Lecture 4 2/18/2004, CURVES Math21aHOMEWORK FOR WEDNESDAY: Section 10.5: 4,10,26,42 (can assume 4.1)PARAMETRIC PLANE CURVES. If x(t), y(t) are functions of one variable, defined on the parameter inter-val I = [a, b], then ~r(t) = hf(t), g(t)i is a parametric curve in the plane. The functions x(t), y(t) are calledcoordinate functions.PARAMETRIC SPACE CURVES. If x(t), y(t), z(t) are functions, then ~r(t) = hx(t), y(t), z(t)i is a space curve.Always think of the parameter t as time. For every fixed t, we have a point (x(t), y(t), z(t)) in space. As tvaries, we move along the curve.EXAMPLE 1. If x(t) = t, y(t) = t2+ 1, we can write y(x) = x2+ 1 and the curve is a graph.EXAMPLE 2. If x(t) = cos(t), y(t) = sin(t), then ~r(t) follows a circle.EXAMPLE 3. If x(t) = cos(t), y(t) = sin(t), z(t) = t, then ~r(t) describes a spiral.EXAMPLE 4. If x(t) = cos(2t), y(t) = sin(2t), z(t) = 2t, then we have the same curve as in example 3 but wetraverse it faster. The parameterization changed.EXAMPLE 5. If x(t) = cos(−t), y(t) = sin(−t), z(t) = −t, then we have the same curve as in example 3 butwe traverse it in the opposite direction.EXAMPLE 6. If P = (a, b, c) and Q = (u, v, w) are points in space, then ~r(t) = ha+t(u−a), b+t(v−b), c+t(w−c)idefined on t ∈ [0, 1] is a line segment connecting P with Q.ELIMINATION: Sometimes it is possible to eliminate the parameter t and write the curve using equations(one equation in the plane or two equations in space).EXAMPLE: (circle) If x(t) = cos(t), y(t) = sin(t), then x(t)2+ y(t)2= 1.EXAMPLE: (spiral) If x(t) = cos(t), y(t) = sin(t), z(t) = t, then x = cos(z), y = sin(z). The spiral is theintersection of two graphs x = cos(z) and y = sin(z).CIRCLE HEART LISSAJOUS SPIRAL(cos(t), 3 sin(t)) (1 + cos(t))(cos(t), sin(t)) (cos(3t), sin(5t))et/10(cos(t), sin(t))TRIFOLIUM EPICYCLESPRING (3D) TORAL KNOT (3D)− cos(3t)(cos(t), sin(t))(cos(t) + cos(3t)/2, sin(t) +sin(3t)/2)(cos(t), sin(t), t)(cos(t) + cos(9t)/2, sin(t) +cos(9t)/2, sin(9t)/2)WHERE DO CURVES APPEAR? Objects like particles, celestial bodies, orquantities change in time. Their motion is described by curves. Examplesare the motion of a star moving in a galaxy, or data changing in time like(DJIA(t),NASDAQ(t),SP500(t))Strings or knots are closed curves in space.Molecules like RNA or proteins can be modeled as curves.Computer graphics: surfaces are represented by mesh of curves.Typography: fonts represented by Bezier curves.Space time A curve in space-time describes the motion of particles.Topology Examples: space filling curves, boundaries of surfaces or knots.DERIVATIVES. If ~r(t) = hx(t), y(t), z(t)i is a curve, then ~r0(t) = hx0(t), y0(t), z0(t)i = h ˙x, ˙y, ˙zi is called thevelocity. Its length |~r0(t)| is called speed and ~v/|~v| is called direction of motion. The vector ~r00(t) is calledthe acceleration. The third derivative ~r000is called the jerk.The velocity vector ~r(t) is tangent to the curve at ~r(t).EXAMPLE. If ~r(t) = hcos(3t), sin(2t), 2 sin(t)i, then ~r0(t) = h−3 sin(3t), 2 cos(2t), 2 cos(t)i, ~r00(t) =h−9 cos(3t), −4 sin(2t), −2 sin(t)i and ~r000(t) = h27 sin(3t), 8 cos(2t), −2 cos(t)i.WHAT IS MOTION? The paradoxon of Zeno of Elea: ”When looking ata body at a specific time, the body is fixed. Being fixed at each instant, thereis no motion”. While one might wonder today a bit about Zeno’s thoughts,there were philosophers like Kant, Hume or Hegel, who thought seriously aboutZeno’s challenges. Physicists continue to ponder about the question: ”what istime and space?” Today, the derivative or rate of change is defined as a limit(~r(t + dt) − ~r(t))/dt, where dt approaches zero. If the limit exists, the velocityis defined.EXAMPLES OF VELOCITIES.Person walking: 1.5 m/sSignals in nerves: 40 m/sPlane: 70-900 m/sSound in air: Mach1=340 m/sSpeed of bullet: 1200-1500 m/sEarth around the sun: 30’000 m/sSun around galaxy center: 200’000 m/sLight in vacuum: 300’000’000 m/sEXAMPLES OF ACCELERATIONS.Train: 0.1-0.3 m/s2Car: 3-8 m/s2Space shuttle: ≤ 3G = 30m/s2Combat plane (F16) (blackout): 9G=90 m/s2Ejection from F16: 14G=140 m/s2.Free fall: 1G = 9.81 m/s2Electron in vacuum tube: 1015m/s2DIFFERENTIATION RULES.The rules in one dimensions (f + g)0= f0+ g0(cf)0= cf0, (f g)0= f0g + fg0(Leibniz), (f(g))0= f0(g)g0(chain rule) generalize for vector-valued functions: (~v + ~w)0= ~v0+ ~w0, (c~v)0= c~v0, (~v · ~w)0= ~v0· ~w + ~v · ~w0(~v × ~w)0= ~v0× ~w + ~v × ~w0(Leibniz), (~v(f(t)))0= ~v0(f(t))f0(t) (chain rule). The Leibniz rule for the triple dotproduct [~u,~v, ~w] = ~u · (~v × ~w) is d/dt[~u,~v, ~w] = [~u0, ~v, ~w] + [~u,~v0, ~w] + [~u,~v, ~w0] (see homework).INTEGRATION. If ~r0(t) and ~r(0) is known, we can figure out ~r(t)by integration ~r(t) = ~r(0) +Rt0~r0(s) ds.Assume we know the acceleration ~a(t) = ~r00(t) as well as initial ve-locity and position ~r0(0) and ~r(0). Then ~r(t) = ~r(0) + t~r0(0) +~R(t),where~R(t) =Rt0~v(s) ds and ~v(t) =Rt0~a(s) ds.EXAMPLE. Shooting a ball. If ~r00(t) = h0, 0, −10i, ~r0(0) =h0, 1000, 2i, ~r(0) = h0, 0, hi, then ~r(t) = h0, 1000t, h + 2t − 10t2/2i.50 100 150 200 250
View Full Document
Unlocking...