Vector Functions

1. Vector Functions Vector Function Definitions Position Vector: r(t) = OP = f(t)I + g(t)j +h(t)k, where OPis a vector from the origin to point P(x,y,z) on the curve Derivative:r’(t) = lim = i + j + k IndefiniteIntegral:  r(t) dt = R(t) + C, where R is the antiderivative of r Definite Integral:r(t) dt = (f(t) dt )i + (g(t) dt)j + (h(t) dt )k Velocity:v = (vector) Speed: Speed = | v | (scalar) Direction of motion at time t: (unit vector) Acceleration: a = = (vector) Length of a smooth curve: L =  ( )2 + ( )2 + ( )2dt =  | v | dt Curvature (k):k = (scalar) (where T is the unit tangent vector of the smooth curve) Principal Unit Normal N: N = (unit vector to T) Binormal vector B: B = T x N (unit vector to T and N Frenet Frame TNB: Frame of mutually orthogal unit vectors traveling along a curve in space Tortion (t):t=– . N = (scalar) (where x = dx/dt, x = d2x/dt2, x = d3x/dt3, etc.) Tangential and Normal Components of Acceleration (a): a = aT T + aN N aT = = | v | aN = k( )2 = k | v |2 (scalars) aN = | a |2 – aT2 Ideal Projectile motion Ideal Projectile Motion: r =(v0 cos a)t i + [(v0 sin a)t – ½ gt2 ] j (g = 9.8 m/sec2 or 32 ft/sec2 ) Maximum height: ymax = Flight time:t = Range:R = sin 2a Firing from (x0, y0): r =(x0 + (v0 cos a)t i + [y0 + (v0 sin a)t – ½ gt2 ] j r(t + Dt) – r(t) Dt df dt dg dt dh dt (v0 sin a )2 Dt_> 0 2g 2v0 sin a g v02 b b b b g a a a a drdt Arc Length, Unit Tangent Vector T, Curvature Arc Length Parameter with Base Point P(t 0): s(t) =  [x’(t )]2+ [y’(t)]2 + [z’(t)]2dt =  | v(t ) | dt Speed on a Smooth Curve: Speed = = | v(t) | Unit Tangent Vector:T=== Curvature: k == Vector Formula for Curvature: k= Radius of curvature (r): r = 1/k Principal Unit Normal: N == dvdt d2rdt2 t t t0 t0 b b dx dt dy dt dz dt ds dt Vector Function Differentiation Rules u, v (differentiable vector functions of t); C (constant vector); c (scalar); f (differentiable scalar function) Constant function rule: C = 0 Scalar multiple rules: [ c u(t) ] = c u’(t) [ f(t) u(t) ] = f’(t)u(t) + f(t) u’(t) Sum Rule: [u(t) + v(t)] = u’(t) + v’(t) Difference Rule: [u(t) – v(t)] = u’(t) – v’(t) Dot Product Rule: [u(t) .v(t)] = u’(t) .v(t) + u(t) .v’(t) Chain Rule: [u(f(t)] = f’(t) u’(t) Constant length Rule: If | u(t) | = c, then u(t) . u’(f(t)) = 0 (orthogonal) a a dT ds drds dr/dt ds/dt v | v | 1 | v | dT dt 1dTkds d dt d dt d dt d dt d dt d dt d dt dBds v | v | | v x a |2 dTds ... d2s dt2 d dt ds dt x y z x y z x y z | v x a | | v |3 1dtkds dT / dt | dT / dt | Calculus III – Thomas Chapter 13 sectinos 1-5 R. M. E. Revised 3-04-07