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17PS18PSMatching In Exercise 5-8, match the vector field with its graph. [The graphs are labeled (a), (b), (c), and (d).] F(x,y)=yiMatching In Exercise 5-8, match the vector field with its graph. [The graphs are labeled (a), (b), (c), and (d).] F(x,y)=xjMatching In Exercise 5-8, match the vector field with its graph. [The graphs are labeled (a), (b), (c), and (d).] F(x,y)=yixjMatching In Exercise 5-8, match the vector field with its graph. [The graphs are labeled (a), (b), (c), and (d).] F(x,y)=xi+3yjSketching a Vector Field In Exercises 914, find F and sketch several representative vectors in the vector field. F(x,y)=i+jSketching a Vector Field In Exercises 914, find F and sketch several representative vectors in the vector field. F(x,y)=yi2xj7ESketching a Vector Field In Exercises 914, find F and sketch several representative vectors in the vector field. F(x,y)=yi+xjSketching a Vector Field In Exercises 914, find F and sketch several representative vectors in the vector field. F(x,y,z)=i+j+kSketching a Vector Field In Exercises 914, find F and sketch several representative vectors in the vector field. F(x,y,z)=xi+yj+zkGraphing a Vector Field Using Technology In Exercises 1518, use a computer algebra system to graph several representative vectors in the vector field. F(x,y)=18(2xyi+y2j)12E13E14E15E16E17E18E19E20E21E22E23E24ETesting for a Conservative Vector Field In Exercises 2936, determine whether the vector field is conservative. F(x,y)=xy2i+x2yjTesting for a Conservative Vector Field In Exercises 2936, determine whether the vector field is conservative. F(x,y)=1x2(yixj).27E28E29E30E31E32EFinding a Potential Function In Exercises 3342, determine whether the vector field is conservative. If it is, find a potential function for the vector field. F(x,y)=yi+xj34E35EFinding a Potential Function In Exercises 3744, determine whether the vector field is conservative. If it is, find a potential function for the vector field. F(x,y)=xex2y(2yi+xj)37EFinding a Potential Function In Exercises 3744, determine whether the vector field is conservative. If it is, find a potential function for the vector field. F(x,y)=1y2(yi2xj)Finding a Potential Function In Exercises 37-44, determine whether the vector field is conservative. If it is, find a potential function for the vector field. F(x,y)=2yxix2y2jFinding a Potential Function In Exercises 3744, determine whether the vector field is conservative. If it is, find a potential function for the vector field. F(x,y)=xi+yjx2+y2Finding a Potential Function In Exercises 3342, determine whether the vector field is conservative. If it is, find a potential function for the vector field. F(x,y)=ex(cosyisinyj)42E43EFinding the Curl of a Vector Field In Exercises 45-48, find the curl of the vector field at the given point. F(x,y,z)=x2zi2xzj+yzk ; (2,1,3).45E46E47E48E49E50E51E52E53E54E55EFinding a Potential Function In Exercises 51-56, determine whether the vector field is conservative. If it is, find a potential function for the vector field. F(x,y,z)=xx2+y2i+yx2+y2j+kFinding the Divergence of a Vector Field In Exercises 57-60, find the divergence of the vector field. F(x,y)=x2i+2y2j58E59E60E61E62E63E64EVector Field Define a vector field in the plane and in space. Give some physical examples of vector fields. Vector Field Define a vector field in the plane and in space. Give some physical examples of vector fields.CONCEPT CHECK Conservative Vector Field What is a conservative vector field? How do you test whether a vector field is conservative in the plane and in space?57095-15.1-67E-Question-Digital.docx Curl Define the curl of a vector field.68ECurl of a Cross Product In Exercises 69 and 70, find curl(FG)=(FG) F(x,y,z)=i+3xj+2ykG(x,y,z)=xiyj+zk70E71E72E73E74EDivergence of the Curl of a Vector Field In Exercises 75 and 76, find div(curlF)=(F). F(x,y,z)=xyzi+yj+zk76EProof In parts (a) - (h), prove the property for vector fields F and G and scalar function f. (Assume that the required partial derivatives are continuous.) curl(F+G)=curlF+curlG curl(f)=(f)=0 div(F+G)=divF+divG div(FG)=(curlF)GF(curlG) [f+(F)]=(F) (fF)=f(F)+(f)F div(fF)=fdivF+fF div(curlF)=078E79E80E81ETrue or False? In Exercises 7982, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If F is a vector field and curl F = 0, then F is irrotational but not conservative.Earths magnetic field A cross section of Earths magnetic field can be represented as a vector field in which the center of Earth is located at the origin and the positive y-axis points in