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All Textbook Solutions for Calculus (MindTap Course List)
The Mean Value Theorem for double integrals says that if f is a continuous function on a plane region D that is of type I or II, then there exists a point (x0,y0) in D such that Df(x,y)dA=f(x0,y0)A(D) Use the Extreme Value Theorem 14.7.8 and Property 15.2.11 of integrals to prove this theorem. Use the proof of the single-variable version in Section 5.5 as a guide.59Ea Evaluate D1(x2+y2)n/2dA, where n is an integer and D is the region bounded by the circles with center the origin and radii r and R, 0rR. b For what values of n does the integral in part a have a limit as r0+? c Find E1(x2+y2+z2)n/2dV, where E is the region bounded by the spheres with center the origin and radii r and R,0rR. d For what values of n does the integral in part c have a limit as r0+?If x denotes the greatest integer in x, evaluate the integral Rx+ydA where R={(x,y)|1x3,2y5}.Evaluate the integral 0101emax{x2,y2}dydx where max {x2,y2} means the larger of the numbers x2 and y2.3PIf a, b, and c are constant vectors, r is the position vector xi+yj+zk, and E is given by the inequalities 0ar,0br,0cr, show that E(ar)(br)(cr)dV=()28|a(bc)|5PLeonhard Euler was able to find the exact sum of the series in Problem 5. In 1736 he proved that n11n2=26 In this problem we ask you to prove this fact by evaluating the double integral in Problem 5. Start by making the change of variables x=uv2y=u+v2 This gives a rotation about the origin through the angle /4. You will need to sketch the corresponding region in the uv-plane. Hint: If, in evaluation the integral, you encounter either of the expressions (1sin)/cos or (cos)/(1+sin), you might like to use the identity cos=sin((/2)) and the corresponding identity for sin.7PShow that 0arctanxarctanxxdx=2ln by first expressing the integral as an iterated integral.9P10P11PEvaluate limnn2i=1nj=1n21n2+ni+j.The plane xa+yb+zc=1a0,b0,c0 cuts the ellipsoid x2a2+y2b2+z2c21 into two pieces. Find the volume of the smaller piece.110 Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. F(x,y)=0.3i0.4jSketch the vector field F by drawing a diagram like Figure 5 or Figure 9. F(x,y)=12xi+yjSketch the vector field F by drawing a diagram like Figure 5 or Figure 9. F(x,y)=12i+(yx)j110 Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. F(x,y)=yi+(x+y)j110 Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. F(x,y)=yi+xjx2+y2110 Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. F(x,y)=yixjx2+y27E8E9E10E11EMatch the vector fields F with the plots labelled IIV. Give reasons for your choices. F(x,y)=y,xyMatch the vector fields F with the plots labelled IIV. Give reasons for your choices. F(x,y)=y,y+2Match the vector fields F with the plots labeled IIV. Give reasons for your choices. F(x,y)=cos(x+y),xMatch the vector fields F on 3 with the plots labelled IIV. Give reasons for your choices. F(x,y,z)=i+2j+3kMatch the vector fields F on 3 with the plots labeled IIV. Give reasons for your choices. F(x,y,z)=i+2j+zkMatch the vector fields F on 3 with the plots labelled IIV. Give reasons for your choices. F(x,y,z)=xi+yj+3kMatch the vector fields F on 3 with the plots labelled IIV. Give reasons for your choices. F(x,y,z)=xi+yj+zk19E20E21EFind the gradient vector field of f. f(s,t)=2s+3tFind the gradient vector field of f. f(x,y,z)=x2+y2+z2Find the gradient vector field of f. f(x,y,z)=x2yey/zFind the gradient vector field f of f and sketch it. f(x,y)=12(xy)2Find the gradient vector field f of f and sketch it. f(x,y)=12(x2y2)27EPlot the gradient vector field of f together with a contour map of f. Explain how they are related to each other. f(x,y)=cosx2sinyMatch the functions f with the plots of their gradient vector fields labeled IIV. Give reasons for your choices. f(x,y)=x2+y2Match the functions f with the plots of their gradient vector fields labeled IIV. Give reasons for your choices. f(x,y)=x(x+y)Match the functions f with