This differential geometry lesson defines the speed of a parameterized curve as the magnitude (norm) of the velocity vector, s = |v(t)| = |dr/dt|, distinguishing speed as a scalar quantity from velocity as a vector with direction, and derives this via the position vector r(t), its derivative (the velocity vector), and the vector's length. It then draws a formal distinction between physics and differential geometry: in physics, r(t) describes an object's motion through time and yields velocity, speed, and acceleration; in differential geometry, the same derivative operations define intrinsic, purely geometric properties of a curve (tangent vector, its norm, curvature) that exist independent of time, mass, or force, with true velocity/speed/acceleration only meaningful when the parameter is specifically interpreted as time.
This differential geometry lesson defines the speed of a parameterized curve as the magnitude (norm) of the velocity vector, s = |v(t)| = |dr/dt|, distinguishing speed as a scalar quantity from veloc…