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All Textbook Solutions for Calculus (MindTap Course List)

27E Mass In Exercises 25-28, find the total mass of the wire with density whose shape is modeled by r. r(t)=2costi+2sintj+3tk,0t2,(x,y,z)=k+z(k0)Evaluating a Line Integral of a Vector Field In Exercises 29-34, evaluate cFdr. F(x,y)=xi+yjC:r(t)=(3t+1)i+tj,0t1 30E 31E Evaluating a Line Integral of a Vector Field In Exercises 29-34, evaluate cFdr. F(x,y)=3xi+4yjC:r(t)=ti+4t2j,2t2 Evaluating a Line Integral of a Vector Field In Exercises 29-34, evaluate cFdr. F(x,y,z)=xyi+xzj+yzkC:r(t)=ti+t2j+2tk,0t1 34E Evaluating a Line Integral of a Vector Field Using Technology In Exercises 35 and 36, use a computer algebra system to evaluate cFdr. F(x, y, z) = x2zi+6yj+yz2kC:r(t)=ti+t2j+lntk,1t3 36E Work In Exercises 37-42, find the work done by the force field F on a particle moving along the given path. F(x,y)=xi+2yjC:x=t,y=t3from(0,0)to(2,8)Work In Exercises 37-42, find the work done by the force field F on a particle moving along the given path. F(x,y)=x2ixyjC:x=cos3t,y=sin3tfrom(1,0)to(0,1)Work In Exercises 37-42, find the work done by the force field F on a particle moving along the given path. F(x,y)=xi+yj C: counterclockwise around the triangle with vertices (0, 0), (1, 0), and (0, 1)Work In Exercises 3742, find the work done by the force field F on a particle moving along the given path. F(x, y) = -yi-yj C: counterclockwise around the semicircle y=4x2 from (2, 0) to (-2,0).41E Work In Exercises 3742, find the work done by the force field F on a particle moving along the given path. F(x,y,z) = yzi+xzj+xyk C: line from (0, 0, 0) to (5, 3, 2).Work In Exercises 43-46, determine whether the work done along the path C is positive, negative, or zero. Explain.Work In Exercises 43-46, determine whether the work done along the path C is positive, negative, or zero. Explain.Work In Exercises 43-46, determine whether the work done along the path C is positive, negative, or zero. Explain.Work In Exercises 43-46, determine whether the work done along the path C is positive, negative, or zero. Explain.Evaluating a Line Integral of a Vector Field In Exercises 47 and 48, evaluate CFdr for each curve. Discuss the orientation of the curve and its effect on the value of the integral. F(x,y)=x2i+xyj (a) C1:r1(t)=2ti+(t1)j,1t3 (b) C2:r2(t)=2(3t)i+(2t)j,0t2 48E 49E 50E 51E 52E Evaluating a Line Integral in Differential Form In Exercises 53-56, evaluate the line integral along the path C given by x=2t,y=4t, where 0t1. C(x+3y2)dy 54E 55E Evaluating a Line Integral in Differential Form In Exercises 53-56, evaluate the line integral along the path C given by x=2t,y=4t, where 0t1. C(yx)dx+5x2y2dy Evaluating a Line Integral in Differential Form In Exercises 5764, evaluate C(2xy)dx+(x+3y)dy. C: x axis from x=0 to x=5 Evaluating a Line Integral in Differential Form In Exercises 5764, evaluate C(2xy)dx+(x+3y)dy. C: y-axis from y=0 to y=2 Evaluating a Line Integral in Differential Form In Exercises 5764, evaluate C(2xy)dx+(x+3y)dy. C: line segments from (0, 0) to (3, 0) and (3, 0) to (3, 3).Evaluating a Line Integral in Differential Form In Exercises 5764, evaluate C(2xy)dx+(x+3y)dy. C: line segments from (0, 0) to (0,-3) and (0,-3) to (2,-3).Evaluating a Line Integral in Differential Form In Exercises 5764, evaluate C(2xy)dx+(x+3y)dy. C: arc on y=1x2 from (0, 1) to (1, 0).62E 63E 64E Lateral Surface Area In Exercises 65-72, find the area of the lateral surface (see figure) over the curve C in the xy-plane and under the surface z=f(x,y) where Lateral surface area=Cf(x,y)ds. f(x,y)=h, C: line from (0, 0) to (3, 4)66E Lateral Surface Area In Exercises 65-72, find the area of the lateral surface (see figure) over the curve C in the xy-plane