the plots of their gradient vector fields labeled IIV. Give reasons for your choices. f(x,y)=(x+y)2Match the functions f with the plots of their gradient vector fields labeled IIV. Give reasons for your choices. f(x,y)=sinx2+y2A particle moves in a velocity field V(x,y)=x2,x+y2. If it is at position (2,1) at time t=3, estimate its location at time t=3.01.34EThe flow lines or streamlines of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines. a Use a sketch of the vector field F(x,y)=xiyj to draw some flow lines. From your sketches, can you guess the equations of the flow lines? b If parametric equations of a flow line are x=x(t), y=y(t), explain why these functions satisfy the differential equations dx/dt=x and dy/dt=y. Then solve the differential equations to find an equation of the flow line that passes through the point (1,1).a Sketch the vector field F(x,y)=i+xj and then sketch some flow lines. What shape do these flow lines appear to have? b If parametric equations of the flow lines are x=x(t), y=y(t), what differential equations do these functions satisfy? Deduce that dy/dx=x. c If a particle starts at the origin in the velocity field given by F, find an equation of the path it follows.Evaluate the line integral, where C is the given curve. cyds,C:x=t2,y=2t,0t3Evaluate the line integral, where C is the given curve. c(x/y)ds,C:x=t3,y=t4,1t2Evaluate the line integral, where C is the given curve. cxy4ds,C is the right half of the circle x2+y2=16.4EEvaluate the line integral, where C is the given curve. c(x2y+sinx)dy, C is the arc of the parabola y=x2 from (0,0) to (,2)Evaluate the line integral, where C is the given curve. Cexdx, C is the arc of the curve x=y3 from (1,1) to (1,1)Evaluate the line integral, where C is the given curve. c(x+2y)dx+x2dy, C consists of line segments from (0,0) to (2,1) and from (2,1) to (3,0)Evaluate the line integral, where C is the given curve. cx2dx+y2dy, C consists of the arc of the circle x2+y2=4 from (2,0) to (0,2) followed by the line segment from (0,2) to (4,3)Evaluate the line integral, where C is the given curve. cx2yds, C:x=cost,y=sint,z=t,0t/2Evaluate the line integral, where C is the given curve. Cy2zds, C is the line segment from (3,1,2) to (1,2,5)Evaluate the line integral, where C is the given curve. cxeyzds, C is the line segment from (0,0,0) to (1,2,3)Evaluate the line integral, where C is the given curve. C(x2+y2+z2)ds, C:x=t,y=cos2t,z=sin2t,0t2Evaluate the line integral, where C is the given curve. cxyeyzdy,C:x=t,y=t2,z=t3,0t114EEvaluate the line integral, where C is the given curve. cz2dx+x2dy+y2dz, C is the line segment from (1,0,0) to (4,1,2)16ELet F be the vector fields shown in the figure. a If C1 is the vertical line segment from (3,3) to (3,3), determine whether C1Fdr is positive, negative, or zero. b If C2 is the counterclockwise-oriented circle with radius 3 and center the origin, determine whether C2Fdr is positive, negative, or zero.The figure shows a vector field F and two curves C1 and C2. Are the line integrals of F over C1 and C2 positive, negative, or zero? Explain.19E20E21EEvaluate the line integral CFdr, where C is given by the vector function r(t). F(x,y,z)=xi+yj+xyk, r(t)=costi+sintj+tk,0t23E24E25E26EUse a graph of the vector field F and the curve C to guess whether the line integral of F over C is positive, negative or zero. Then evaluate the line integral. F(x,y)=(xy)i+xyj, C is the arc of the circle x2+y2=4 traversed counter-clockwise from (2,0) to (0,2)Use a graph of the vector field F and the curve C to guess whether the line integral of F over C is positive, negative or zero. Then evaluate the line integral. F(x,y)=xx2+y2i+yx2+y2j, C is the parabola y=1+x2 from (1,2) to (1,2)a Evaluate the line integral CFdr, where F(x,y)=ex1i+xyj and C is given by r(t)=t2i+t3j,0t1. b Illustrate part a by using a graphing calculator or computer to graph C and the vectors from the vector field corresponding to t=0,1/2, and 1 as in Figure 13.a Evaluate