the direction of the magnetic north pole. The equation for this field is F(x,y)=M(x,y)i+N(x,y)j = m(x2+y2)5/2[3xyi+(2y2x2)j] where m is the magnetic moment of Earth. Show that this vector field is conservative.Finding a Piecewise Smooth Parametrization In Exercises 38, find a piecewise smooth parametrization of the path C. (There is more than one correct answer.)2EFinding a Piecewise Smooth Parametrization In Exercises 38, find a piecewise smooth parametrization of the path C. (There is more than one correct answer.)Finding a Piecewise Smooth Parametrization In Exercises 38, find a piecewise smooth parametrization of the path C. (There is more than one correct answer.)Finding a Piecewise Smooth Parametrization In Exercises 38, find a piecewise smooth parametrization of the path C. (There is more than one correct answer.)Finding a Piecewise Smooth Parametrization In Exercises 38, find a piecewise smooth parametrization of the path C. (There is more than one correct answer.)Evaluating a Line Integral In Exercises 19-22, evaluate the line integral along the given path. CxydsC:r(t)=4ti+3tj0t1Evaluating a Line Integral In Exercises 19-22, evaluate the line integral along the given path. C3(xy)dsC:r(t)=ti+(2t)j0t2Evaluating a Line Integral In Exercises 19-22, evaluate the line integral along the given path. C(x2+y2+z2)dsC:r(t)=sinti+costj+2k0t2Evaluating a Line Integral In Exercises 19-22, evaluate the line integral along the given path. C2xyzdsC:r(t)=12ti+5tj+84tk0t1Evaluating a Line Integral In Exercises 9-12, (a) find a parametrization of the path C , and (b) evaluate C(x2+y2)ds. C: line segment from (0, 0) to (1, 1)Evaluating a Line Integral In Exercises 9-12, (a) find a parametrization of the path C , and (b) evaluate C(x2+y2)ds. C: line segment from (0, 0) to (2, 4)Evaluating a Line Integral In Exercises 9-12, (a) find a parametrization of the path C , and (b) evaluate C(x2+y2)ds. C: counterclockwise around the circle x2+y2=1 from (1, 0) to (0, 1)14EEvaluating a Line Integral In Exercises 15–18, (a) find a parametrization of the path C, and (b) evaluate
along C.
15. C: x-axis from x = 0 to x = 1
16E17E18EEvaluating a Line Integral In Exercises 17 and 18, (a) find a piecewise smooth parametrization of the path C shown in the figure and (b) evaluate C(2x+y2z)ds.Evaluating a Line Integral In Exercises 17 and 18, (a) find a piecewise smooth parametrization of the path C shown in the figure, and (b) evaluate c(2x+y2z)dsMass In Exercises 23 and 24, find the total mass of a spring with density in the shape of the circular helix r(t)=2costi+2sintj+tk,0t4 (x,y,z)=12(x2+y2+z2)Mass In Exercises 23 and 24, find the total mass of a spring with density in the shape of the circular helix r(t)=2costi+2sintj+tk,0t4 (x,y,z)=z23E24E25E26EEvaluating a Line Integral of a Vector Field In Exercises 29-34, evaluate cFdr. F(x,y)=xi+yjC:r(t)=(3t+1)i+tj,0t128E29EEvaluating a Line Integral of a Vector Field In Exercises 29-34, evaluate cFdr. F(x,y)=3xi+4yjC:r(t)=ti+4t2j,2t2Evaluating 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,0t132EEvaluating 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,1t334EWork 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).39EWork 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,0t246E47E48E49E50EEvaluating 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)dy52E53E54EEvaluating a Line Integral in Differential Form In Exercises 5764, evaluate C(2xy)dx+(x+3y)dy. C: x axis from x=0 to x=5Evaluating a Line Integral in Differential Form In Exercises 5764, evaluate C(2xy)dx+(x+3y)dy. C: y-axis from y=0 to y=2Evaluating 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).60E61E62ELateral 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)64ELateral 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)67E68E69ELateral 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=471E72E73E74E75E76E77E78E79E80E81EHOW 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=01t2dtTrue 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=0.85ETrue or False? In Exercises 8386, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. IfCFTds=0,thenFandTareorthogonal.87ELine 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,0/257095-15.3-2E-Question-Digital.docx Evaluating a Line Integral for Different ParametrizationsIn Exercises 14, show that the value of CFdr is the same for each parametric representation of C. F(x,y)=(x2+y2)ixj(a)r1(t)=ti+tj,0t4(b)r2(w)=w2i+wj,0w23E4ETesting for Conservative Vector Fields In Exercises 5–10, determine whether the vector field is conservative.
