MATH 32A Lecture Notes - Lecture 7: Osculating Plane, Osculating Circle, Parametric Equation
Document Summary
Standard notation for vvfs of curves: r(t) = position of some object at time t. v(t) = r"(t) = velocity. a(t) = r""(t) = acceleration. ||v|| = speed, magnitude of v(t), how fast the object is moving, is scalar. s(t) = integral from t o to t of ||v(u)||du = distance travelled along the curve from time t o to time t. It is a unit vector in the direction of the velocity, or the direction in which the object is moving, also is the instantaneous direction of motion. Recall: unit vector is vector divided its magnitude. Curvature: how curvy r(t) is, or how sharp it is. It is pretty much the change in the direction (not magnitude) of the velocity, so it is equal to t"(t). Curvature of a curve at a particular point is the rate of change of direction per unit of distance traveled along the curve.