the line integral CFdr, where F(x,y,z)=xizj+yk and C is given by r(t)=2ti+3tjt2k,1t1. b Illustrate part a by using a computer to graph C and the vectors from the vector field corresponding to t=1 and 12 as in Figure 13.Find the exact value of Cx3y3zds, where C is the curve with parametric equations x=etcos4t,y=etsin4t,z=et,0t2.a Find the work done by the force field F(x,y)=x2i+xyj on a particle that moves once around the circle x2+y2=4 oriented in the counter-clockwise directions. b Use a computer algebra system to graph the force field and circle on the same screen. Use the graph to explain your answer to part a.A thin wire is bent into the shape of a semicircle x2+y2=4,x0. If the linear density is a constant k, find the mass and center of mass of the wire.A thin wire has the shape of the first-quadrant part of the circle with center the origin and radius a. If the density function is p(x,y)=kxy, find the mass and center of mass of the wire.a Write the formulas similar to Equations 4 for the center of mass (x,y,z) of a thin wire in the shape of a space curve C if the wire has density function p(x,y,z). b Find the center of mass of a wire in the shape of the helix x=2sint,y=2cost,z=3t,0t2, if the density is a constant k.Find the mass and center of mass of a wire in the shape of the helix x=t,y=cost,z=sint,0t2, if the density at any points is equal to the square of the distance from the origin.37E38EFind the work done by the force field F(x,y)=xi+(y+2)j in moving an object along an arch of the cycloid r(t)=(tsint)i+(1cost)j0t2Find the work done by the force field F(x,y)=x2i+yexj on a particle that moves along the parabola x=y2+1 from (1,0) to (2,1).41E42E43EAn object with mass m moves with position function r(t)=asinti+bcostj+ctk,0t/2 Find the work done on the object during this time period.A 160-lb man carries a 25-lb can of paint up a helical staircase that encircles a silo with a radius of 20 ft. If the silo is 90 ft high and the man makes exactly three complete revolutions climbing to the top, how much work is done by the man against gravity?46Ea Show that a constant force field does zero work on a particle that moves once uniformly around the circle x2+y2=1. b Is this also true for a force field F(x)=kx where k is a constant and x=x,y?48EIf C is a smooth curve given by a vector function r(t),atb and v is a constant vector, show that Cvdr=v[r(b)r(a)]50EAn object moves along the curve C shown in the figure from 1, 2 to 9, 8. The lengths of the vectors in the force field F are measured in newtons by the scales on the axes. Estimate the work done by F on the object.Experiments show that a steady current I in a long wire produces a magnetic field B that is tangent to any circle that lies in the plane perpendicular to the wire and whose center is the axis of the wire as in the figure. Ampres Law relates the electric current to its magnetic effects and states that CBdr=0I where I is the net current that passes through any surface bounded by a closed curve C, and 0 is a constant called the permeability of free space. By taking C to be a circle with radius r, show that the magnitude B=|B| of the magnetic field at a distance r from the center of the wire is B=0I2rThe figure shows a curve C and a contour map of a function f whose gradient is continuous. Find Cfdr.2EDetermine whether or not F is a conservative vector field. If it is, find a function f such that F=f. F(x,y)=(xy+y2)i+(x2+2xy)jDetermine whether or not F is a conservative vector field. If it is, find a function f such that F=f. F(x,y)=(y22x)i+2xyjDetermine whether or not F is a conservative vector field. If it is, find a function f such that F=f. F(x,y)=y2exyi+(1+xy)exyj6E7E8E9EDetermine whether or not F is a conservative vector field. If it is, find a function f such that F=f. F(x,y)=(lny+y/x)i+(lnx+x/y)jThe figure shows the vector field F(x,y)=2xy,x2 and three curves that start at 1, 2 and end at 3, 2. a Explain why CFdr has the same value for all three curves. b