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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
66EIdentities Prove the following identities. Assume that is a differentiable scalar-valued function and F and G are differentiable vector fields, all defined on a region of R3. 67.(F)=F+F (Product Rule)Identities Prove the following identities. Assume that is a differentiable scalar-valued function and F and G are differentiable vector fields, all defined on a region of R3. 68.(F)=(F)+(F) (Product Rule)69E70E71E72E73EGradients and radial fields Prove that for a real number p. with r=x,y,z,(1|r|p)=pr|r|p+2.75EDescribe the surface r(u, v) = 2cosu,2sinu,v,for 0 u and 0 v 1.Describe the surface r(u, v) = vcosu,vsinu,v, for 0 u and 0 v 10.Describe the surface r(u, v) = 4sinucosv,4sinusinv,4cosu, for 0 u /2 and 0 v .4QCExplain why explicit description for a cylinder x2 + y2 = a2 cannot be used for a surface integral over a cylinder and a parametric description must be used.Explain why the upward flux for the radial field in Example 8 is greater for the hemisphere than for the paraboloid.Give a parametric description for a cylinder with radius a and height h, including the intervals for the parameters.2EGive a parametric description for a sphere with radius a, including the intervals for the parameters.4EExplain how to compute the surface integral of a scalar-valued function f over a sphere using a parametric description of the sphere.Explain what it means for a surface to be orientable.Describe the usual orientation of a closed surface such as a sphere.Why is the upward flux of a vertical vector field F = 0, 0, 1 across a surface equal to the area of the projection of the surface in the xy-plane?Parametric descriptions Give a parametric description of the form r(u,v)=x(u,v),y(u,v),z(u,v) for the following surfaces. The descriptions are not unique. Specify the required rectangle in the uv-plane. 11.The plane 2x 4y + 3z = 16Parametric descriptions Give a parametric description of the form r(u,v)=x(u,v),y(u,v),z(u,v) for the following surfaces. The descriptions are not unique. Specify the required rectangle in the uv-plane. 12.The cap of the sphere x2+y2+z2=16, for 22z4Parametric descriptions Give a parametric description of the form r(u,v)=x(u,v),y(u,v),z(u,v) for the following surfaces. The descriptions are not unique. Specify the required rectangle in the uv-plane. 13.The frustum of the cone z2 = x2 + y2, for 2 z 8Parametric descriptions Give a parametric description of the form r(u,v)=x(u,v),y(u,v),z(u,v) for the following surfaces. The descriptions are not unique. Specify the required rectangle in the uv-plane. 14.The cone z2 = 4 (x2 + y2), for 0 z 4Parametric descriptions Give a parametric description of the form r(u,v)=x(u,v),y(u,v),z(u,v) for the following surfaces. The descriptions are not unique. Specify the required rectangle in the uv-plane. 15.The portion of the cylinder x2 + y2 = 9 in the first octant, for 0 z 3Parametric descriptions Give a parametric description of the form r(u,v)=x(u,v),y(u,v),z(u,v) for the following surfaces. The descriptions are not unique. Specify the required rectangle in the uv-plane. 16.The cylinder y2 + z2 = 36, for 0 x 9Identify the surface Describe the surface with the given parametric representation. 17.r(u,v)=u,v,2u+3v1, for 1u3,2v4Identify the surface Describe the surface with the given parametric representation. 18.r(u,v)=u,u+v,2uv, for 0u2,0v2Identify the surface Describe the surface with the given parametric representation. 19.r(u,v)=vcosu,vsinu,4v, for 0u,0v3Identify the surface Describe the surface with the given parametric representation. 20.r(u,v)=v,6cosu,6sinu, for 0u2,0v2Surface area using a parametric description Find the area of the following surfaces using a parametric description of the surface. 21.The half-cylinder {(r,,z):r=4,0,0z7}Surface area using a parametric description Find the area of the following surfaces using a parametric description of the surface. 22.The plane z = 3 x 3y in the first octantSurface area using a parametric description Find the area of the following surfaces using a parametric description of the surface. 23.The plane z = 10 x y above the square |x|2,|y|2Surface area using a parametric description Find the area of the following surfaces using a parametric description of the surface. 