8/15/2006 FINAL EXAM PRACTICE 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. If you need additional paper, write your nameon it.• Justify your answers. Answers without derivation can not be given credit.• Do not detach pages from this exam packet or unstaple the packet.• Please write neatly. Answers which are illegible fo r the grader can not be given credit.• No notes, books, calculators, computers, or other electronic aids can be allowed.• You have 180 minutes time t o complete your wo r k.11 202 103 104 105 106 107 108 109 1010 1011 1012 1013 1014 1015 10Total: 160Problem 1) (20 point s)1)T FFor any two nonzero vectors ~v, ~w the vector ~v − ~w is perpendicular to ~v × ~w.2)T FThe cross product satisfies the law (~u ×~v) × ~w = ~u ×(~v × ~w).3)T FIf the curvature of a smooth curve ~r(t) in space is defined and zero for allt, then the curve is part of a line.4)T FThe curve ~r(t) = (1 − t)A + tB, t ∈ [0, 1] connects the point A with thepoint B.25)T FFor every c, the function u(x, t) = (2 cos(ct) + 3 sin(ct)) sin(x) is a solutionto the wave equation utt= c2uxx.6)T FThe length of the curve ~r(t) = (t, sin(t)), where t ∈ [0, 2π] isR2π0q1 + cos2(t) dt.7)T FLet (x0, y0) be the maximum o f f(x, y) under the constraint g(x, y) = 1.Then fxx(x0, y0) < 0.8)T FThe function f(x, y, z) = x2− y2− z2decreases in the direction(2, −2, −2)/√8 at the point (1, 1, 1).9)T FAssume~F is a vector field satisfying |~F ( x, y, z)| ≤ 1 everywhere. For everycurve C : ~r(t) with t ∈ [0, 1 ], t he line int egra lRC~F ·~dr is less or equal thanthe arc length of C.10)T FLet~F be a vector field and C is a curve which is a flow line, thenRC~F ·~dr > 0.11)T FThe divergence of the gra dient of any f (x, y, z) is always zero.12)T FFor every function f, one has div(curl(grad(f))) = 0.13)T FIf for two vector fields~F and~G one has curl(~F ) = curl(~G), then~F =~G + (a, b, c ) , where a, b, c are constants.14)T FFor every vector field~F the identity grad(div(~F ) ) =~0 holds.15)T FIf a nonempty quadric surface g(x, y, z) = ax2+ by2+ cz2= 5 can becontained inside a finite box, then a, b, c ≥ 0.16)T FIf~F is a vector field in space then the flux of~F through a ny closed surfaceS is 0.17)T FIf div(~F )(x, y, z) = 0 for all (x, y, z), then curl(~F ) = (0, 0 , 0) for all (x, y, z).18)T FThe flux of the vector field~F (x, y, z) = (y + z, y, −z) through the boundaryof a solid region E is equal to the volume of E.19)T FIf in spherical coordinates the equation φ = α (with a constant α) definesa plane, then α = π/2.20)T FFor every function f(x, y, z), there exists a vector field~F such that div(~F ) =f.3Problem 2) (10 point s)-2 -1 1 2-2-112-1 -0.5 0.5 1-1-0.50.51I II-1 -0.5 0.5 1-1-0.50.51-1 -0.5 0.5 1-1-0.50.51III IVFor the sign of the curl or divergence, where either + (positive), − (negative) or 0 for zero.The vector fields are considered on the square [−1/2, 1/2]x[−1/2, 1/2] in this problem.4Enter I,II,III,IV here Vector field curl sign divergence signF ( x, y) = (x, y2)F ( x, y) = (1 −y, x)F ( x, y) = (y − x, −y)F ( x, y) = (−x, y3)Problem 3) (10 point s)Mark with a cross in the column below ”conservative” if a vector fields is conservative(that is if curl(~F )(x, y, z) = (0, 0, 0) for all points (x, y, z)). Similarly, mark the fieldswhich are incompressible (that is if div(~F )(x, y, z) = 0 for all (x, y, z)). No justificationsare needed.Vectorfield conserva tive incompressiblecurl(~F ) =~0 div(~F ) = 0~F ( x, y, z) = (−5, 5, 3)~F ( x, y, z) = (x, y, z)~F ( x, y, z) = (−y, x, z)~F ( x, y, z) = (x2+ y2, xyz, x − y + z)~F ( x, y, z) = (x −2yz, y − 2zx, z − 2xy)Problem 4) (10 point s)Let E be a parallelogram in three dimensional space defined by two vectors ~u and ~v.a) (3 points) Express the diagonals of the parallelogram as vectors in terms of ~u and ~v.b) (3 points) What is the relation between the length of the crossproduct of the diagonalsand t he area of the parallelogram?5c) (4 points) Assume that the diagonals are perpendicular. What is the relation betweenthe lengths of the sides of the parallelogram?Problem 5) (10 point s)Find the volume of the largest rectangular box with sides parallel to the coordinate planesthat can be inscribed in the ellipsoidx24+y29+z225= 1.Problem 6) (10 point s)EvaluateZ80Z2y1/3y2ex2x8dxdy.Problem 7) (10 point s)EvaluateR RD2xyx2+y2dxdy, where D is the intersection of the annulus 1 ≤ x2+ y2≤ 2 withthe second quadrant {x ≤ 0, y ≥ 0 }.Problem 8) (10 point s)a) (3 points) Find all the critical points of the function f(x, y) = −(x4− 8x2+ y2+ 1).b) (3 points) Classify the critical points.c) (2 points) Locate the local and absolute maxima of f.d) (2 points) Find the equation for the tangent plane to the graph of f at each absolutemaximum.6Problem 9) (10 point s)Find t he areaR RR1 dxdy of the 10 legged ”sea star” R, enclosed by the polar curver(θ) = 2 + sin(10 θ) ,where θ ∈ [0, 2π]. The photo to the r ig ht shows a real sea star.Problem 10) (10 points)Find the volume of the intersection of the interior of the one sided hyperboloid x2+y2−z2≤1 with the solid ball enclosed by the sphere x2+ y2+ z2≤ 9.-4-2 0 24-4-20247Problem 11) (10 points)Let the curve C be parametrized by ~r(t) = (t, sin t, t2cos t) for 0 ≤ t ≤ π. Let f(x, y, z) =z2ex+2y+ x2and~F = ∇f. FindRC~F · d~r.Problem 12) (10 points)a) Find the linear a pproximation L(x, y) of f(x, y) =√4 + 2x2+ 4y2at the point (x, y) =(2, 1).b) Find the equation for the tangent line to the level curve of f (u, v) at (2, 1).Problem 13) (10 points)Find the line integral of t he vector field~F (x, y) = (x30+ y, y50+ x) along the path~r(t) = (4 sin(π sin(t)) + sin(10t), t) with 0 ≤ t ≤ π/2 .12 340.250.50.7511.251.5Problem 14) (10 points)8Evaluate the line integral of the vector field~F ( x, y) = (y2, x2) in the clockwise directionaround the triangle in the xy-plane defined by the points (0, 0), (1 , 0) and (1, 1) in two ways:a) (5 points) by evaluating the three line integrals.b) (5 points) using Greens theorem.Problem 15) (10 points)Use Stokes theorem to evaluate the line integral of~F ( x, y, z) = (−y3, x3, −z3) along thecurve ~r(t) = (cos(t), sin(t), 1 − cos(t) − sin(t)) …
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