What is this common value?a Find a function f such that F=f and b use part a to evaluate CFdr along the given curve C. F(x,y)=(3+2xy2)i+2x2yj, C is the arc of the hyperbola y=1/x from 1, 1 to (4,14)a Find a function f such that F=f and b use part a to evaluate CFdr along the given curve C. F(x,y)=x2y3i+x3y2j, C:r(t)=t32t,t3+2t,0t1a Find a function f such that F=f and b use part a to evaluate CFdr along the given curve C. F(x,y)=(1+xy)exyi+x2exyj, C:r(t)=costi+2sintj,0t/2a Find a function f such that F=f and b use part a to evaluate CFdr along the given curve C. F(x,y,z)=yzi+xzj+(xy+2z)k, C is the line segment from (1,0,2) to (4,6,3)16E17E18EShow that the line integral is independent of path and evaluate the integral. C2xeydx+(2yx2ey)dy, C is any path from (1,0) to (2,1)Show that the line integral is independent of path and evaluate the integral. Csinydx+(xcosysiny)dy, C is any path from (2,0) to (1,)Suppose youre asked to determine the curve that requires the least work for a force field F to move a particle from one point to another point. You decide to check first whether F is conservative, and indeed it turns out that it is. How would you reply to the request?22EFind the work done by the force field F in moving an object from P to Q. F(x,y)=x3i+y3j;P(1,0),Q(2,2)Find the work done by the force field F in moving an object from P to Q. F(x,y)=(2x+y)i+xj;P(1,1),Q(4,3)Is the vector field shown in the figure conservative? Explain.Is the vector field shown in the figure conservative? Explain.If F(x,y)=sinyi+(1+xcosy)j, use a plot to guess whether F is conservative. Then determine whether your guess is correct.Let F=f, where f(x,y)=sin(x2y). Find curves C1 and C2 that are not closed and satisfy the equation. a C1Fdr=0 b C2Fdr=0Show that if the vector field F=Pi+Qj+Rk is conservative and P, Q, R have continuous first-order partial derivatives, then Py=QxPz=RxQz=RyUse Exercise 29 to show that the line integral Cydx+xdy+xyzdz is not independent of path.Determine whether or not the given set is a open, b connected, and c simply-connected. {(x,y)|0y3}Determine whether or not the given set is a open, b connected, and c simply-connected. {(x,y)|1|x|2}Determine whether or not the given set is a open, b connected, and c simply-connected. {(x,y)|1x2+y24,y0}34E35Ea Suppose that F is an inverse square force field, that is, F(r)=cr|r|3 for some constant c, where r=xi+yj+zk. a Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d2 from these points to the origin. b An example of an inverse square field is the gravitational field F=(mMG)r/|r|3 discussed in Example 16.1.4. Use part a to find the work done by the gravitational field when the earth moves from aphelion at a maximum distance of 1.52108 km from the sun to perihelion at a minimum distance of 1.47108 km. Use the values m=1.521024 kg, M=1.521030 kg, and G=6.671011 N3m2/kg2. c Another example of an inverse square field is the electric force field F=qQr/|r|3 discussed in Example 16.1.5. Suppose that an electron with a charge of 1.61019 C is located at the origin. A positive unit charge is positioned a distance 1012 m from the electron and moves to a position half that distance iron: the electron. Use part a to find the work done by the electric force field. Use the value =8.985109.Evaluate the line integral by two methods: a directly and b using Greens Theorem. cy2dx+x2ydy, C is the rectangle with vertices (0,0), (5,0), (5,4), and (0,4)Evaluate the line integral by two methods: a directly and b using Greens Theorem. cydxxdy, C is the circle with center the origin and radius 4Evaluate the line integral by two methods: a directly and b using Greens Theorem. cxydx+x2y3dy, C is the triangle with vertices (0,0), (1,0), and (1,2)Evaluate the line integral by two methods: a directly and b using Greens Theorem. cx2y2dx+xydy, C consists of the arc of the parabola y=x2 from (0,0) to(1,1)and the line segments from (1,1) to (0,1), and from (0,0)Use Greens Theorem to evaluate the line