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HARVARD MATH 21A - Review

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MATH 21a REVIEW, 5/14/2003 Math 21a, O. KnillYour lucky BINGO numbers are below. During the presentation, we will pick randomly 5 of 9 mathematiciansrelevant for multivariable calculus. If one of your mathematicians appear, cross it off. There is a big change thatone student gets all 5 right. If you have collected 4 mathematicians, raise your voice.9 7 5 6 81. Geometry of Spacecoordinates and vectors in the plane and in spacev = (v1, v2, v3), w = (w1, w2, w3), v + w = (v1+ w1, v2+ w2, v3+ w3)dot product v.w = v1w1+ v2w2+ v3w3= |v||w| cos(α)cross product, v.(v × w) = 0, w.(v × w) = 0, |v × w| = |v||w| sin(α)triple cross product u · (v × w) volume of parallelepipedparallel vectors v × w = 0, orthogonal vectors v · w = 0scalar projection compw(v) = v · w/|w|, vector projection projw(v) = (v ·w)w/|w|2completion of square technique: example x2− 4x + y2= 1 is equivalent to(x − 2)2+ y2= −3distance d(P, Q) = |~P Q| =p(P1− Q1)2+ (P2− Q2)2+ (P3− Q3)22. Lines, Planes, Functionssymmetric equation of line(x−x0)a=(y−y0)b=z−z0cplane ax + by + cz = dparametric equation for line ~x = ~x0+ t~vparametric equation for plane ~x = ~x0+ t~v + s ~wswitch from parametric to implicit descriptions for lines and planesdomain and range of functions f(x, y)graph G = {(x, y, f(x, y))}intercepts: intersections of G with coordinate axestraces: intersections with coordinate planesgeneralized traces: intersections with {x = c}, {y = c} or {z = c}quadrics: ellipsoid, paraboloid, hyperboloids, cylinder, cone, parabolic hyper-boloidplane ax + by + cz = d has normal ~n = (a, b, c)line(x−x0)a=y−y0b=z−z0ccontains ~v = (a, b, c)sets g(x, y, z) = c describe surfaces, example graphs g(x, y, z) = z − f(x, y)linear equation 2x + 3y + 5z = 7) defines planequadratic equation i.e. x2− 2y2+ 3z2= 4 defines quadric surfacedistance point-plane: d(P, Σ) = |(~P Q) · ~n|/|~n|distance point-line: d(P, L) = |(~P Q) × ~u|/|~u|distance line-line: d(L, M ) = |(~P Q) · (~u × ~v)|/|~u × ~v|finding plane through three points P, Q, R: find first normal vector3. Curvesplane and space curves ~r(t)velocity ~r0(t), Acceleration ~r00(t)unit tangent vector~T (t) = ~r0(t)/|~r0(t)|unit normal vector~N(t) =~T0(t)/|~T0(t)|binormal vector~B(t) =~T (t) ×~N(t)curvature κ(t) = |~T0(t)|/|~r0(t)|arc lengthRba|~r0(t)| dt~r0(t) is tangent to the curve~v = ~r0then ~r =Rt0~v dt + ~cκ(t) =|r0(t)×r00(t)||r0(t)|3ddt(~v(t) · ~w(t)) = ~v0(t) · ~w(t) + ~v(t) · ~w0(t)T, N, B are unit vectors which are perpendicular to each otherfind parameterizations of basic curves (i.e. intersections of surfaces)4. Surfacespolar coordinates (x, y) = (r cos(θ), r sin(θ))cylindrical coordinates (x, y, z) = (r cos(θ), r sin(θ), z)spherical coordinates (x, y, z) = (ρ cos(θ) sin(φ), ρ sin(θ) sin(φ), ρ cos(φ))g(r, θ) = 0 polar curve, especially r = f(θ), polar graphsg(r, θ, z) = 0 cylindrical surface, especially r = f(z, θ) or r = f(z) surface ofrevolutiong(ρ, θ, φ) = 0 spherical surface especially ρ = f(θ, φ)f(x, y) = c level curves of f(x, y)g(x, y, z) = c level surfaces of g(x, y, z)circle: x2+ y2= r2, ~r(t) = (r cos t, r sin t).ellipse: x2/a2+ y2/b2= 1 , ~r(t) = (a cos