8/15/2006 FINAL EXAM Maths 21a, O. Knill, Summer 2006Name:• Start by printing your name in the above box.• Try to answer each question on the same page as the question is asked. If needed, usethe back or the next empty page for work.• Do not detach pages from this exam packet or unstaple the packet.• Please try to write neatly. Answers which a r e illegible for the gra der can not be givencredit.• No notes, books, calculators, computers, or other electronic aids are allowed.• Problems 1-3 do not require any justifications. For the rest of the problems you have toshow your work. Answers without derivation can not be given credit.• You have 180 minutes time to complete your wo r k.1 202 103 104 105 106 107 108 109 1010 1011 1012 1013 10Total: 1401Problem 1) (20 points)1)T FIf ~u + ~v + ~w =~0 then ~u · (~v × ~w) = 0.2)T FR50Rπ0r dθ dr is half the area of a disc radius 5 in the plane.3)T FIf a vector field~F (x, y) satisfies curl(F )(x, y) = 0 f or all points (x, y) in theplane, then~F is conservative.4)T FIf the acceleration of a parameterized curve ~r(t) = (x(t), y(t), z(t)) is zerothen the curve ~r(t) is a line.5)T FA circle of radius 1/2 has a smaller curvature than a circle of radius 1.6)T FThe curve ~r(t) = (−sin(t), cos(t)) for t ∈ [0, π] is half a circle.7)T FThe function u(t, x) = sin(x + t) is a solution of the partial differentialequation utx+ u = 08)T FThe length of a curve ~r(t) in space parameterized on a ≤ t ≤ b is the valueof the integralR21|~T′(t)| dt, where~T (t) is the unit tangent vector.9)T FLet (x0, y0) be the maximum of f(x, y) under the constraint g(x, y) = 1.Then the gradient of g at (x0, y0) is parallel to the gradient of f at (x0, y0).10)T FAt a point which is not a critical point, the directional derivativeD~vf(x0, y0, z0) can take both the negative and the positive sign.11)T FIf a vector field~F (x, y) is a gradient field, we always can find a curve C forwhich the line integralRC~F ·~dr is positive.12)T FIf C is a closed level curve o f a function f (x, y) and~F = (fx, fy) is thegradient field of f, thenRC~F ·~dr = 0.13)T FThe divergence of a gradient vector field~F (x, y, z) = ∇f (x, y, z) is a lwayszero.14)T FThe line integral of the vector field~F (x, y, z) = hx2, y2, z2i along a linesegment from (0, 0, 0) to (1, 1, 1) is 1.15)T FIf~F (x, y) = (x2− y, x) and C : ~r( t) = hqcos(t),qsin(t)i parameterizes theboundary of t he region R : x4+ y4≤ 1, thenRCF · ds is twice the area ofR.16)T FThe flux of the vector field~F (x, y, z) = h0, y, 0i through the boundary S ofa solid sphere E is equal to the volume the sphere.17)T FThe quadratic surface −x2+ y2+ z2= 5 is a one-sheeted hyperboloid.18)T FIf~F is a vector field in space and S is the bo undary of a sphere then theflux of~curl(~F ) through S is 0.19)T FIf div(~F )(x, y, z) = 0 for all (x, y, z) and S is a torus surface, then the fluxof~F through S is zero.20)T FIn spherical coordinates, the equation ρ cos(φ) = ρ cos(θ) sin(φ) defines aplane.2Problem 2) (10 points)I II III IVV VI VII VIIIEnter I,II,III,IV,V,VI,VII,VI II here Equationx2− y2+ z2= 1~r(t) = hcos(3t), sin(2t)iz = f (x, y) = cos(3x) + sin(2y)~F (x, y) = h−y/√x2+ y2, x/√x2+ y2icos(3x) + sin(2y) = 1~F (x, y, z) = h−y, x, 1i~r(u, v) = hcos( 3 u), sin(2u), vi{(x, y) ∈ R2| |x2− y2| = 1 }Furthermore, fill in the peoples names, Green, Stokes, Gauss, Fubini, Clairot. If there isno name associated to the theorem, write the name of the theorem.Formula Name of the theoremRC~F ·~dr =R RScurl(~F ) ·dSfxy(x, y) = fy x(x, y)RC~F · dr =R RRcurl(~F ) dxdyRba∇f(~r(t)) ·~r′(t)) dt = f(~r(b)) − f(~r(a))R RSF · dS =R R REdiv(F ) dVRbaRdcf(x, y) dxdy =RdcRbaf(x, y) dydx3Problem 3) (10 points)In this problem, vector fields~F are written as~F = hP, Qi. We use abbreviations curl(F ) =Qx−Py. When stating curl(F ) = 0, we mean that curl(F )(x, y) = 0 vanishes for all (x, y).Similarly, we say div(F ) if div(F )(x, y) = Px(x, y) + Qy(x, y) = 0 for all x, y.Check the box which match the formulas o f the vector fields with the corresponding pictureI,II,III or IV and mark also the places, indicating the vanishing of curl(F ).Vectorfield I II III IV curl(F ) = 0 div(F ) = 0~F (x, y) = h1, xi~F (x, y) = h3y, −3xi~F (x, y) = h7, 2i~F (x, y) = hx, yiI II-1-0.5 0.51 1.5-1-0.50.51-1-0.5 0.51-1-0.50.51III IV-1-0.5 0.51-1-0.50.51-3 -2-1 12 3-3-2-11234Problem 4) (10 points)a) (5 points) What is t he area of the triangle A, B, P , where A = (1, 1, 1), B = (1, 2, 3)and P = (3, 2, 4)?b) (5 points) Find the distance between the point the point P and the line L passingthrough the points A with B.Problem 5) (10 points)The height of the gro und near the Simplon pass in Switzerland is given by the functionf(x, y) = −x −y33−y22+x22.There is a lake in that area as you can see in the photo.a) (7 points) Find and classify all the critical points of f and tell from each of them,whether it is a local maximum, a local minimum or a saddle point.b) (3 points) For any pair of two different critical points A, B found in a) let Ca,bbe theline segment connecting the points, evaluate the line integralRCa,b∇f~dr.Photo of the lake in the Swiss alps near the Simplon mountain pass.Problem 6) (10 points)a) (4 points) Find the linearization L(x, y, z) of f(x, y, z) = 2 + z − sin(−x − 3y) at thepoint P = (0, π, 2).5b) (4 points) Find the equation of the tangent plane at that point P = (0, π, 2).c) (2 points) Estimate f(0.001, π, 2.02) using the linearization.Problem 7) (10 points)Find the volume of the wedge shaped solid that lies above the xy-plane and below theplane z = x and within the solid cylinder x2+ y2≤ 9.xyzxProblem 8) (10 points)The distance from a point (x, y) to the line y = x in the plane is given by f(x, y) =(y −x)/√2. Use the Lagrange method to find the point (x, y) on the parabolag(x, y) = x2− y = −2which is closest to the line.6Problem 9) (10 points)a) (5 points) A ribbon of a girl is modeled as a surface S which is parameterized by~r(t, s) = (s cos(t), sin(t), t), where t ∈ [0, 2π] a nd s ∈ [0, 1]. Find the surface area of thisribbon S.b) (5 points) Part of the boundary of the ribbon is obtained when fixing s = 1. It is acurve in space. Find the arc length of this curve ~r(t), parametrized from t = 0 to 2π.Painting: ”Young Girl with Blue …
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