integral along the given positively oriented curve. cyexdx+2exdy, C is the rectangle with vertices (0,0), (3,0), (3,4), and (0,4)Use Greens Theorem to evaluate the line integral along the given positively oriented curve. c(x2+y2)dx+(x2y2)dy, C is the rectangle with vertices (0,0), (2,1), and (0,1)Use Greens Theorem to evaluate the line integral along the given positively oriented curve. c(y+ex)dx+(2x+cosy2)dy, C is the boundary of the region enclosed by the parabolas y=x2 and x=y2Use Greens Theorem to evaluate the line integral along the given positively oriented curve. cy4dx+2xy3dy, C is the ellipse x2+2y2=2Use Greens Theorem to evaluate the line integral along the given positively oriented curve. cy3dxx3dy, C is the circle x2+y2=4Use Greens Theorem to evaluate the line integral along the given positively oriented curve. c(1y3)dx+(x3+ey2)dy, C is the boundary of the region between the circles x2+y2=4 and x2+y2=9Use Greens Theorem to evaluate cFdr. Check the orientation of the curve before applying the theorem. F(x,y)=ycosxxysinx,xy+xcosx, C is the triangle from (0,0), to (0,4) (2,0), to (0,0)12E13E14EVerify Greens Theorem by using a computer algebra system to evaluate both the line integral and the double integral. P(x,y)=x3y4, Q(x,y)=x5y4, C consists of the line segment from (/2,0), to (/2,0) followed by the arc of the curve y=cosx from (/2,0), to (/2,0)Verify Greens Theorem by using a computer algebra system to evaluate both the line integral and the double integral. P(x,y)=2xx3y5, Q(x,y)=x3y8, C is the ellipse 4x2+y2=417EA particle starts at the origin, moves along the x-axis to (5,0), then along the quarter-circle x2+y2=25, x0, y0 to the point (0,5), and then down the y-axis back to the origin. Use Greens Theorem to find the work done on this particle by the force field F(x,y)=sinx,siny+xy+13x3.19EIf a circle C with radius 1 rolls along the outside of the circle x2+y2=16, a fixed point P on C traces out a curve called an epicycloid, with parametric equations x=5costcos5t, y=5sintsin5t. Graph the epicycloid and use 5 to find the area it encloses.21E22EUse Exercise 22 to find the centroid of a quarter-circular region of radius a.24EA plane lamina with constant density (x,y)= occupies a region in the xy-plane bounded by a simple closed path C. Show that its moments of inertia about the axes are Ix=3cy3dx Iy=3cx3dy26EUse the method of Example 5 to calculate CFdr, where F(x,y)=2xyi+(y2x2)j(x2+y2)2 and C is any positively oriented simple closed curve that encloses the origin.28EIf F is the vector field of Example 5, show that CFdr=0 for every simple closed path that does not pass through or enclose the origin.Complete the proof of the special case of Greens Theorem by proving Equation 3. cQdy=DQxdA(3)31EFind a the curl and b the divergence of the vector field. F(x,y,z)=xy2z2i+x2yz2j+x2y2zkFind a the curl and b the divergence of the vector field. F(x,y,z)=x3yz2j+y4z3zkFind a the curl and b the divergence of the vector field. F(x,y,z)=xyezi+yzexkFind a the curl and b the divergence of the vector field. F(x,y,z)=sinyzi+sinzxj+sinxykFind a the curl and b the divergence of the vector field. F(x,y,z)=x1+zi+y1+xj+z1+ykFind a the curl and b the divergence of the vector field. F(x,y,z)=ln(2y+3z)i+ln(x+3z)j+ln(x+2y)kFind a the curl and b the divergence of the vector field. F(x,y,z)=exsiny,eysinz,ezsinxFind a the curl and b the divergence of the vector field. F(x,y,z)=arctan(xy),arctan(yz),arctan(zx)The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. In other words, F is independent of z and its z-component is 0. a Is div F positive, negative, or zero? Explain. b Determine whether curl F=0. If not, in which direction does curl F point?The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. In other words, F is independent of z and its z-component is 0. a Is div F positive, negative, or zero? Explain. b Determine whether curl F=0. If not, in which direction does curl F point?The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. In