Naala Brewer Brewer MAT 267 Spring 2008WeBWorK assignment number Section 13.7 is due : 06/13/2004 at 08:11am MST.The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are makingsome kind of mistake. Usually you can attempt a problem as many times as you want before the due date. However, if you arehaving trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA’s or your professor forhelp. Don’t spend a lot of time guessing – it’s not very efficient or effective.Give 4 or 5 significant digits for (floating point) numerical answers. For most problems when entering numerical answers,you can if you wish enter elementary expressions such as 2 ∧3 instead of 8, sin(3 ∗pi/2) instead of -1, e ∧(ln(2)) instead of 2,(2+tan(3)) ∗(4−sin(5)) ∧6−7/8 instead of 27620.3413, etc. Here’s the list of the functions which WeBWorK understands.You can use the E-mail instructor button on each problem page to send e-mail to the professors.1. (1 pt) EvaluateZ ZSp1+ x2+ y2dS where S is the heli-coid: r(u,v) = ucos(v)i+usin(v)j+vk, with 0 ≤u ≤4,0 ≤v ≤4πCorrect Answers:• 318.3480555637662. (1 pt) Let S be the part of the plane 4x+ 3y+ z = 3 whichlies in the first octant, oriented upward. Find the flux of thevector field F = 3i+ 2j+ 2k across the surface S.Correct Answers:• 7.53. (1 pt)Find the flux of F(x,y,z) = (3xy2,3x2y,z3) out of the sphereof radius 5 centered at the origin. Hint: Use spherical coordi-nates and be mindful of the orientation.The flux is given by the integral:55RbaRdcf(θ,φ)dθdφ where:a = , b = , c = , d = , andf(θ,φ) =(use variables ”t” for theta and ”p” for phi).The value of the integral is .Correct Answers:• 0• 3.14159265358979• 0• 6.28318530717959• 6*cos(t)ˆ2*sin(t)ˆ2*sin(p)ˆ5+cos(p)ˆ4*sin(p)• 23561.94490192344. (1 pt) A fluid has density 2 and velocity field v = −yi+xj+ 4zk.Find the rate of flow outward through the sphere x2+y2+z2= 1Correct Answers:• 33.51029333333335. (1 pt) Use Gauss’s law to find the charge enclosedby the cube with vertices (±1,±1,±1) if the electric field isE(x,y,z) = 6xi+ 5yj+ 2zk.ε0Correct Answers:• 1046. (1 pt) The temperature u in a star of conductivity 1is inversely proportional to the distance from the center: u =1√x2+y2+z2.If the star is a sphere of radius 5, find the rate of heat flow out-ward across the surface of the star.Correct Answers:• 12.56637061435927. (1 pt) Suppose F is a radial force field, S1is a sphere of ra-dius 7 centered at the origin, and the flux integralZ ZS1F·dS =7.Let S2be a sphere of radius 35 centered at the origin, and con-sider the flux integralZ ZS2F·dS.(A) If the magnitude of F is inversely proportional to thesquare of the distance from the origin,what is the value ofZ ZS2F·dS?(B) If the magnitude of F is inversely proportional to the cubeof the distance from the origin, what is the value ofZ ZS2F·dS?Correct Answers:• 7• 1.48. (1 pt)Let M be the closed surface that consists of the hemisphereM1: x2+ y2+ z2= 1, z ≥0,and its baseM2: x2+ y2≤ 1, z = 0.Let E be the electric field defined by E = (3x, 3y, 3z). Find theelectric flux across M. Write the integral over the hemisphere1using spherical coordinates, and use the outward pointing nor-mal.ZZM1E·dS =ZbaZdcf(θ,φ)dθdφ,wherea = , b = , c = , d = ,Using t for θ and p for φ,f(θ,φ) =RRM1E·dS =RRM2E·dS = , soRRME·dS = .Correct Answers:• 0• 1.5707963267949• 0• 6.28318530717959• 3*[sin(p)ˆ3+sin(p)*cos(p)ˆ2]• 18.8495559215388• 0• 18.8495559215388Generated by the WeBWorK systemcWeBWorK Team, Department of Mathematics, University of