and under the surface z=f(x,y) where Lateral surface area=Cf(x,y)ds. f(x,y)=xy, C: x2+y2=1 line from (1, 0) to (0, 1)Lateral Surface Area In Exercises 65-72, find the area of the lateral surface (see figure) over the curve C in the xy-plane and under the surface z=f(x,y) where Lateral surface area=Cf(x,y)ds. f(x,y)=x+y, C: x2+y2=1 line from (1, 0) to (0, 1)69E 70E 71E Lateral Surface Area In Exercises 65-72, find the area of the lateral surface (see figure) over the curve C in the xy-plane and under the surface z=f(x,y) where Lateral surface area=Cf(x,y)ds. f(x,y)=x2y2+4,C:x2+y2=4 73E 74E 75E 76E 77E 78E 79E 80E 81E Line Integrals Let F(x,y)=2xi+xy2j and consider the curve y=x2 from (0,0) to (2,4) in the xy-plane. Set up and evaluate line integrals of the forms CFdr and CMdx+Ndy. Compare your results. Which method do you prefer? Explain.83E HOW DO YOU SEE IT? For each of the following, determine whether the work done in moving an object from the first to the second point through the force field shown in the figure is positive, negative, or zero. Explain your answer. (In the figure, the circles have radii 1, 2, 3, 4, 5, and 6.) (a) From (-3,-3) to (3, 3) (b) From (-3, 0) to (0, 3) (c) From (5, 0) to (0, 3)True or False? In Exercises 85 and 86, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If C is given by x=t,y=t, where 0t1, then Cxyds=01t2dt True or False? In Exercises 85 and 86, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If C2=C1, then C1f(x,y)ds+C2f(x,y)ds=0.87E CONCEPT CHECK Fundamental Theorem of Line Integrals Explain how to evaluate a line integral using the Fundamental Theorem of Line Integrals.Independence of Path What does it mean for a line integral to be independent of path? State the method for determining whether a line integral is independent of path.Line Integral of a Conservative Vector Field In Exercises 3-8, (a) show that F is conservative and (b) verify that the value of CFdr is the same for each parametric representation of C. F(x,y)=x2i+yj (i) C1:r1(t)=ti+t2j,0t1 (ii) C2:r2()=sini+sin2j,0/2 4E 5E 6E Line Integral of a Conservative Vector Field In Exercises 3-8, (a) show that F is conservative and (b) verify that the value of CFdr is the same for each parametric representation of C. F(x,y,z)=y2zi+2xyzj+xy2k (i) C1:r1(t)=ti+2tj+4tk,0t1 (ii) C2:r2()=sini+2sinj+4sink,0/2 Line Integral of a Conservative Vector Field In Exercises 3-8, (a) show that F is conservative and (b) verify that the value of CFdr is the same for each parametric representation of C. F(x,y,z)=2yzi+2xzj+2xyk (i) C1:r1(t)=ti4tj+t2k,0t3 (ii) C2:r2(s)=s2i+43s4j+s4k,0s3 In Exercises 918, Using the Fundamental Theorem of Line Integrals In Exercises 9-18, evaluate CFdr using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. F (x, y) = 3yi+3xj C: Smooth curve from (0, 0) to (3, 8).Using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. F (x, y) = 2(x+y)i+2(x+y)j C: line segment from (1, 1) to (3, 2).11E 12E Using the Fundamental Theorem of Line Integrals In Exercises 9-18, evaluate cFdr using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. F(x,y)=exsinyi+excosyjC:cycloidx=sin,y=1cosfrom(0,0)to(2,0)14E Using the Fundamental Theorem of Line Integrals In Exercises 9-18, evaluate CF.dr F (x, y, z) = (z+2y)i+(2xz)j+(xy)k (a). C1: line segment from (0, 0, 0) to (1, 1, 1). (b). C2: line segment from (0, 0, 0) to (0, 0, 1) and from (0, 0, 1) to (1, 1, 1). (c). C3: line segment from (0, 0, 0) to (1, 0, 0), from (1, 0, 0) to (1, 1, 0) and from (1, 1, 0) to (1, 1, 1).16E Using the Fundamental Theorem of Line Integrals In Exercises 918, evaluate CF.dr using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. F (x, y, z) = sinxi+zj+yk C: Smooth curve from (0, 0, 0) to (2,3,4).18E 19E Finding Work in a Conservative