other words, F is independent of z and its z-component is 0. a Is div F positive, negative, or zero? Explain. b Determine whether curl F=0. If not, in which direction does curl F point?Let f be a scalar field and F a vector field. State whether each expression is meaningful. If not, explain why. If so, state whether it is a field or a vector field. a curl f b grad f c div F d curlgrad f e grad F f graddiv F g divgrad f h graddiv f i curlcurl F j divdiv F k (gradf)(divF) l divcurlgrad fDetermine whether or not the vector field is conservative. If it is conservative, find a function f such that F=f. F(x,y,z)=y2z3i+2xyz3j+3xy2z2kDetermine whether or not the vector field is conservative. If it is conservative, find a function f such that F=f. F(x,y,z)=xyz4i+x2z4j+4x2yz3k15EDetermine whether or not the vector field is conservative. If it is conservative, find a function f such that F=f. F(x,y,z)=i+sinzj+ycoszk17EDetermine whether or not the vector field is conservative. If it is conservative, find a function f such that F=f. F(x,y,z)=exsinyzi+zexcosyzj+yexcosyzkIs there a vector field G on 3 such that curl G=xsiny,cosy,zxy? Explain.20EShow that any vector field of the form F(x,y,z)=f(x)i+g(y)j+h(z)k where f, g, h are differentiable functions, is irrational.22E23EProve the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F,FG, and FG are defined by (fF)(x,y,z)=f(x,y,z)F(x,y,z)(FG)(x,y,z)=F(x,y,z)G(x,y,z)(FG)(x,y,z)=F(x,y,z)G(x,y,z) curl(F+G)=curlF+curlG25EProve the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F,FG, and FG are defined by (fF)(x,y,z)=f(x,y,z)F(x,y,z)(FG)(x,y,z)=F(x,y,z)G(x,y,z)(FG)(x,y,z)=F(x,y,z)G(x,y,z) curl(fF)=fcurlF+(f)F27EProve the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F,FG, and FG are defined by (fF)(x,y,z)=f(x,y,z)F(x,y,z)(FG)(x,y,z)=F(x,y,z)G(x,y,z)(FG)(x,y,z)=F(x,y,z)G(x,y,z) div(fg)=0Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F,FG, and FG are defined by (fF)(x,y,z)=f(x,y,z)F(x,y,z)(FG)(x,y,z)=F(x,y,z)G(x,y,z)(FG)(x,y,z)=F(x,y,z)G(x,y,z) curl(curlF)=grad(divF)2FLet r=xi+yj+zk and r=|r|. Verify each identity. a r=3 b (rr)=4r c 2r3=12rLet r=xi+yj+zk and r=|r|. Verify each identity. a r=r/r b r=0 c (1/r)=r/r3 d lnr=r/r2Let r=xi+yj+zk and r=|r|. If F=r/rp, find div F. Is there a value of p for which div F=0?Use Greens Theorem in the form of Equation 13 to prove Greens first identity: Df2gdA=cf(g)ndsDfgdA Where D and C satisfy the hypotheses of Greens Theorem and the appropriate partial derivatives of f and g exist and are continuous. The quantity gn=Dng occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.34ERecall from Section 14.3 that a function g is called harmonic on D if it satisfies Laplaces equation, that is 2g=0 on D. Use Greens first identity with the same hypotheses as in Exercise 33 to show that if g is harmonic on D, then CDngds=0. Here Dng is the normal derivative of g defined in Exercise 33. Use Greens Theorem in the form of Equation 13 to prove Greens first identity: Df2gdA=cf(g)ndsDfgdA Where D and C satisfy the hypotheses of Greens Theorem and the appropriate partial derivatives of f and g exist and are continuous. The quantity gn=Dng occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.36EThis exercise demonstrates a connection between the curl vector and rotations. Let B be a rigid body rotating about the z-axis. The rotation can be described by the vector w=k, where co is the angular speed of B, that is, the tangential speed of any point P in B divided by the distance d. from the axis of rotation. Let r=(x,y,z) be the position vector of P. a By considering the angle in the figure, show that the velocity field of B is given by v=wr b Show that v=yi+xj. c Show that curl v = 2w.Maxwells equations relating the electric field E and magnetic field H as they vary with time in a region containing no charge and no current can be stated as follows: div E=0div H=0 curlE=1cHtcurlH=1cEt where c is the speed of light. Use these equations to