24.The hemisphere x2 + y2 + z2 = 100, for z 0Surface area using a parametric description Find the area of the following surfaces using a parametric description of the surface. 25.A cone with base radius r and height h, where r and h are positive constantsSurface area using a parametric description Find the area of the following surfaces using a parametric description of the surface. 26.The cap of the sphere x2+y2+z2=4, for 1 z 2Surface integrals using a parametric description Evaluate the surface integral Sf(x,y,z)dS using a parametric description of the surface. 27.f(x,y,z)=x2+y2, where S is the hemisphere x2+y2+z2=36, for z 0Surface integrals using a parametric description Evaluate the surface integral Sf(x,y,z)dS using a parametric description of the surface. 28.f(x,y,z)=y, where S is the cylinder x2+y2=9,0z3Surface integrals using a parametric description Evaluate the surface integral Sf(x,y,z)dS using a parametric description of the surface. 29.f(x,y,z)=x, where S is the cylinder x2+z2=1,0y3Surface integrals using a parametric description Evaluate the surface integral Sf(x,y,z)dS using a parametric description of the surface. 30.f(,,)=cos, where S is the part of the unit sphere in the first octantSurface area using an explicit description Find the area of the following surfaces using an explicit description of the surface. 29. The part of plane z = 2x + 2y + 4 over the region R bounded by the triangle with vertices (0, 0), (2, 0), and (2, 4)Surface area using an explicit description Find the area of the following surfaces using an explicit description of the surface.
The part of the plane z = x + 3y + 5 over the region R = {(x, y): 1 ≤ x2 + y2 ≤ 4}
Surface area using an explicit description Find the area of the following surfaces using an explicit description of the surface. 31.The cone z2=4(x2+y2), for 0 z 4Surface area using an explicit description Find the area of the following surfaces using an explicit description of the surface. 32. The trough z=12x2, for 1 x 1, 0 y 4Surface area using an explicit description Find the area of the following surfaces using an explicit description of the surface. 32.The paraboloid z = 2 (x2 + y2), for 0 z 8Surface area using an explicit description Find the area of the following surfaces using an explicit description of the surface. 34. The part of the hyperbolic paraboloid z = 3 + x2 y2 above the sector R = {(r, ): 0 r 2, 0 /2}.Surface integrals using an explicit description Evaluate the surface integral Sf(x,y,z)dS using an explicit representation of the surface. 35.f(x, y, z) = xy; S is the plane z = 2 x y in the first octant.Surface integrals using an explicit description Evaluate the surface integralSf(x,y,z)dSusing an explicit representation of the surface. 36. f(x, y, z)=x2+y2; S is the paraboloid z=x2 + y2. for 0 z 1.Surface integrals using an explicit description Evaluate the surface integral Sf(x,y,z)dS using an explicit representation of the surface. 37.f(x,y,z)=25x2y2; S is the hemisphere centered at the origin with radius 5, for z 0.Surface integrals using an explicit description Evaluate the surface integral Sf(x,y,z)dS using an explicit representation of the surface. 38.f(x,y,z)=ez; S is the plane z = 8 x 2y in the first octant.Average Values 39. Find the average temperature on that part of the plane 2x + 2y + z = 4 over the square 0x1,0y1.where the temperature is given by T(x, y, z) = e2x+y+z3Average values 40.Find the average squared distance between the origin and the points on the paraboloid z = 4 x2 y2, for z 0.Average values 41.Find the average value of the function f(x, y, z) = xyz on the unit sphere in the first octant.Average values 42.Find the average value of the temperature function T(x,y,z)=10025z on the cone z2=x2+y2, for 0 z 2.Surface integrals of vector fields Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface. 43.F = 0, 0, 1 across the slanted face of the tetrahedron z = 4 x y in the first octant; normal vectors point upward.Surface integrals of vector fields Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface. 44.F = x, y, z across the slanted face of the tetrahedron z = 10 2x 5y in the first octant; normal vectors point upward.Surface integrals of vector