Force Field In Exercises 19-22, (a) show that CFdr independent of' path and (b) calculate the work done by the force field F on an object moving along a curve from P to Q. F(x,y)=2xyix2y2jP(1,1),Q(3,2)Finding Work in a Conservative Force Field In Exercises 19-22, (a) show that CFdr independent of' path and (b) calculate the work done by the force field F on an object moving along a curve from P to Q. F(x,y,z)=3i+4yjsinzkP(0,1,2),Q(1,4,)22E 23E Evaluating a Line Integral In Exercises 23-32, evaluate CFdr along each path. (Hint: If F is conservative, the I integration may be easier on an alternative path.) F(x,y)=yexyi+xexyj (a) C1:r1(t)=ti(t3)j,0t3 (b) C1:: The closed path consisting of line segments from (0, 3) to (0. 0), from (0, 0) to (3,0), and then from (3, 0) to (0. 3)25E Evaluating a Line Integral In Exercises 23-32, evaluate CFdr along each path. (Hint: If F is conservative, the integration may be easier on an alternative path.) C(2x3y+1)dx(3x+y5)dy Evaluating a Line Integral In exercises 2332, evaluate CF.dr along each path. (Hint: If F is conservative, the integration may be easier on an alternative path.) F(x, y, z) = yzi+zxj+xyk (a). C1:r1(t)=ti+2j+tk,0t4 (b). C2:r2(t)=t2i+tj+t2k,0t2 Evaluating a Line Integral In Exercises 23-32, evaluate CFdr along each path. (Hint: If F is conservative, the I integration may be easier on an alternative path.) (28) F(x,y,z)=i+zj+yk (a).C1:r1(t)=costi+sintj+t2k,0t (b).C2:r2(t)=(12t)i+2tk,0t1 29E 30E 31E 32E 33E 34E Work A zip line is installed 50 meters above ground level. It runs to a point on the ground 50 meters away from the base of the installation. Show that the work done by the gravitational force field for a 175pound person moving the length of the zip line is the same for each path. C1:r1(t)=ti+(50t)j C2:r2(t)=ti+150(50t)2j 36E 37E 38E 39E HOW DO YOU SEE IT? Consider the force field shown in the figure. To print an enlarged copy of the graph. Give a verbal argument that the force field is not conservative because you can identify two paths that require different amounts of work to move an object from (4,0) to (3,4). Of the two paths, which requires the greater amount of work? Give a verbal argument that the force field is not conservative because you can find a closed curve C such that CFdr0.Graphical Reasoning In Exercises 41 and 42, consider the force field shown in the figure. Is the force field conservative? Explain why or why not.Graphical Reasoning In Exercises 41 and 42, consider the force field shown in the figure. Is the force field conservative? Explain why or why not.43E 44E 45E 46E 47E Kinetic and Potential Energy The kinetic energy of an object moving through a conservative force field is decreasing at a rate of 15 units per minute. At what rate is the potential energy changing? Explain.49E CONCEPT CHECK WritingWhat does it mean for a curve to be simple? What does it mean for a plane region to be simply connected?Green's Theorem Explain the usefulness of Green's Theorem.3E AreaDescribe how to find the area of a plane region bounded by a piecewise smooth simple closed curve that is oriented counterclockwise.Verifying Greens TheoremIn Exercises 58, verify Greens Theorem by evaluating both integrals cy2dx+x2dy=R(NxMy)dA for the given path. C: boundary of the region lying between the graphs of y=x and y=x2.Verifying Greens TheoremIn Exercises 58, verify Greens Theorem by evaluating both integrals cy2dx+x2dy=R(NxMy)dA for the given path. C: boundary of the region lying between the graphs of y=x and y=x.Verifying Greens TheoremIn Exercises 58, verify Greens Theorem by evaluating both integrals cy2dx+x2dy=R(NxMy)dA for the given path. C: square with vertices (0,0),(1,0),(1,1), and (0,1)8E 9E 10E 11E 12E 13E Evaluating a Line Integral Using Greens