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22E23E24E25ERepresenting a Surface Parametrically In Exercises 17-26, find a vector-valued function whose graph is the indicated surface. Thecylinder4x2+y2=1627ERepresenting a Surface Parametrically In Exercises 1726, find a vector-valued function whose graph is the indicated surface. The ellipsoid x29+y24+z21=129E30E31E32E33E34E35E36E37EFinding 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,02,0v140E41E42E43E57095-15.5-44E-Question-Digital.docx Area In Exercises 3946, find the area of the surface over the given region. Use a computer algebra system to verify your results. Thetorusr(u,v)=(a+bcosv)cosui+(a+bcosv)sinuj+bsinvk,whereab,0u2,and0v245E46ERepresenting a Cone Parametrically Show that the cone in Example 3 can be represented parametrically by r(u,v)=ucosvi+usinvj+uk, where u0 and 0v2.50E51EDifferent 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)53E54E55E56EArea 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.58E59E60E57095-15.6-39E-Question-Digital.docx WRITING ABOUT CONCEPTS Surface Integral Define a surface integral of the scalar function f over a surface z = g(x, y). Explain how to evaluate the surface integral.40E41EEvaluating a surface Integral In Exercise 58, evaluate S(x2y+z)dS S:z=4x, 0x4, 0y32EEvaluating a surface Integral In Exercise 58, evaluate S(x2y+z)dS S:z=2, x2+y214EEvaluating a Surface Integral In Exercises 9 and 10. evaluate sxydS. S:z=3xy first octant6E9E10E7E8E11EMass In Exercises 13 and 14, find the mass of the surface lamina S of density . S:z=a2x2y2,(x,y,z)=kzEvaluating a Surface Integral In Exercises15-18, evaluate sf(x,y)dS. f(x,y)=y+5S:r(u,v)=ui+vj+2vk0u1,0v214E15EEvaluating a Surface Integral In Exercises 15-18, evaluate sf(x,y)dS. f(x,y)=x+yS:r(u,v)=4ucosvi+4usinvj+3uk0u4,0vEvaluating a Surface Integral In Exercises 19-24, evaluate sf(x,y,z)dS f(x,y,z)=x2+y2+z2S:z=x+y,x2+y2118EEvaluating a Surface Integral In Exercises 19-24.evaluate sf(x,y,z)dS f(x,y,z)=x2+y2+z2S:z=x2+y2,x2+y24Evaluating 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+y21Evaluating a Surface Integral In Exercises 19-24, evaluate sf(x,y,z)dS f(x,y,z)=x2+y2+z2S:x2+y2=9,0x3,0y3,0z9Evaluating a Surface Integral In Exercises 19-24, evaluate sf(x,y,z)dS f(x,y,z)=x2+y2+z2S:x2+y2=9,0x3,0zxEvaluating 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 octant24EEvaluating 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,z026EEvaluating 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+y24Evaluating 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=a2x2y2Evaluating 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=0Evaluating 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=137E38E31E32E33E34E35E36EHOW DO YOU SEE IT? Is the surface shown in the figure orient able? Explain why or why not.43EClassifying 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?1EVerifying the Divergence Theorem In Exercises 1–6, verify the Divergence Theorem by evaluating
as a surface integral and as a triple integral.
2.
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 planesVerifying 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 = 0Verifying 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=4Using 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=aUsing 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=aUsing 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=0Using 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=0Using 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=912E13E14E15E16E17E18E19E21EHOW 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?23E24E25E26E27EProof 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?1E2E3E4E5EVerifying Stokess Theorem In Exercises 3-6, verify Stokess Theorem by evaluating CFdr as a line integral and as a double integral. F(x,y,z)=(y+z)i+(xz)j+(xy)kS:z=1x2y2Verifying Stokess Theorem In Exercises 3-6, verify Stokess Theorem by evaluating CFdr as a line integral and as a double integral. F(x,y,z)=xyzi+yj+zkS:6x+6y+z=12,firstoctantVerifying Stokes Theorem In Exercises 3-6, verify Stokes Theorem by evaluating cFdr as a line integral and as a double integral. F(x,y,z)=z2i+x2j+y2k S:z=y2, 0xa, 0ya9E10E11E12E13EUsing Stokess TheoremIn Exercises 716, use Stokess Theorem to evaluate CFdr. In each case, C is oriented counterclockwise as viewed from above. F(x,y,z)=x2i+z2jxyzkS:z=4x2y2Using Stokess Theorem In Exercises 7-16, use Stokess Theorem to evaluate CFdr . In each case, C is oriented counterclockwise as viewed from above. F(x,y,z)=lnx2+y2i+arctanxyj+k S:z=92x3y over r=2sin2 in the first octantUsing Stokess Theorem In Exercises 7-16, use Stokess Theorem to evaluate CFdr . In each case, C is oriented counterclockwise as viewed from above. F(x,y,z)=yzi+(23y)j+(x2+y2)k,x2+y216S:thefirst-octantprotionofx2+z2=16overx2+y2=16Using Stokes Theorem In Exercises 7-16, use Stokes Theorem to evaluate cFdr. In each case, C is oriented counterclockwise as viewed from above. F(x,y,z)=xyzi+yj+zk S: z=x2, 0xa, 0yaUsing Stokes Theorem In Exercises 7-16, use Stokes Theorem to evaluate cFdr. In each case, C is oriented counterclockwise as viewed from above. F(x,y,z)=xyzi+yj+zk, x2+y2a2 S: the first-octant portion of z=x2 over x2+y2=a219E20E23EHOW DO YOU SEE IT? Let S1 be the portion of the paraboloid lying above the xy-plane, and let S2 be the hemisphere, as shown in the figures. Both surfaces are oriented upward. For a vector field F(x,y,z) with continuous partial derivatives, does S1(curlF).NdS1=S2(curlF).NdS2? Explain your reasoning.25ESketching a Vector Field In Exercises 1 and 2, find F and sketch several representative vectors in the vector field. Use a computer algebra system to verify your results. F(x,y,z)=xi+j+2kSketching a Vector Field In Exercises 1 and 2, find F and sketch several representative vectors in the vector field. Use a computer algebra system to verify your results. F(x,y)=i2yj3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13REDivergence and Curl In Exercises 19-26, find (a) the divergence of the vector field and (b) the curl of the vector field. F(x,y,z)=y2jz2kDivergence and Curl In Exercises 19-26, find (a) the divergence of the vector field and (b) the curl of the vector field. F(x,y,z)=(cosy+ycosx)i+(sinxxsiny)j+xyzk16RE17RE18RE19RE20RE21RE22REEvaluating a Line Integral In Exercises 27-30, evaluate the line integral along the given path(s). c(x2+y2)dsC:r(t)=(1sint)i+(1cost)j,0t224RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE57095-15-47RE-Question-Digital.docx Evaluating a Line Integral In Exercises 4550, use Greens Theorem to evaluate the line integral. Cxy2dx+x2ydyC:x=4cost,y=4sint48RE49RE50RE51RE52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE1PSHeat Flux Consider a single heat source located at the origin with temperature T(x,y,z)=25x2+y2+z2 Calculate the heat flux across the surface S={ (x,y,z):z=1x2y2,x2+y21 } as shown in the figure. Repeat the calculation in part (a) using the parametrization x=sinucosv, y=sinusinv, z=cosu Where 0u2 and 0v2.3PS4PS5PS6PS7PS8PS9PS10PS11PS12PS13PS