fields Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface. 45.F = x, y, z across the slanted surface of the cone z2 = x2 + y2, for 0 z 1; normal vectors point upward.Surface integrals of vector fields Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface. 46.F = ey, 2z, xy across the curved sides of the surface S={(x,y,z):z=cosy,|y|,0x4}; normal vectors point upward.Surface integrals of vector fields Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface. 47.F=r/|r|3 across the sphere of radius a centered at the origin, where r = x, y, z; normal vectors point outward.Surface integrals of vector fields Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface. 48.F = (y, x, 1) across the cylinder y = x3, for 0 x 1, 0 z 4; normal vectors point in the general direction of the positive y-axis.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.If the surface S is given by {(x,y,z):0x1,0y1,z=10}, then Sf(x,y,z)dS=0101f(x,y,10)dxdy. b.If the surface S is given by {(x,y,z):0x1,0y1,z=x}, then Sf(x,y,z)dS=0101f(x,y,z)dxdy. c.The surface r = (v cos u, v sin u, v2), for 0u,0v2, is the same as the surface r=vcos2u,vsin2u,v, for 0u/2,0v4. d.Given the standard parameterization of a sphere, the normal vectors tu tv are outward normal vectors.Miscellaneous surface integrals Evaluate the following integrals using the method of your choice. Assume normal vectors point either outward or upward. 50.Sln|r|ndS, where S is the hemisphere x2 + y2 + z2 = a2, for z 0, and where r = (x, y, z)Miscellaneous surface integrals Evaluate the following integrals using the method of your choice. Assume normal vectors point either outward or upward. 51.S|r|dS, where S is the cylinder x2 + y2 = 4, for 0 z 8, and where r = x, y, zMiscellaneous surface integrals Evaluate the following integrals using the method of your choice. Assume normal vectors point either outward or upward. 52.SxyzdS, where S is that part of the plane z = 6 y that lies in the cylinder x2 + y2 = 4Miscellaneous surface integrals Evaluate the following integrals using the method of your choice. Assume normal vectors point either outward or upward. 53.S(x,0,z)x2+z2ndS, where S is the cylinder x2 + z2 = a2, |y|2Cone and sphere The cone z2 = x2 + y2 for z 0, cuts the sphere x2 + y2 + z2 = 16 along a curve C. a.Find the surface area of the sphere below C, for z. 0. b.Find the surface area of the sphere above C. c.Find the surface area of the cone below C, for z 0.Cylinder and sphere Consider the sphere x2 + y2 + z2 = 4 and the cylinder (x 1)2 + y2 = 1, for z 0 Find the surface area of the cylinder inside the sphere.Flux on a tetrahedron Find the upward flux of the field F = x, y, z across the plane x/a+y/b+z/c=1 in the first octant. Show that the flux equals c times the area of the base of the region. Interpret the result physically.Flux across a cone Consider the field F = x, y, z and the cone z2=(x2+y2)/a2 for 0 z 1. a.Show that when a = 1, the outward flux across the cone is zero. Interpret the result. b.Find the outward flux (away from the z-axis), for any a 0. Interpret the result.Surface area formula for cones Find the general formula for the surface area of a cone with height h and base radius a (excluding the base).Surface area formula for spherical cap A sphere of radius a is sliced parallel to the equatorial plane at a distance a h from the equatorial plane (see figure). Find the general formula for the surface area of the resulting spherical cap (excluding the base) with thickness h.Radial fields and spheres Consider the radial field F=r/|r|p, where r = x, y, z and p is a real number. Let S be the sphere of radius a centered at the origin. Show that the outward flux of F across the sphere is 4/ap3. It is instructive to do the calculation using both an explicit and parametric description of the sphere.Heat flux The heat flow vector field for conducting objects is F = kT, where T(x, y, z) is the temperature in the object and k 0 is a constant that depends on the material. Compute the outward flux of F across the following surfaces S for the given temperature distributions. Assume k = 1. 61.T(x,y,z)=100exy; S consists of the faces of the cube |x|1,|y|1,|z|1.Heat flux The heat flow vector field for conducting objects is F = kT, where T(x, y, z) is the temperature in the object and k 0 is a constant that depends on the material. Compute the outward flux of F across the following surfaces S for the given temperature distributions. Assume k = 1. 