In multivariable calculus, the shape and speed of motion along a curve are properties intrinsic to the curve itself, independent of any coordinate system, and can be captured by the tangential-normal (and, in space, binormal) vector frame rather than fixed Cartesian basis vectors. The unit tangent vector T is defined as the derivative of the position vector with respect to arc length (dR/dS), computed practically as dR/dt divided by its magnitude; the unit normal N is the derivative of T with respect to arc length divided by its magnitude, giving dT/dS = κN, where curvature κ = |dφ/dS| measures the rate of change of the tangent direction. Velocity and acceleration decompose naturally into T and N components (V = (ds/dt)T; A = (d²s/dt²)T + κ(ds/dt)²N), and in three dimensions the binormal B = T × N completes an orthonormal frame whose rate of change (torsion, dB/dS) measures the curve's twist out of its osculating plane.
In multivariable calculus, the shape and speed of motion along a curve are properties intrinsic to the curve itself, independent of any coordinate system, and can be captured by the tangential-normal…