TheoremIn Exercises 1114, use Greens Theorem to evaluate the line integral c(yx)dx+(2xy)dy for the given path. C: boundary of the region lying inside the semicircle y=25x2 and outside the semicircle y=9x2 15E 16E 17E Evaluating a Line Integral Using Greens Theorem In Exercises 1524, use Greens Theorem to evaluate the line integral. c(x2y2)dx+2xydy Evaluating a Line Integral Using Greens TheoremIn Exercises 1524, use Greens Theorem to evaluate the line integral. cexcos2ydx2exsin2ydy C:x2+y2=a2 Evaluating a Line Integral Using Greens TheoremIn Exercises 1524, use Greens Theorem to evaluate the line integral. c2arctanyxdx+ln(x2+y2)dy C:x=4+2cos,y=4+sin 21E 22E 23E 24E 25E Work In Exercises 25-28, use Greens Theorem to calculate the work done by the force F on a particle that is moving counterclockwise around the closed path C, F(x,y)=(ex3y)i+(ey+6x)jC:r=2cos 27E 28E Area In Exercises 29-32, use a line integral to find the area of the region R. R: region bounded by the graph of x2+y2=4 30E 31E 32E Using Green's Theorem to Verify a Formula In Exercises 33 and 34, use Greens Theorem to verify the line integral formula(s). The centroid of the region having area A bounded by the simple closed path C has coordinates x=12ACx2dy and y=12ACy2dx.34E 35E 36E 37E 38E 39E 40E 41E 42E 43E HOW DO YOU SEE IT? The figure shows a region R bounded by a piecewise smooth simple closed path C. (a) Is R simply connected? Explain. (b) Explain why Cf(x)dx+g(y)dy=0, where f and g are differentiable functions.45E Greens Theorem: Region with a Hole Let R be the region inside the ellipse x=4cos, y=3sin and outside the circle x=5cos, y=5sin and outside the circle x=2cos, y=2sin. Evaluate the line integral C(3x2y+1)dx+(x3+4x)dy where C=C1+C2 is the boundary of R, as shown in the figure.47E 48E 49E 50E 51E 52E 53E 54E CONCEPT CHECK Parametric Surface Explain how a parametric surface is represented by a vector-valued function and how the vector valued function is used to sketch the parametric surface.2E 3E Matching In Exercises 3-8, match the vector-valued function with its graph. [The graphs are labelled (a), (b), (c), (d), (e), and (f).] r(u,v)=ucosvi+usinvj+uk a). b). c). d). e). f).5E Matching In Exercises 3-8, match the vector-valued function with its graph. [The graphs are labelled (a), (b), (c), (d), (e), and (f).] a). b). c). d). e). f). r(u,v)=vi+cosuj+sinuk 7E Matching In Exercises 3-8, match the vector-valued function with its graph. [The graphs are labelled (a), (b), (c), (d), (e), and (f).] a). b). c). d). e). f). r(u,v)=ui+14v3j+vk 9E 10E 11E Sketching a Parametric Surface In Exercises 9-12, find the rectangular equation for the surface by eliminating the parameters from the vector- valued function. Identify the surface and sketch its graph. r(u,v)=3cosvcosui+3cosvsinuj+5sinvk 13E 14E 15E 16E 17E 18E 19E 20E 21E Representing a Surface Parametrically In Exercises 17-26, find a vector-valued function whose graph is the indicated surface. Thecylinder4x2+y2=16 23E Representing a Surface Parametrically In Exercises 1726, find a vector-valued function whose graph is the indicated surface. The ellipsoid x29+y24+z21=1 25E 26E 27E 28E 29E Representing a Surface Revolution ParametricallyIn Exercises 2732, write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. FunctionAxis of Revolution x=z2,2z5 z -axis 31E 32E 33E 34E 35E Finding a Tangent Plane In Exercises 33-36, find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. r(u,v)=2ucosvi+2usinhvj+12u2k,(4,0,2)Finding Surface Area In Exercises 37-42, find the area of the surface over the given region. Use a computer algebra system to verify your results. r(u,v)=4uivj+vk,02,0v1 38E 39E Finding Surface Area In Exercises 3742, find the area of the surface over the given region. Use a computer