62.T(x,y,z)=100ex2y2z2; S is the sphere x2 + y2 + z2 = a263EFlux across a cylinder Let S be the cylinder x2 + y2 = a2, for L z L. a.Find the outward flux of the field F = x, y, 0 across S. b.Find the outward flux of the field F=x,y,0(x2+y2)p/2=r|r|p across S. where |r| is the distance from the z-axis and p is a real number. c.In part (b), for what values of p is the outward flux finite as a (with L fixed)? d.In part (b), for what values of p is the outward flux finite as L (with a fixed)?Flux across concentric spheres Consider the radial fields F=x,y,z(x2+y2+z2)p/2=r|r|p, where p is a real number. Let S consist of the spheres A and B centered at the origin with radii 0 a b, respectively. The total outward flux across S consists of the flux out of S across the outer sphere B minus the flux into S across the inner sphere A. a.Find the total flux across S with p = 0. Interpret the result. b.Show that for p = 3 (an inverse square law), the flux across S is independent of a and b.Mass and center of mass Let S be a surface that represents a thin shell with density . The moments about the coordinate planes (see Section 13.6) are Myz=Sx(x,y,z)dS,Mxz=Sy(x,y,z)dS, and Mxy=Sz(x,y,z)dS. The coordinates of the center of mass of the shell are x=Myzm,y=Mxzm,z=Mxym, where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible. 66.The constant-density hemispherical shall x2+y2+z2=a2,z0Mass and center of mass Let S be a surface that represents a thin shell with density . The moments about the coordinate planes (see Section 13.6) are Myz=Sx(x,y,z)dS,Mxz=Sy(x,y,z)dS, and Mxy=Sz(x,y,z)dS. The coordinates of the center of mass of the shell are x=Myzm,y=Mxzm,z=Mxym, where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible. 67.The constant-density cone with radius a, height h, and base in the xy-planeMass and center of mass Let S be a surface that represents a thin shell with density . The moments about the coordinate planes (see Section 13.6) are Myz=Sx(x,y,z)dS,Mxz=Sy(x,y,z)dS, and Mxy=Sz(x,y,z)dS. The coordinates of the center of mass of the shell are x=Myzm,y=Mxzm,z=Mxym, where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible. 68.The constant-density half cylinder x2+z2=a2,h/2yh/2,z0Mass and center of mass Let S be a surface that represents a thin shell with density . The moments about the coordinate planes (see Section 13.6) are Myz=Sx(x,y,z)dS,Mxz=Sy(x,y,z)dS, and Mxy=Sz(x,y,z)dS. The coordinates of the center of mass of the shell are x=Myzm,y=Mxzm,z=Mxym, where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible. 69.The cylinder x2+y2=a2,0z2, with density (x,y,z)=1+zOutward normal to a sphere Show that |tutv|=a2sinufor a sphere of radius a defined parametrically by r(u, v) = a sin u cos v, a sin u sin v, a cos u, where 0 u and 0 v 2.Special case of surface integrals of scalar-valued functions Suppose that a surface S is defined as z = g(x, y) on a region R. Show that tx ty = zx, zy, 1 and that Sf(x,y,z)dS=Rf(s,y,g(x,y))zx2+zy2+1dA.Surfaces of revolution Suppose y = f(x) is a continuous and positive function on [a, b]. Let S be the surface generated when the graph of f on [a, b] is revolved about the x-axis. a.Show that S is described parametrically by r(u, v) = u, f(u) cos v, f(u) sin v, for a u b, 0 v 2 . b.Find an integral that gives the surface area of S. c.Apply the result of part (b) to find the area of the surface generated with f(x) = x3, for 1 x 2. d.Apply the result of part (b) to find the area of the surface generated with f(x) = (25 x2)1/2, for 3 x 4.Rain on roofs Let z = s(x, y) define a surface over a region R in the xy-plane, where z 0 on R. Show that the downward flux of the vertical vector field F = 0, 0, 1 across S equals the area of R. Interpret the result physically.Surface area of a torus a.Show that a torus with radii R r (see figure) may be described parametrically by r(u, v) = (R + r cos u) cos v, (R + r cos u) sin v, r sin u, for 0 u 2, 0 v 2 . b.Show that the surface area of the torus is 42Rr.Surfaces of revolution single variable Let f be the difference and positive on the interval [a, b]. Let S be the surface generated