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HARVARD MATH 21A - General Tips

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FIRST MIDTERM REVIEW, 3/10/2004 Math 21aGENERAL TIPS.• Review the online quizzes.• Make list of facts on a sheet of paper.• Fresh up short-term memory before test.• Review homework. Find error patterns.• Ask questions:”Ask a question and you’re a fool forthree minutes; do not ask a questionand you’re a fool for the rest of yourlife.” - Chinese Proverb• During the exam: read the questions carefully. Wrong understanding could lead you to solve an other problem:There was a college student trying to earn some pocket money by going from house to house offering todo odd jobs. He explained this to a man who answered one door. ”How much will you charge to paintmy porch?” asked the man. ”Forty dollars.” ”Fine” said the man, and gave the student the paint andbrushes. Three hours later the paint-splattered lad knocked on the door again. ”All done!”, he says, andcollects his money. ”By the way,” the student says, ”That’s not a Porsche, it’s a Ferrari.”MIDTERM TOPICS.• Properties of dot, cross and triple product• Orthogonality, parallel, vector projection• Parametrized Lines and Planes• Given line and plane, find intersection• Given plane and plane, find intersection• Given line and point, find plane• Given two points, find line• Given three points, find plane• Distances: point-line, line-line,point-plane• Distinguish and analyse curves• Determine curves from acceleration• Know Keplers laws, polar form of ellipse• Recognize functions f(x, y) of two variables.• Tangent lines, tangent curves• Distance between two lines• Distance between two planes• Angle between two vectors• Angle between two planes• Area of parallelogram, triangle in space• Volume of parallelepiped• Distinguish contour maps, graphs• Compute velocity, acceleration, speed• Integrate from velocity to get position• Find length of curves• Level curves, level surfaces• Directional derivative• Chain rule• Implicit differentiation• Tangent planesVECTORS.Two points P = (1, 2, 3), Q = (3, 4, 6) define a vec-tor ~v =~P Q = h2, 2, 3i. If ~v = λ ~w, then the vec-tors are parallel if ~v · ~w = 0, then the vectors arecalled orthogonal. For example, (1, 2, 3) is par-allel to (−2, −4, −6) and orthogonal to (3, −2, 1).The addition, subtraction and scalar multiplicationof vectors is done componentwise. For example:(3, 2, 1) − 2((1, 1, 1) + (−1, −1, 0)) = (3, 2, −1).A nonzero vector ~v and a point P = (x0, y0, z0) de-fine a line ~r(t) = P + t~v. Two nonzero, non-parallelvectors ~v, ~w and a point P define a plane P + t~v +s~s.The vector ~n = ~v × ~w = (a, b, c) is orthogonal tothe plane. Points on the line satisfy the symmet-ric equationx−x0a=y−y0b=z−z0c. Points on theplane satisfy an equation ax + by + cz = d, whered = ax0+ by0+ cz0. Using the dot product for pro-jection and the vector product to get orthogonal vec-tors, one can solve many geometric problems in 3D.DOT PRODUCT (is scalar)~v · ~w = ~w · ~v commutative|~v · ~w| = |~v||~w| cos(α) angle(a~v) · ~w = a(~v · ~w) linearity(~u + ~v) · ~w == ~u · ~w + ~v · ~w distributivity{1, 2, 3}.{3, 4, 5} in Mathematicaddt(~v · ~w) = (ddt~v) · ~w + (~v ·ddt~w) product ruleCROSS PRODUCT (is vector)~v × ~w = − ~w × ~v anti-commutative|~v × ~w| = |~v|| ~w| sin(α) angle(a~v) × ~w = a(~v × ~w) linearity(~u + ~v) × ~w = ~u × ~w + ~v × ~w distributivityCross[{1, 2, 3}, {3, 4, 5}] Mathematicaddt(~v × ~w) = (ddt~v) × ~w + ~v × (ddt~w) product rulePROJECTIONS.Vector projection:proj~v( ~w) =(~v· ~w)~v|~v|2.Is a vector parallel to ~w.Scalar projection:comp~v( ~w) = |proj~v( ~w)| =|~v·~w||~v|the length of the projectedvector.Applications:• Distance P +t~v, Q+s ~w is scalarprojection of~P Q onto ~v × ~w.• Distance P, Q+t~v+s ~w is scalarprojection of~P Q onto ~n = ~v × ~w.SURFACES {f(x, y, z) = c}.Examples are graphs, where f(x, y, z) = z − g(x, y) = 0 or planes, where f(x, y, z) = ax + by + cz = c.Surfaces can be analyzed by looking at intersections with planes parallel to the coordinate planes. For graphs,the traces f(x, y) = c are contour lines. Most important fact:The gradient ∇f (x0, y0, z0) is normal to the surface f(x, y, z) = c containing (x0, y0, z0).SURFACES EXAMPLES• sphere x2+ y2+ z2= 1• cylinder x2+ y2= 1• ellipsoid x2/a2+ y2/b2+ z2/c2= 1• cone x2+ y2− z2= 0• plane ax + by + cz = d• one sheeted hyperboloid x2+ y2− z2= 1• two sheeted hyperboloid x2+ y2− z2= −1• paraboloid x2+ y2− z = 0• hyperbolic paraboloid x2− y2− z = 0• graph of function g(x, y) − z = 0can be identified using traces, the intersections with planes.CURVES.~r(t) = (x(t), y(t), z(t)), t ∈ [a, b] defines a curve. By differentiation, we obtain the velocity ~r0(t) andacceleration ~r00(t). If we integrate the speed |~r0(t)| over the interval [a, b], we obtain the length of thecurve.Rbapx0(t)2+ y0(t)2+ z0(t)2dtExample: ~r(t) = (1, 3t2, t3), ~r0(t) = (0, 6t, 3t2), so that |~r0(t)| = 3t(4 + t2). The length of the curve between 0and 1 isR103t(4 + t2) dt = 6t2+ 3t44|10= 6 ·34.DIRECTIONAL DERIVATIVE.For any vector ~v and a function f(x, y), define D~vf(x, y) = ∇f(x, y) · ~v. Unlike in many textbooks:We do not divide by |~v| to compute the directional derivative.CHAIN RULE.We have seen that d/dtf(~r(t)) = ∇f(~r(t)) · ~r0(t). This works also, if ~r(t, s) is a function of two variables:ft(x(t, s), y(t, s)) = ∇f(~r(t, s)) · ~rt(t, s).fs(x(t, s), y(t, s)) = ∇f(~r(t, s)) · ~rs(t, s).Other variables: w(u, v) function of u, v, where u, v are functions of x and eventually of


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