algebra system to verify your results. r(u,v)=(a+bcosv)cosui+(a+bcosv)sinuj+bsinvk,ab,0u2,0v2 41E 42E 43E 44E 45E 46E Representing a Cone Parametrically Show that the cone in Example 3 can be represented parametrically by r(u,v)=ucosvi+usinvj+uk, where u0 and 0v2.48E 49E Different Views of a Surface Use a computer algebra system to graph the vector-valued function r(u,v)=ucosvi+usinvj+vk,0u,0v from each of the points (10.0, 0), (0, 0, 10), and (10, 10, 10)51E 52E 53E 54E Area Use a computer algebra system to graph one turn of the spiral ramp r(u,v)=ucosvi+usinvj+2vk, where 0u3 and 0v2. Then analytically find the area of one turn of the spiral ramp.56E 57E 58E CONCEPT CHECK Surface Integral Explain how to set up a surface integral given that you will project the surface onto the xz-plane.CONCEPT CHECK Surface Integral For what conditions does the surface integral over S yield the surface area of S?3E 4E Evaluating a surface Integral In Exercise 58, evaluate S(x2y+z)dS S:z=4x, 0x4, 0y3 6E Evaluating a surface Integral In Exercise 58, evaluate S(x2y+z)dS S:z=2, x2+y21 8E Evaluating a Surface Integral In Exercises 9 and 10. evaluate sxydS. S:z=3xy first octant 10E 11E 12E 13E Mass In Exercises 13 and 14, find the mass of the surface lamina S of density . S:z=a2x2y2,(x,y,z)=kz Evaluating a Surface Integral In Exercises15-18, evaluate sf(x,y)dS. f(x,y)=y+5S:r(u,v)=ui+vj+2vk0u1,0v2 16E Evaluating a Surface Integral In Exercises 15-18, evaluate sf(x,y)dS. f(x,y)=3yx S:r(u,v)=cosui+sinuJ+vk 0u3,0v1 Evaluating a Surface Integral In Exercises 15-18, evaluate sf(x,y)dS. f(x,y)=x+yS:r(u,v)=4ucosvi+4usinvj+3uk0u4,0v Evaluating a Surface Integral In Exercises 19-24, evaluate sf(x,y,z)dS f(x,y,z)=x2+y2+z2S:z=x+y,x2+y21 20E Evaluating a Surface Integral In Exercises 19-24.evaluate sf(x,y,z)dS f(x,y,z)=x2+y2+z2S:z=x2+y2,x2+y24 Evaluating a Surface Integral In Exercises 19-24, evaluate sf(x,y,z)dS f(x,y,z)=x2+y2+z2S:z=x2+y2,(x1)2+y21 Evaluating a Surface Integral In Exercises 19-24, evaluate sf(x,y,z)dS f(x,y,z)=x2+y2+z2S:x2+y2=9,0x3,0y3,0z9 Evaluating a Surface Integral In Exercises 19-24, evaluate sf(x,y,z)dS f(x,y,z)=x2+y2+z2S:x2+y2=9,0x3,0zx Evaluating a Flux Integral In Exercises 25-30, find the flux of F across S, SFNdS where N is the upward unit normal vector to S. F(x,y,z)=3zi4j+yk; S:z=1xy, first octant 26E Evaluating a Flux Integral In Exercises 25-30, find the flux of F across S, SFNdS where N is the upward unit normal vector to S. F(x,y,z)=xi+yj+zk;S:z=1x2y2,z0 28E Evaluating a Flux Integral In Exercises 25-30, find the flux of F across S, SFNdS where N is the upward unit normal vector to S. F(x,y,z)=4i-3j+5kS:z=x2+y2,x2+y24 Evaluating a Flux Integral In Exercises 25-30, find the flux of F across S, SFNdS where N is the upward unit normal vector to S. F(x,y,z)=xi+yj-2zkS:z=a2x2y2 Evaluating a Flux Integral In Exercises 31 and 32, find the flux of F over the closed surface. (Let N be the outward unit normal vector of the surface.) F(x,y,z)=(x+y)i+yj+kS:z=16x2y2,z=0 Evaluating a Flux Integral In Exercises 31 and 32, find the flux of F over the closed surface. (Let N be the outward unit normal vector of the surface.) F(x,y,z)=4xyi+z2j+yzk S: unit cube bounded by the planes x=0,x=1,y=0 y=1,z=0,z=1 33E 34E 35E 36E 37E 38E 39E 40E EXPLORING CONCEPTS Using Different Methods Evaluate S(x+2y)dS where S is the first-octant portion of the plane 2x+2y+z=4 by projecting 5 onto (a) the xy-plane, (b) the xz-plane, and (c) the yz-plane. Verify that all answers are the same.HOW DO YOU SEE IT? Is the surface shown in the figure orient able? Explain why or why not.43E CONCEPT CHECK Using Different Methods Suppose that a solid region Q is bounded by z=x2+y2 and z=2, as shown in the figure. What methods can you use to evaluate