8/5/2010 FINAL EXAM Maths 21a, O. Knill, Summer 2010Name:• Start by printing your name in the above box.• Try to answer each question o n 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 o r unstaple the packet.• Please try to write neatly. Answers which are illegible for the grader 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. Even correct answers without derivation can no t 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 FThe line ~r(t) = ht, t, −ti is is contained in the pla ne x + y + z = 1 .2)T FThe quadratic surface x2+ y2= z2is an elliptic paraboloid.3)T FIf~T (t),~B(t),~N(t) are the unit tangent, normal and binormal vectors o f acurve with ~r′(t) 6= 0 everywhere, then~T (t) ·~B(t) ×~N(t) is always equal to1 or −1.4)T FIf |~u ×~v| = 0, then Proj~v(~u) = ~u.5)T FThere is a vector field~F (x, y) which has the property curl(~F ) = div(~F ) = 1.where curl(~F )(x, y) = Qx(x, y) − Py(x, y) and div(~F )(x, y) = Px(x, y) +Qy(x, y).6)T FThe acceleration vector ~r′′(t) = hx(t), y(t)i is always in the plane spannedby the vector ~r(t) and the velocity vector ~r′(t).7)T FFor every curve on the unit sphere, the curvature is constant and equal to1.8)T FIf a smooth function f(x, y) has no maximum nor minimum, then it doesnot have a critical point.9)T FThe linearization L(x, y) of a cubic function f(x, y) = x3+y3is the functionL(x, y) = 3x2+ 3y2.10)T FIf~F (x, y) is a gradient field~F = ∇f and ~r(t) is a flow line satisfying~r′(t) =~F (~r(t)) thenddtf(~r(t)) = |F |2(~r(t)).11)T FIf f + g and f − g have a common critical point (a, b), then this point is acritical point of both f and g.12)T FAssume a vector field~F (x, y, z) is a gradient field, thenRC~F ·~dr = 0 whereC is the intersection of x2+ y2= 1 with z = 1.13)T FIf the flux of vector field is zero through any surface S in space, then thedivergence of the field is zero everywhere in space.14)T FThe curl of a gradient field~F (x, y, z) = ∇f(x, y, z) is zero, if f (x, y, z) =√x10+ y10z2.15)T FThe line integral of the curl of a vector field~F (x, y, z) = hx, y, zi along acircle in the xy− plane is zero.16)T FFor a solid E which is rot ationally symmetric around the z-axes, the integralRRRE√x2+ y2dxdydz is equal to the volume of the solid.17)T FThe curvature of the curve ~r(t) = h1+2 cos(1+t), 1+2 sin(1+t)i is constantequal to 1/2 everywhere.18)T FThe directional derivative of f(x, y, z) = div(~F (x, y, z)) of the divergence o fthe vector field~F = hP, Q, Ri in the direction ~v = h1, 0, 0i is Pxx+Qxy+Rxz.19)T FR2π0R2π0r dθ dr =R2π0R2π01 dxdy.20)T FThe set {φ = π, ρ > 0 } in spherical coordinates is the negative z-axis.2Problem 2) (10 points) No justifications are necessary.a) (4 points) Match the vector fields with the definitions1 23 4Enter vector field1-4~F (x, y) = hx + y, x − yi~F (x, y) = h0 , xi~F (x, y) = h−y, xi~F (x, y) = hx, 0ib) (3 p oints) Match the pa r tial differential equations (PDE’s) with their names1) Wave equation2) Heat equation3) Transport equation4) Burgers equationEnter 1-4 PDEut− uxx= 0utt− uxx= 0ut− ux= 0c) (3 point s) Match the curves1234Enter 1-4 Parametrized curve~r(t) = hcos(4t), sin(7t)i~r(t) = h√t sin(t),√t cos(t)i~r(t) = h|cos(4t)|, |sin(7t)|i~r(t) = ht3, t4i3Problem 3) (10 points) No justifications are necessarya) (6 points) Check the boxes which apply. Leave the other boxes empty. The expression”involves XYZ” means that the formulation of the statement contains the object XYZsomewhere.Statement involves involves involvesa curve a surface a vector fieldStokes theoremDivergence theoremLagrange equationsFund. theorem line integra lsSurface area formulaCurvature formulab) (4 p oints) Match the objects with their definitions1 23 4Enter 1-4 object definition~r(t) = hcos(3t), sin ( t), cos(2t)icos(3x) + sin(y) + cos(2z) = 1~r(t, s) = hcos(3t), sin(s), cos(2t)i~F (x, y, z) = hcos(3x), sin(y), cos(2z)i4Problem 4) (10 points)a) (5 Points) Write down a parametrization ~r(t) of the linewhich is perpendicular to the plane x + 2y + z = 0 and whichpasses through the origin.b) (5 points) Find the distance of this line to the point (3, 4, 5).Problem 5) (10 points)Find the place where the curlf(x, y) = curl(~F )(x, y) = Qx(x, y) −Py(x, y)of the vector field~F (x, y) = hP (x, y), Q(x, y)i = h x + y2+ y, x2y + 2x + y2iis maximal under the constraint that the divergenceg(x, y) = div(~F )(x, y) = Px(x, y) + Qy(x, y) .is equal to 1 . Find the functions f, g and solve the problemusing Lagra nge.Problem 6) (10 points)a) (5 points) Find the surface area of the surface~r(s, t) = hs cos(t), s sin(t), tiwith 1 ≤ s ≤ 2, 0 ≤ t ≤ 4π.b) (5 points) Find the arc length of the curve~r(t) = hcos(t), sin(t), tiwith 0 ≤ t ≤ 4π.Hint. You can use without derivation the during lecture derived integralR√x2+ 1 dx =5(x√x2+ 1 + arcsinh(x))/2 and you can leave terms like arcsinh(2).Problem 7) (10 points)Find the volume of the solid given in spherical coo r dina tes asρ(φ, θ) ≤ cos2(φ) .with 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π.Problem 8) (10 points)Find the fluxRRS~F ·~dS of the vector field~F (x, y, z) = hx3, y3, z + ( 1 − x2− y2)(1 − z2)ithrough the boundary S of the solid cylinderE : x2+ y2≤ 1, z2≤ 1 .The boundary of S of the solid E is oriented outwards asusual.Problem 9) (10 points)Where on the sphere is the functionf(φ, θ) = sin(φ) + sin(θ)extremal? Find all maxima, minima and saddle points of fas well as the global maxima and minima.Remark. The variables φ, θ are the usual spherical coordi-nates variables. You are welcome of course to write f(x, y) =sin(x) + sin(y) and look for solutions 0 ≤ x ≤ π, 0 ≤ y < 2π.Problem 10) (10 point s)6The ”sin-log” function sin(x)/ log(x) has no known antideriva-tive. Determined to overcome this obstacle, we neverthelessintegra t eZ10Zeeysin(x)log(x)dxdy .Note that log( x) denotes the nat ur al logarithm. The ln nota-tion is for greenhorns.Problem 11) (10 point s)Find the line integralZ10π0~F ·~dr ,where~F (x, y, z) = hx2+ 1, y2, z3+ x2i and where ~r(t) is thespiral ~r(t) = hcos(t), sin(t), ti.Problem 12) (10 point s)Find the
View Full Document
Unlocking...