t, b sin t)sphere: x2+ y2+ z2= r2, ~r(u, v) = (r cos u sin v, r sin u sin v, r cos v)ellipsoid: x2/a2+ y2/b2+ z2/c2= 1, ~r(u, v) = (a cos u sin v, b sin u sin v, c cos v)line: ax + by = d, ~r(t) = (t, d/b − ta/b)plane: ax + by + cz = d , ~r(u, v) = ~r0+ u~v + v ~w, (a, b, c) = ~v × ~wsurface of revolution: r(θ, z) = f(z), ~r(u, v) = (f(v) cos(u), f (v) sin(u), v)graph: g(x, y, z) = z − f(x, y) = 0, ~r(u, v) = (u, v, f(u, v))5. Partial Derivativesfx(x, y) =∂∂xf(x, y) partial derivativepartial differential equation PDE: F (f, fx, ft, fxx, ftt) = 0ft= fxxheat equationftt− fxx= 0 1D wave equationfx− ft= 0 transport equationfxf − ft= 0 Burger equationfxx+ fyy= 0 Laplace equationL(x, y) = f(x0, y0)+fx(x0, y0)(x−x0)+fy(x0, y0)(y−y0) linear approximationtangent line: L(x, y) = L(x0, y0), ax + by = d with a = fx(x0, y0), b =fy(x0, y0), d = ax0+ by0tangent plane: L(x, y, z) = L(x0, y0, z0)estimate f(x, y, z) by L(x, y, z) near (x0, y0, z0)f(x, y) differentiable if fx, fyare continuousfxy= fyxClairot’s theorem~ru(u, v), ~rvtangent to surface ~r(u, v)6. Chain Rule∇f(x, y) = (fx, fy), ∇f(x, y, z) = (fx, fy, fz), gradientDvf = ∇if · v directional derivativeddtf(~r(t)) = ∇f (~r(t)) · ~r0(t) chain rule∇f(x0, y0, z0) is orthogonal to the level surface f(x, y, z) = c which contains(x0, y0, z0).ddtf(~x + t~v) = Dvf by chain rulex−x0fx(x0,y0,z0)=y−y0fy(x0,y0,z0)=z−z0fz(x0,y0,z0)normal line to surface f (x, y, z) = c at(x0, y0, z0)(x − x0)fx(x0, y0, z0) + (y − y0)fy(x0, y0, z0) + (z − z0)fz(x0, y0, z0) = 0 tangentplane at (x0, y0, z0)directional derivative is maximal in the ~v = ∇f directionf(x, y) increases, if we walk on the xy-plane in the ∇f directionpartial derivatives are special directional derivativesif Dvf(~x) = 0 for all ~v, then ∇f(~x) =~0implicit differentiation: f(x, y(x)) = 0, fx1 + fyy0(x) = 0 gives y0(x) = −fx/fy7. Extrema∇f(x, y) = (0, 0), critical point or stationary pointD = fxxfyy− f2xydiscriminant or Hessian determinantf(x0, y0) ≥ f (x, y) in a neighborhood of (x0, y0) local maximumf(x0, y0) ≤ f (x, y) in a neighborhood of (x0, y0) local minimum∇f(x, y) = λ∇g(x, y), g(x, y) = c, λ Lagrange multipliertwo constraints: ∇f = λ∇g + µ∇h, g = c, h = dSecond derivative test: ∇f = (0, 0), D > 0, fxx< 0 local max, ∇f = (0, 0), D >0, fxx> 0 local min, ∇f = (0, 0), D < 0 saddle8. Double IntegralsR RRf(x, y) dA double integralRbaRdcf(x, y) dydx integral over rectangleRbaRg2(x)g1(x)f(x, y) dydx type I regionRdcRh2(y)h1(y)f(x, y) dxdy type II regionR RRf(r, θ)r drdθ polar coordinatesR RR|~ru× ~rv| dudv surface areaRbaRdcf(x, y) dydx =RdcRbaf(x, y) dxdy FubiniR RR1 dxdy area of region RR RRf(x, y) dxdy volume of solid bounded by graph(f) xy-plane9. Triple IntegralsR R RRf(x, y, z) dV triple integralRbaRdcRvuf(x, y, z) dydx integral over rectangular boxRbaRg2(x)g1(x)Rh2(x,y)h1(x,y)f(x, y) dzdydx type I regionf(r, θ, z)rdzdrdθ cylindrical coordinatesR R RRf(ρ, θ, z)ρ2sin(φ)dzdrdθ spherical coordinates∂(x,y)∂(u,v)= xuyv− xvyuJacobian of T (u, v) = (x(u, v), y(u, v))∂(x,y,z)(u,v,w)= [∇x, ∇y, ∇z] Jacobian of T (u, v, w) = (x, y, z)RbaRdcRvuf(x, y, z) dzdydx =RvuRdcRbaf(x, y, z) dxdydz FubiniV =R R RR1 dV volume of solid RM =R R RRρ(x, y, z) dV mass of solid R


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