SFNdS, where F=2xi+3yjz2k? Which method do you prefer?Classifying a Point in a Vector Field How do you determine whether a point (x0,y0,z0) in a vector field is a source, a sink, or incompressible?Verifying the Divergence Theorem In Exercises 38, verify the Divergence Theorem by evaluating sFNdS as a surface integral and as a triple integral. F(x,y,z)=2xi2yj+z2k S: cube bonded by the planes x=0,x=1,y=0,y=1,z=0,z=1.4E Verifying the Divergence Theorem In Exercises 38, verify the Divergence Theorem by evaluating SFNds as a surface integral and as a triple integral. F(x,y,z)=(2xy)i(2yz)j+zk S: surface bounded by the plane 2x+4y+2z=12 and the coordinate planes Verifying the Divergence Theorem In Exercises 38, verify the Divergence Theorem by evaluating sFNds as a surface integral and as a triple integral. F(x,y,z)=xyi+zj+(x+y)k S: surface bounded by the planes y = 4 and z = 4 x and the coordinate plane.Verifying the Divergence Theorem In Exercises 38, verify the Divergence Theorem by evaluating sFNdS as a surface integral and as a triple integral. F(x,y,z)=xzi+zyj+2z2k S: surface bounded by z=1x2y2 and z = 0 Verifying the Divergence Theorem In Exercises 38, verify the Divergence Theorem by evaluating sFNdS as a surface integral and as a triple integral. F(x,y,z)=xy2i+yx2j+ek S: surface bounded by z=x2+y2 and z=4 Using the Divergence Theorem In Exercises 9-18, use the Divergence Theorem to evaluate SFNdS and find the outward flux of F through the surface of the solid S bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x,y,z)=x2i+y2j+z2kS:x=0,x=a,y=0,y=a,z=0,z=a Using the Divergence Theorem In Exercises 9-18, use the Divergence Theorem to evaluate SFNdS and find the outward flux of F through the surface of the solid S bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x,y,z)=x2z2i2yj+3xyzkS:x=0,x=a,y=0,y=a,z=0,z=a Using the Divergence Theorem In Exercises 9-18, use the Divergence Theorem to evaluate SFNdS and find the outward flux of F through the surface of the solid S bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x,y,z)=x2i2xyj+xyz2kS:z=a2x2y2,z=0 Using the Divergence Theorem In Exercises 9-18, use the Divergence Theorem to evaluate SFNdS and find the outward flux of F through the surface of the solid S bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x,y,z)=xyi+yzjyzkS:z=a2x2y2,z=0 Using the Divergence Theorem In Exercises 9-18, use the Divergence Theorem to evaluate SFNdS and find the outward flux of F through the surface of the solid S bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x,y,z)=xi+yj+zkS:x2+y2+z2=9 14E 15E 16E 17E 18E Classifying a Point In Exercises 19-22, a vector field and a point in the vector field are given. Determine whether the point is a source, a sink, or incompressible. F(x,y,z)=2i+yj+k,(2,2,1)Classifying a Point In Exercises 19-22, a vector field and a point in the vector field are given. Determine whether the point is a source, a sink, or incompressible. F(x,y,z)=exixy2j+lnzk,(0,3,1)21E 22E 23E 24E 25E HOW DO YOU SEE IT? The graph of a vector field F is shown. Does the graph suggest that the divergence of F at P is positive, negative, or zero?27E 28E 29E 30E 31E Proof In Exercises 31 and 32, prove the identity, assuming that Q, S, and N meet the conditions of the Divergence Theorem and that the required partial derivatives of the scalar functions f and g are continuous. The expressions DNf and DNg are the derivatives in the direction of the vector N and are defined by DNf=fNandDNg=gN. Q(f2gg2f)dV=S(fDNggDNf)dS [ Hint: Use Exercise 31 twice.]CONCEPT CHECK Stokess Theorem Explain the benefit of Stokess Theorem when the boundary of the surface is a piecewise curve.Curl What is the physical interpretation of curl?3E

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