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Math21a, ReviewSpring 2006, Part IOliver Knill, May 20061Part I: GeometryDa Vinci: 1452-1519Harvard college founded: 16262r(t)3Vectors 1) Space ....45QPDetermined by two points P,Q. Can be placed anywhere in spacebut translatedvectors areconsidered “equal”Vectors6Parallel and “identical”7vitruvian man8vuu+vu-v2vAddition, Scaling9Application10vuu . v = |u| |v| cos(t)= u v + u v + u v1 1 2233Dot Product 11vucos(t) = u . v /( |u| |v| )u.u = |u|2orthogonalu . v =0Angle, Length,Orthogonality12vuThree vectors u,v,u-v form a triangle. We know |u|,|v|,|u-v|. This determines the angles of the triangle.u-vFrom vectors to angles13vu|(u-v)| = (u-v).(u-v)2=|u| +|v| - 2(u.v) we know the dot product.2 2and so the angle. From|(u-v)| 2- |u| -|v| 2 2-2 |u| |v|cos(t)=u-v14vuu x v = u x vCross Product 15+-+=Cross Product 16Length is |u| |v| sin(t)tLength of Cross Product 17vu | u x v | = |u| |v| sin(t)and area.Cross Product 18Pvr(t,s) = P +t v+s wwParametric equationn = (a,b,c) = v x wa x + b y + c z = dPlanes19Q=(0,0,1)Dot Product Problem 1A=(0,0,0)B=(0,1,0)C=(1,1,0)What are thepoints, forwhich thedistance to Qis equal to thedistance to theplane throughA,B and C?20Vector andvwproj (w)vScalar Projection21Vectors wproj (w)vvw=(3,1,-5)Example: find the vector projection of w onto vv=(1,1,0)The vector projection ofw onto v is4v/2 = 2v22vwproj (w)vProjections23PvLr(t) = P +t vParametrized Lines24Distance Formulas25PQnDistance Point-Plane26PQvDistance Point-Line27PQvDistance Line-LinewvxwProject Q-P onto vxw28vuDistance FormulasPQ29Distance Formulas30| |= VolumeParallelepiped31Pax+by+cz=dvPlane and Line32PQCan get v as a cross product.Plane-Plane33One Mars year = 586 daysSpirit, Dec 200534Find the distance from the tip of the cydonia pyramide on Mars with coordinates (1,-1,3) to the surface modeled as the plane x+2y+2z=1. Cydonia ProblemCydonia ProblemCydonia pyramide Problem 335http://www.geocities.com/cafemomus/thanks-fgump.gifCydonia ProblemCydonia Problem2) Curves362) Curves37Curves38r(t) = <x(t),y(t),z(t)>Parametric CurvesVelocity: vectorSpeed: length of velocity vector39r(s) = (x(s),y(s),z(s)) R(t)=r(s) + t r’(s)Tangent Line40r ’’(t) known at all times, r’(0) known, r(0) known, then r(t)= r(0)+r’(0) t + r’’(t) t /2 is knownEven MS research is aware of this principle and built a prototype which knows where you are:Integratation41Integrate speedover parameter interval to obtain arc length.r’(t) velocity|r’(t)| speedr’(t)Arc Length42Find the length of the curvefrom t=0 to t=1. A frisbee flies on the following curve:The Frisbie Problem43Yale college has claimed to be the place where the frisbie was invented. The school has argued that a Yale undergraduate named Elihu Frisbie grabbed a passing collection tray from the chapel and flung it out into the campus, thereby becoming the inventor of the Frisbie and winning glory for Yale. By the way...44evenso Yale has “Lux and Veritas”in their emblem, it is no accident that“lux” is above “veritas....this story is not true. The frisbie wasinvented at HarvardBut ... 45Lets go back in time46Cambridge 1775 ...471854....48Cambridge 1877 ...4950Today’s lecture is here51The fact is:From 1871 to 1958, the Frisbie Baking company made pies that were sold to many New England colleges. Hungry Harvard students soon discovered that the empty pie tins could be tossed and cought, providing endless hours of game and sport. 52George Frisbie Hoar graduated from Harvard University in 1846. Frisbie was often teased because his name could be found on every pie. One beautiful march day as today,at the exactly same spot as you sit now, a plate was thrown athim with the words: One of the students was The frisbie was inventedFrisbie, catch the Frisbie! 53Note the transitionfrom PRISBY to FRISBIE 1854 187754George Frisbie Hoar (1826-1904), studied later law at Harvard Law School and served on the Mass state senate and the United States House of Representatives. 55Frisbies are documented at Harvard even as entertainment for dogs56and the event is celebrated still each year at themath department on pi day: 3/14 at 1:5957Find the length of the curvefrom t=0 to t=1. Back to the problem58Angular momentum conservationApplication:Product Rules59Opportunity has photographed a “bunny”-shaped yellow object of about4-5 centimetersdiameter.The Mars-bug riddle60Curvature ProblemAn mars bug flies along the path r(t) = (50 cos(t), 50 sin(t),10) The mars rover travels along the pathr(t) = (2 cos(t), 2 sin(t),10) Which path has larger curvature? 61Unit tangent vectors etc6263BTN6465FormulasTNBUnit tangent, unit normal and binormal vectors ore normal to each othersphere of radius 1/curvature66http://www.geocities.com/cafemomus/thanks-fgump.gifCydonia ProblemCydonia ProblemThats all I have to say about that.673) Functions etc68Functions of more variables Functions of more 69Functions of more curvesContour or Level70and their Gradients71Surfaces722 ways to represent surfaces•Implicit surface g(x,y,z)=c•Parametric surface r(u,v) 73Surfaces you should know:SpheresGraphsSurfaces of revolutionPlanes7475Surface Area(x,y,z)=r(u,v)RS76http://www.geocities.com/cafemomus/thanks-fgump.gifCydonia ProblemCydonia ProblemThats all I have to say about that.77Derivatives78Chain Rule79Implicit Differentiationg(x,y,z) = 0 defines z = f(x,y)80Example: 81Gradients and TangentsCrucial: the normal vector(a,b,c) is the gradient. The equation of the plane isax+by+cz=dwhere d is found at the end.82EstimationLinear ApproximationTangent planeA Trinity: 83Richard Feynmans trickMovie: “Infinity” (1996)84EstimationThe cube root of 1729.02 is closeto the cube root of 1728, which is 12=x0.We have f(x)=f(1728)+f’(1728) 1.02 = 12+1.02/(144 *3) =12.002368586PDE’s Equations in which partial derivatives appear: PDE’s 87PDE’s PDE examples appearing in this course are Clairot’s theorem: or identities like curl(grad(f))=0or div(curl(F))=0 which arebased on Clairot. 88PDE’s You should be able to verify that certain functionssatisfy a PDE like waveheattransport89A subject already of interest to Da Vinci 90http://www.geocities.com/cafemomus/thanks-fgump.gifCydonia ProblemCydonia ProblemThats all I have to say about that.91Extrema 92Local Max D>0, f <0Local Min D>0 f >0Saddle D<0xxxxSecond Derivative Test93Problem 6: Find all critical points of and classify them. 94Problem 5: extremize 95Quiz coming upWin a DVD.96DVD preview97Are you
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