5/23/2006, THIRD PRACTICE FINAL Math 21a, Spring 2006Name:MWF 10 Samik Ba suMWF 10 Joachim KriegerMWF 11 Matt LeingangMWF 11 Veronique GodinTTH 10 Oliver KnillTTH 115 Thomas Lam• Please mark the box to the left which lists your section.• Do not detach pages from this exam packet or unstaplethe packet.• Show yo ur work. Answers without r easoning can notbe given credit except for the True/False and multiplecho ice problems.• Please write neatly.• Do not use notes, books, calculators, computers, or otherelectronic aids.• Unspecified functions are assumed to be smooth and de-fined everywhere unless stated otherwise.• You have 180 minutes time to complete your wo r k.• The Biochem section can ignore problems with vectorfields and line integrals.1 202 103 104 105 106 107 108 109 1010 1011 1012A 1013A 1014A 1012B 1013B 1014B 10Total: 140Problem 1) True/False questions (20 points)1)T FFor any two nonzero vectors ~v, ~w the vector ((~v × ~w) ×~v) ×~v) is parallel to~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.5)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 of 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], the line integra lRC~F ·~dr is less or equal thanthe arc length of C.10)T FLet~F be a vector field which coincides with the unit normal vector~N foreach point on a curve C. ThenRC~F ·~dr = 0.11)T FIf for two vector fields~F and~G o ne has curl(~F ) = curl(~G), then~F =~G + (a, b, c), where a, b, c are constants.12)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 .13)T FIf div(~F )(x, y, z) = 0 for all (x, y, z), then curl(~F ) = (0, 0, 0) for all (x, y, z).14)T FIf in spherical coordinates the equation φ = α (with a constant α) definesa plane, then α = π/2.TF PROBLEMS FOR REGULAR AND PHYSICS SECTIONS:15)T FThe divergence of the gradient of any f(x, y, z) is always zero.16)T FFor every vector field~F the identity grad(div(~F )) =~0 holds.17)T FFor every function f, one has div ( curl(grad(f))) = 0.18)T FIf~F is a vector field in space then the flux of~F through any closed surfaceS is 0.19)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.20)T FFor every function f(x, y, z), there exists a vector field~F such that div(~F ) =f.TF PROBLEMS FOR BIOCHEM SECTIONS:21)T FThe following reasoning is correct: tossing 3 unbiased coins, the possiblenumbers of heads appearing are 0, 1, 2, and 3. Therefore each of theseevents has probability 1/4.22)T FTwo events A, B for which P (B) > 0 are independent if and only ifP (A|B) = P (A). .23)T FFor two independent random va riables X, Y one has the following identitiesfor the variance D(X) − D(Y ) = D(X − Y ).24)T FLet A, B be arbitrary events. If P (A|B) = P (B|A) then P (A) = P (B).25)T FThe probability that from 6 random coins to ssed, 6 show tail is smaller thanthe probability that 5 show tail. Each coin has probability 1/2 to show tail.26)T FIf you throw 2 dice and yo u know the first one shows the number 1, thenthe chance that the second one shows 1 is less than 1/6.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 conserva t ive(that is if curl(~F )(x, y, z) = (0, 0, 0) for all points (x, y, z)). Similarly, mark the fieldswhich are incompressible (t hat is if div(~F )(x, y, z) = 0 for all (x, y, z)). No justificationsare 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 diagonalsand the area of the parallelogram?c) (4 points) Assume that the diagonals are perpendicular. What is the relation betweenthe lengths of the sides of the parallelogram?Problem 5) (10 points)Find the volume o f the largest rectangular box with sides parallel to the coordinate planesthat can be inscribed in the ellipsoidx24+y29+z225= 1.Problem 6) (10 points)EvaluateZ80Z2y1/3y2ex2x8dxdy.Problem 7) (10 points)In this problem we evaluateR RD(x−y )4(x+y )4dxdy, where D is the triangular region bounded bythe x and y axis and the line x + y = 1.a) (3 points) Find the region R in the uv-plane which is transformed into D by the changeof variables u = x − y, v = x + y. (It is enough to draw a carefully labeled picture of R.)b) (3 points) Find the Jacobian∂(x,y)∂(u,v)of the transformation (x, y) = (u+v2,v−u2).c) (4 points) EvaluateR RD(x−y )4(x+y )4dxdy using the above defined change of variables.Hint. The general topic of change of variables does not appear this semester. You can solvethe problem nevertheless, when given the formula∂(x,y)∂(u,v)= xyyv− xvyufor the integrationfactor (analoguous to r when changing to polar coor dinates, or ρ2sin(φ) when going tospherical coordinates). The integral in c ) becomes thenR RRu4/v4dudv. The region R isthe triangle bounded by the edges (0, 0), (1, 1), (−1, 1).Problem 8) (10 points)a) (3 points) Find all the critical points of the function f(x, y) = −(x4− 8x2+ y2+
View Full Document
Unlocking...