Physics 197 Lecture 25: Vector Calculus

• School: WUSTL
• Department: Physics
• Course: Physics 197
• Professor: Ferrar

Document Summary

Time derivative: df/dt = lim as (cid:791)t 0 [f(t + (cid:791)t) - f(t)]/(cid:791)t. (cid:791)(vector q) = (vector q)(t + (cid:791)t) - (vector q)(t: (cid:791)(vector q) = [qx(t + (cid:791)t) - qx(t)] D(vector q)/dt: lim as (cid:791)t 0 ((cid:791)vector q/(cid:791)t) = lim as (cid:791)t 0 (1/(cid:791)t) [qx(t + (cid:791)t) - qx(t)] [qz(t + (cid:791)t) - qz(t): d(vector q)/dt = [dqx/dt] Vector v = d(vector r)/dt = small displacement/short time interval. Instantaneous velocity: vector v = lim as (cid:791)t 0 [(vector r)(t + (cid:791)t) - vector r(t)]/(cid:791)t. Vector v = d(vector r)/dt = lim as (cid:791)t 0 (1/(cid:791)t) [x(t + (cid:791)t) - x(t)] = [dx/dt] !1 (cid:791)t, where (cid:791)t is some nite interval of time; d(vector r)/dt, dx/dt, dy/dt, dz/dt now of cially represent time derivatives. X-velocity: vx =dx/dt (same for all directional components) V = mag(vector v) = (vx^2 + vy^2 + vz^2)^1/2 = ((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2)^1/2.