FIRST MIDTERM REVIEW Math 21aGENERAL TIPS.• Do at least one practice exams, online TF questions.• 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 to do oddjobs. He explained this to a man who answered one door. ”How much will you charge to paint my porch?” askedthe man. ”Forty dollars.” ”Fine” said the man, and gave the student the paint and brushes. Three hours laterthe paint-splattered lad knocked on the door again. ”All done!”, he says, and collects 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• Orthogonal, parallel, projection• Parametrized Lines and Planes• Switch between parameterization and equations• Given line and plane, find intersection• Given plane and plane, find intersection• Given line and point, find plane• Given point and point, find line• Given three points, find plane• Distances: point-line, line-line,point-plane• Distinguish and analyse curves• Determine curves from acceleration• Calculate curvature,~T ,~N,~B.• Continuity of functions f(x, y).• 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• Identify surfaces in spherical coordinates• Identify surfaces in cylindrical coordinates• Distinguish parametric surfaces, contour maps,quadrics, graphs• Traces, intercepts, generalized traces, grid curves• Compute velocity, acceleration• Find length of curves• Parameterize curves by intersecting two surfaces• Level=contour curves, level=contour surfacesVECTORS.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, nonparallelvectors ~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.Projection:proj~v( ~w) =(~v· ~w)~v|~v|2.Is a vector parallel to ~v.Scalar projection:comp~v( ~w) = |proj~v(~v)| =|~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.{g(x, y, z) = C} define in general surfaces. Examples are graphs, where g(x, y, z) = z − f(x, y) = 0 or planes,where g(x, y, z) = ax + by + cz = C. If g has quadratic or linear terms only, the surface is called a quadric:example x2+ xy + y2= −z2+ 2x = 0. Some surfaces are sometimes easier to describe in cylindrical orspherical coordinates: example sphere: ρ = const or cylinder: r = const.Surfaces can be analyzed by looking at traces, intersections with planes parallel to the coordinate planes. Thisis especially true for graphs, where the traces f(x, y) = C are called contour lines. Examples are isobars,isotherms or topolographical contour lines.QUADRICS CHECKLIST. Quadrics like:• ellipsoid, sphere• cylinder• hyperbolic cylinder• cone• one sheeted hyperboloid• two sheeted hyperboloid• paraboloid• hyperbolic paraboloidcan 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.CURVATURE,~T ,~N,~B~r(t) = (x(t), y(t), z(t)) curve~r0(t) velocity~r00(t) acceleration~T = ~r0(t)/|~r0(t)| unit tangent vector~N =~T0(t)/|~T0(t)| unit normal vector~B =~T ×~N binormal vectorκ(t) =|~T0(t)||r0(t)|=|~r0(t)×~r00(t)||~r0(t)|3curvatureCOORDINATE SYSTEMS.rectangular cylindrical spherical(x, y, z) (r, θ, z) (ρ, θ, φ)x real r ≥ 0 ρ ≥ 0y real θ ∈ [0, 2π) θ ∈ [0, 2π)z real z real φ ∈ [0, π]x = r cos(θ)y = r sin(θ)z = zx = ρ cos(θ) sin(φ)y = ρ sin(θ) sin(φ)z = ρ
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