6E7E8E9E10EEvaluating a Line Integral of a Vector Field In Exercises 1124, find the value of the line integral CFdr. (Hint: If F is conservative, the integration may be easier on an alternative path.) F(x,y)=2xyi+x2j(a)r1(t)=tit2j,0t1(b)r2(t)=ti+t3j,0t112E13E14E15EEvaluating 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)dy17E18EEvaluating a Line Integral of a Vector Field In Exercises 1124, find the value of the line integral CFdr. (Hint: If F is conservative, the integration may be easier on an alternative path.) F(x,y,z)=yzi+xzj+xyk(a)r1(t)=ti+2j+tk,0t4(b)r2(t)=t2i+tj+t2k,0t220E21E22E23E24EUsing the Fundamental Theorem of Line Integrals In Exercises 2534, evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. C(3yi+3xj)drC:smoothcurvefrom(0,0)to(3,8)26E27E28EUsing the Fundamental Theorem of Line Integrals In Exercises 2534, evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. Cexsinydx+excosydyC:cycloidx=sin,y=1cosfrom(0,0)to(2,0)30E31E32EUsing the Fundamental Theorem of Line Integrals In Exercises 2534, evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. Csinxdx+zdy+ydzC:smoothcurvefrom(0,0,0)to(2,3,4)Using the Fundamental Theorem of Line Integrals In Exercises 2534, evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. C6xdx4zdy(4y20z)dzC:smoothcurvefrom(0,0,0)to(3,4,0)Work In Exercises 35 and 36, find the work done by the force field F in moving an object from P to Q.
35.
36E37E38EWork 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)2j40ECONCEPT 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.43. Thing About It Let Find the value of the line integral .
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.47E48E49E50E51EKinetic 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.53EVerifying 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)4E5E6E7E8E9EEvaluating 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=9x211E12E13EEvaluating a Line Integral Using Greens Theorem In Exercises 1524, use Greens Theorem to evaluate the line integral. c(x2y2)dx+2xydyEvaluating a Line Integral Using Greens TheoremIn Exercises 1524, use Greens Theorem to evaluate the line integral. cexcos2ydx2exsin2ydy C:x2+y2=a2Evaluating 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+sin17E18E19E20E21EWork 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=2cos23E24E25EArea In Exercises 25–28, use a line integral to find the area of the region R.
26. R: triangle bounded by the graphs of x = 0, 3x – 2y = 0, and x + 2y = 8
27E28E29E30EUsing 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.32E33E34E35E36E37E38E39E40E41EHOW 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.43EGreens 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.45E46E47E48E49E50E51E52E57095-15.5-47E-Question-Digital.docx Parametric Surface Define a parametric surface.48E1EMatching 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).3EMatching 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+vk5EMatching In Exercises 1–6, match the vector-valued function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
6.
7E8E9ESketching 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+5sinvk11E12E13E14E15E16E17E18E19E20E21E