prove the following: a (E)=1c22Et2 b (H)=1c22Ht2 c 2E=1c22Et2 d 2H=1c22Ht2We have seen that all vector fields of the form F=g satisfy the equation curl F=0 and that all vector fields of the form F=curl G satisfy the equation div f=0 assuming continuity of the appropriate partial derivatives. This suggests the question: are there any questions that all functions of the form f=div G must satisfy? Show that the answer to this question is No by providing that every continuous function f on 3 is the divergence of some vector field. Hint: Let G(x,y,z)=g(x,y,z),0,0, where g(x,y,z)=0xf(t,y,z)dt.Determine whether the points P and Q lie on the given surface. r(u,v)=u+v,u2v,3+uvP(4,5,1),Q(0,4,6)Determine whether the points P and Q lie on the given surface. r(u,v)=1+uv,u+v2,u2v2P(1,2,1)Q(2,3,3)3EIdentify the surface with the given vector equation. r(u,v)=u2i+ucosvj+usinvk5EIdentify the surface with the given vector equation. r(s,t)=3cost,s,sint,1s1Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have u constant and which have v constant. r(u,v)=u2,v2,u+v, 1u1, 1v18E9E10E11E12EMatch the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant. r(u,v)=ucosvi+usinvj+vkMatch the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant. r(u,v)=uv2i+u2vj+(u2v2)kMatch the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant. r(u,v)=(u3u)i+v2j+u2kMatch the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant. x=(1u)(3+cosv)cos4u, y=(1u)(3+cosv)sin4u, z=3u+(1u)sinv17E18EFind da parametric representation for the surface. The plane through the origin that contains the vectors i j and j kFind da parametric representation for the surface. The plane that passes through the point (0,1,5) and contains the vectors 2,1,4 and 3,2,5Find da parametric representation for the surface. The part of the hyperboloid 4x24y2z2=4 that lies in front of the yz-planeFind da parametric representation for the surface. The part of the ellipsoid x2+2y2+3z2=1 that lies to the left of the xz-planeFind da parametric representation for the surface. The part of the sphere x2+y2+z2=4 that lies above the cone z=x2+y2Find da parametric representation for the surface. The part of the cylinder x2+z2=9 that lies above the xy-plane and between the planes y=4 and y=4Find da parametric representation for the surface. The part of the sphere x2+y2+z2=36 that lies between the planes z=0 and z=33Find da parametric representation for the surface. The part of the plane z=x+3 that lies inside the cylinder x2+y2=127E28E29E30Ea What happens to the spiral tube in Example 2 see Figure 5 if we replace cos u by sin u and sin u by cos u? b What happens if we replace cos u by cos 2u and sin u by sin 2u?32EFind an equation of the tangent plane to the given parametric surface at the specified point. x=u+v, y=3u2, z=uv; (2,3,0)Find an equation of the tangent plane to the given parametric surface at the specified point. x=u2+1, y=v3+1, z=u+v; (5,2,3)35E36E37E38E39E40EFind the area of the surface. The part of the plane x+2y+3z=1 that lies inside the cylinder x2+y2=3Find the area of the surface. The part of the cone z=x2+y2 that lies between the plane y=x and the cylinder y=x2Find the area of the surface. The surface z=23(x3/2+y3/2), 0x1, 0y1Find the area of the surface. The part of the surface z=42x2+y that lies above the triangle with vertices (0,0), (1,0), and (1,1)Find the area of the surface. The part of the surface z=xy that lies within the cylinder x2+y2=1Find the area of the surface. The part of the surface x=z2+y that lies between the planes y=0, y=2, z=0, and z=2Find the area of the surface. The part of the paraboloid y=x2+z2 that lies within the cylinder x2+z2=1648EFind the area of the surface. The surface with parametric equations x=u2, y=uv, z=12v2, 0u1, 0v2Find the area of the surface. The part of the sphere x2+y2+z2=b2 that lies inside the cylinder x2+y2=a2, where 0ab51E52E53E54E55E56E57E58Ea Show that the parametric equations x=asinucosv,y=bsinusinv,z=ccosu,0u,0v2, represent