MATHs 21a CHECKLIST MATERIAL, 8/13/2003 Maths 21a, Oliver Knill• Here is a checklist of material we covered in this course.• If you have worked with the material during the homework, there are not many things to memorize. Anexample might be the parameterization of the sphere or the definition of the curl.• You don’t need to know the formula for the curvature by heart but you should know what the curvaturemeans and what is the curvature of a circle of radius r.• The final exam will cover only material we mentioned in class. The handouts contain usually more materialthen what we covered in class.1. 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 scalar 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: 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 hyperboloidplane 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 other4. 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, i.e. r = f(z, θ) or r = f(z) surface of revolutiong(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 wave equationfx− ft= 0 transport 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. Gradient∇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 tangent plane 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> 0local 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 coordinatesRbaRdcRvuf(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 with
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