in the plane can be described using a moving orthonormal frame of unit vectors, u_r and u_θ, defined relative to polar coordinates rather than the fixed Cartesian fr…