an ellipsoid. b Use the parametric equations in part a to graph the ellipsoid for the case a=1,b=2,c=3 c Set up, but do not evaluate, a double integral for the surface area of the ellipsoid in part b.a Show that the parametric equations x=acoshucosv,y=bcoshusinv,z=csinhu, represent a hyperboloid of one sheet. b Use the parametric equations in part a to graph the hyperboloid for the case a=1,b=2,c=3. c Set up, but do not evaluate, a double integral for the surface area of the part of the hyperboloid in part b that lies between the planes z=3 and z=3.Find the area of the part of the sphere x2+y2+z2=4z that lies inside the paraboloid z=x2+y2.The figure shows the surface created when the cylinder y2+z2=1 intersects the cylinder x2+z2=1. Find the area of this surface.63Ea Find a parametric representation for the torus obtained by rotating about the z-axis the circle in the xz-plane with center (b,0,0) and radius ab. Hint: Take as parameters the angles and shown in the figure. b Use the parametric equations found in part a to graph the torus for several values of a and b. c Use the parametric representation from part a to find the surface area of the torus.Let S be the surface of the box enclosed by the planes x=1,y=1,z=1. Approximate Scos(x+2y+3z)dS by using a Riemann sum as in Definition 1, taking the patches Sij to be the squares that are the faces of the box S and the points Pij* to be the centers of the squares.2E3ESuppose that f(x,y,z)=g(x2+y2+z2), where g is a function of one variable such that g(2)=5. Evaluate Sf(x,y,z)dS, where S is the sphere x2+y2+z2=4.Evaluate the surface integral. S(x+y+z)dS, S is the parallelogram with parametric equations x=u+v,y=uv,z=1+2u+v,0u2,0v1Evaluate the surface integral. SxyzdS, S is the cone with parametric equations x=u+v,y=uv,z=1+2u+v,0u2,0v17EEvaluate the surface integral. S(x2+y2)dS, S is the surface with vector equation r(u,v)=2uv,u2v2,u2+v2,u2+v21Evaluate the surface integral. Sx2yzdS, S is the part of the plane z=1+2x+3y that lies above the rectangle [0,3][0,2]10EEvaluate the surface integral. SxdS, S is the triangular region with vertices (1,0,0),(0,2,0),and(0,0,4)Evaluate the surface integral. SydS, S is the surface z=23(x3/2+y3/2),0x1,0y1Evaluate the surface integral. Sz2dS, S is the part of the paraboloid x=y2+z2 given by 0x1Evaluate the surface integral. Sy2z2dS, S is the part of the cone y=x2+z2 given by 0y515EEvaluate the surface integral. Sy2dS, S is the part of the sphere x2+y2+z2=1 that lies above the cone z=x2+y217EEvaluate the surface integral. S(x+y+z)dS, S is the part of the half-cylinder x2+z2=1,z0, that lies between the planes y=0 and y=2Evaluate the surface integral. SxzdS, S is the boundary of the region enclosed by the cylinder y2+z2=9 and the planes x=0 and x+y=520EEvaluate the surface integral SFdS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive outward orientation. F(x,y,z)=zexyi3zexyj+xyk, S is the parallelogram of Exercise 5 with upward orientation.Evaluate the surface integral SFdS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive outward orientation. F(x,y,z)=zi+yj+xk, S is the helicoid of Exercise 7 with upward orientation.Evaluate the surface integral SFdS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive outward orientation. F(x,y,z)=xyi+yzj+zxk, S is the part of the paraboloid z=4x2y2 that lies above the square 0x1,0y1, and has upward orientationEvaluate the surface integral SFdS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive outward orientation. F(x,y,z)=xiyj+z3k, S is the part of the cone z=x2+y2 between the planes z=1 and z=3 with downward orientation.Evaluate the surface integral SFdS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive outward orientation. F(x,y,z)=xi+yj+z2k, S is the sphere with radius 1 and centre the originEvaluate the surface integral SFdS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive outward orientation. F(x,y,z)=yixj+2zk, S is the hemisphere x2+y2+z2=4,z0, oriented downward