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All Textbook Solutions for Multivariable Calculus

49E 50E 51E Finding 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)=y2z3i+2xyz3j+3xy2z2k 53E Finding 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)=yezi+zexj+xeyk Finding 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)=zyixzy2j+(xy1)k 56E 57E 58E 59E Finding the Divergence of a Vector Field In Exercises 57-60, find the divergence of the vector field. F(x,y,z)=ln(x2+y2)i+xyj+ln(y2+z2)k 61E 62E 63E 64E EXPLORING CONCEPTS Think About It In Exercises 65-67, consider a scalar function f and a vector field F in space. Determine whether the expression is a vector field, a scalar function, or neither. Explain. curl(f).66E 67E HOW DO YOU SEE IT? Several representative vectors in the vector fields F(x,y)=xi+yix2+y2 and G(x,y)=xi-yjx2+y2 are shown below. Match each vector field with its graph. Explain your reasoning.69E Curl of a Cross Product In Exercises 69 and 70, find curl(FG)=(FG) F(x,y,z)=xizkG(x, y, z)=x2i+yj+z2k 71E 72E 73E 74E Divergence of the Curl of a Vector Field In Exercises 75 and 76, find div(curlF)=(F). F(x,y,z)=xyzi+yj+zk 76E 77E 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.CONCEPT CHECK Line integral What is the physical significance of each line integral? a) C1ds b) Cf(x,y,z)ds, where f(x,y,z) is the density of a string of finite length.2E Finding a Piecewise Smooth Parametrization In Exercises 38, find a piecewise smooth parametrization of the path C. (There is more than one correct answer.)4E 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.)8E 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 (1, 1)10E 11E 12E Evaluating a Line Integral In Exercises 1316, (a) find a piecewise smooth parametrization of the path C, and (b) evaluate C(2x+3y)ds. C: line segments from (0, 0) to (1, 0) and (1, 0) to (2, 4).14E 15E 16E 17E 18E 19E 20E Evaluating a Line Integral In Exercises 19-22, evaluate the line integral along the given path. C(x2+y2+z2)dsC:r(t)=sinti+costj+2k0t2 Evaluating a Line Integral In Exercises 19-22, evaluate the line integral along the given path. C2xyzdsC:r(t)=12ti+5tj+84tk0t1 23E 24E 25E 26E 27E 28E 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 32E 33E 34E 35E 36E 37E 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)40E 41E 42E 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.45E 46E 47E 48E 49E 50E 51E 52E 53E 54E 55E 56E 57E 58E 59E 60E 61E 62E 63E 64E 65E 66E 67E 68E 69E 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)=y+1 C: y=1x2 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)=xy C: y=1x2 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)=x2y2+4,C:x2+y2=4 73E 74E Moment of Inertia Consider a wire of density (x,y) given by the space curve C:r(t)=x(t)i+y(t)j, 0tb. The moments of inertia about the x- and y-axes are given by Ix=Cy2(x,y)ds and Iy=Cx2(x,y)ds. In Exercises 75 and 76, find the moments of inertia for the wire of density . A wire along r(t)=acosti+asintj, where 0t2 and a0, with density (x,y)=1.76E 77E 78E 79E 80E 81E 82E 83E 84E 85E 86E 87E 1E 2E 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 7E 8E 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).10E Using the Fundamental Theorem of Line Integrals In Exercises 9-18, evaluate CF.dr F (x, y) = cosxsinyi+sinxcosyj C: line segment from (0,) to (32,2).12E 13E 14E 15E 16E 17E 18E 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)=9x2y2i+(6x3y1)jP(0,0),Q(5,9)20E 21E 22E 23E 24E 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 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) = (2y+x)i+(x2z)j+(2y4z)k (a). C1:r1(t)=ti+t2j+k,0t1 (b). C2:r2(t)=ti+tj+(2t1)2k,0t1 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 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?2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 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 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 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.Greens Theorem: Region with a Hole Let R be the region inside the circle x=5cos, y=5sin and outside the ellipse x=2cos, y=sin.Evaluate the line integral C(ex2/2y)dx+(ey2/2+x)dy where C=C1+C2 is the boundary of R, as shown in the figure.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 1E 2E 3E 4E 5E 6E Matching In Exercises 38, 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)=2cosvcosui+2cosvsinuj+2sinvk 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 12E 13E 14E Graphing a Parametric Surface In Exercises 1316, use a computer algebra system to graph the surface represented by| the vector-valued function. r(u,v)=(usinu)cosvi+(1cosu)sinvj+uk 0u,0v2 16E 17E 18E 19E 20E Representing a Surface Parametrically In Exercises 1726, find a vector-valued function whose graph is the indicated surface. The cylinder x2+y2=25 22E 23E 24E 25E 26E 27E 28E 29E 30E 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 z=cos2y,2y y -axis 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 z=y2+1,0y2 y -axis 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)37E 38E 39E 40E 41E 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)=sinucosvi+uj+sinusinvk,0u,0v2 43E 44E 45E 46E 47E 48E 49E 50E 51E 52E 53E Hyperboloid Find a vector-valued function for the hyperboloid x2+y2z2=1 and determine the tangent plane at (1, 0, 0)55E 56E 57E Mobius Strip The surface shown in the figure is called a Mobius strip and can he represented by the parametric equations x=(a+ucosv2)cosv,y=(a+ucosv2)sinv,z=usinv2 where 1u1,0v2, and a=3. Try to graph other Mobius strips for different values of a using a computer algebra system.CONCEPT CHECK Surface Integral Explain how to set up a surface integral given that you will project the surface onto the xz-plane.2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 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 21E 22E

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