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