8/17/2004 FINAL EXAM PRACTICE Maths 21a, O. Knill, Summer 2004Name:• 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.• Do not detach pages from this exam packet or unstaple the packet.• Please write neatly. Answers which are illegible for the grader can not be given credit.• No notes, books, calculators, computers, or other electronic aids can be allowed.• You have 180 minutes time to complete your work.1 202 103 104 105 106 107 108 109 1010 1011 1012 1013 1014 10Total: 150Problem 1) (20 points)T FFor any two nonzero vectors ~v, ~w the vector ((~v × ~w) ×~v) ×~v) is parallel to~w.T FThe cross product satisfies the law (~u ×~v) × ~w = ~u × (~v × ~w).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.T FThe curve ~r(t) = (1 − t)A + tB, t ∈ [0, 1] connects the point A with thepoint B.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.T FThe length of the curve ~r(t) = (t, sin(t)), where t ∈ [0, 2π] isR2π0q1 + cos2(t) dt.T FLet (x0, y0) be the maximum of f(x, y) under the constraint g(x, y) = 1.Then fxx(x0, y0) < 0.T FThe function f(x, y, z) = x2− y2− z2decreases in the direction(2, −2, −2)/√8 at the point (1, 1, 1).T FAssume~F is a vector field satisfying |~F (x, y, z)| ≤ 1 everywhere. For everycurve C : ~r(t) with t ∈ [0, 1], the line integralRC~F ·~dr is less or equal thanthe arc length of C.T FLet~F be a vector field and C is a curve which is a flow line, thenRC~F ·~dr > 0.T FThe divergence of the gradient of any f (x, y, z) is always zero.T FFor every function f, one has div(curl(grad(f))) = 0.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.T FFor every vector field~Fthe identity grad(div(~F)) =~0holds.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.T FIf~F is a vector field in space then the flux of~F through any closed surfaceS is 0.T FIf div(~F )(x, y, z) = 0 for all (x, y, z), then curl(~F ) = (0, 0, 0) for all (x, y, z).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.T FIf in spherical coordinates the equation φ = α (with a constant α) definesa plane, then α = π/2.T FFor every function f(x, y, z), there exists a vector field~F such that div(~F ) =f.Problem 2) (10 points)Match the equations with the objects. No justifications are needed.I II III IVV VI VII VIIIEnter I,II,III,IV,V,VI,VII,VIII here Equationg(x, y, z) = cos(x) + sin(y) = 1y = cos(x) − sin(x)~r(t) = (cos(t), sin(t))~r(u, v) = (cos(u), sin(v), cos(u) sin(v))~F (x, y, z) = (cos(x), sin(x), 1)z = f(x, y) = cos(x) + sin(y)g(x, y) = cos(x) − sin(y) = 1~F (x, y) = (cos(x), sin(x))Problem 3) (10 points)Mark with a cross in the column below ”conservative” if a vector fields is conservative (thatis if curl(~F )(x, y, z) = (0, 0, 0) for all points (x, y, z)). Similarly, mark the fields which areincompressible (that is if div(~F )(x, y, z) = 0 for all (x, y, z)). No justifications are needed.Vectorfield conservative 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 points)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 diagonals andthe area of the parallelogram?c) (4 points) Assume that the diagonals are perpendicular. What is the relation between thelengths of the sides of the parallelogram?Problem 5) (10 points)Find the volume of the largest rectangular box with sides parallel to the coordinate planes thatcan be inscribed in the ellipsoidx24+y29+z225= 1.Problem 6) (10 points)EvaluateZ80Z2y1/3y2ex2x8dxdy.Problem 7) (10 points)EvaluateR RD2xyx2+y2dxdy, where D is the intersection of the annulus 1 ≤ x2+ y2≤ 2 with thesecond quadrant {x ≤ 0, y ≥ 0 }.Problem 8) (10 points)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 absolute maxi-mum.Problem 9) (10 points)Find the volume of the wedge shaped solid that lies above the xy-plane and below the planez = x and within the cylinder x2+ y2= 4.Problem 10) (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 11) (10 points)a) Find the linear approximation 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 12) (10 points)Evaluate the line integral of the vector field~F (x, y) = (y2, x2) in the clockwise direction aroundthe 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 13) (10 points)Use Stokes theorem to evaluate the line integral of~F (x, y, z) = (−y3, x3, −z3) along the curve~r(t) = (cos(t), sin(t), 1 − cos(t) − sin(t)) with t ∈ [0, 2π].Problem 14) (10 points)Let S be the graph of the function f(x, y) = 2 −x2−y2which lies above the disk {(x, y) | x2+y2≤ 1} in the xy-plane. The surface S is oriented so that the normal vector points upwards.Compute the fluxR RS~F ·~dS of the vectorfield~F = (−4x +x2+ y2− 11 + 3y2, 3y, 7 − z −2xz1 + 3y2)through S using the divergence
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