when the graph of f on [a, b] is revolved about the xaxis. Use Theorem 17.14 to show that the area of S (as given in Section 6.6) is ab2f(x)1+f(x)2dx.Suppose S is a region in the xy-plane with a boundary oriented counterclockwise. What is the normal to S? Explain why Stokes Theorem becomes the circulation form of Greens Theorem.In Example 3a We used the parameterization r(t) = cost,sint,3cos2t+sin2tfor C. Confirm that the parameterization C : r(t) = cost,sint,4cos2t3sin2t also results in an answer of 2.In Example 4, explain why a paddle wheel with its axis aligned with the zaxis does not spin when placed on the yaxis.Explain the meaning of the integral S(F)ndS in Stokes Theorem.Explain the meaning of the integral S(F)ndS in Stokes Theorem.Explain the meaning of Stokes Theorem.Why does a conservative vector field produce zero circulation around a closed curve?Verifying Stokes Theorem Verify that the line integral and the surface integral of Stokes Theorem are equal for the following vector fields, surfaces S. and closed curves C. Assume that C has counterclockwise orientation and S has a consistent orientation. 5.F = y, x, 10; S is the upper half of the sphere x2 + y2 + z2 = 1 and C is the circle x2 + y2 = 1 in the xy-plane.Verifying Stokes Theorem Verify that the line integral and the surface integral of Stokes Theorem are equal for the following vector fields, surfaces S. and closed curves C. Assume that C has counterclockwise orientation and S has a consistent orientation. 6.F = 0, x, y; S is the upper half of the sphere x2 + y2 + z2 = 4 and C is the circle x2 + y2 = 4 in the xy-plane.Verifying Stokes Theorem Verify that the line integral and the surface integral of Stokes Theorem are equal for the following vector fields, surfaces S. and closed curves C. Assume that C has counterclockwise orientation and S has a consistent orientation. 7.F = x, y, z ; S is the paraboloid z = 8 x2 y2, for 0 z 8, and C is the circle x2 + y2 = 8 in the xy-plane.Verifying Stokes Theorem Verify that the line integral and the surface integral of Stokes Theorem are equal for the following vector fields, surfaces S, and closed curves C. Assume that C has counterclockwise orientation and S has a consistent orientation. 8.F = 2z, 4x, 3y; S is the cap of the sphere x2 + y2 + z2 = 169 above the plane z = 12 and C is the boundary of S.Verifying Stokes Theorem Verify that the line integral and the surface integral of Stokes Theorem are equal for the following vector fields, surfaces S. and closed curves C. Assume that C has counterclockwise orientation and S has a consistent orientation. 9.F = y z, z x, x y; S is the cap of the sphere x2 + y2 + z2 = 16 above the plane z=7 and C is the boundary of S.Verifying Stokes Theorem Verify that the line integral and the surface integral of Stokes Theorem are equal for the following vector fields, surfaces S. and closed curves C. Assume that C has counterclockwise orientation and S has a consistent orientation. 10.F = y x, z, y x; S is the part of the plane z = 6 y that lies in the cylinder x2 + y2 = 16 and C is the boundary of S.Stokes Theorem for evaluating line integrals Evaluate the line integral CFdr by evaluating the surface integral in Stokes Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. 11.F = 2y, z, x; C is the circle x2 + y2 = 12 in the plane z = 0.Stokes Theorem for evaluating line integrals Evaluate the line integral CFdr by evaluating the surface integral in Stokes Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. 12.F = y, xz, y; C is the ellipse x2 + y2/4 = 1 in the plane z = 1.Stokes Theorem for evaluating line integrals Evaluate the line integral CFdr by evaluating the surface integral in Stokes Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. 13.F = x2 z2, y, 2xz; C is the boundary of the plane z = 4 x y in the first octant.Stokes Theorem for evaluating line integrals Evaluate the line integral CFdrby evaluating the surface integral in Stokes Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. 14.F = x2 y2, z2 x2, y2 z2; C is the boundary of the square |x| 1, |y| 1 in the plane z = 0.Stokes Theorem for evaluating line integrals Evaluate the line integral CFdrby evaluating the surface integral in Stokes Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. 15.F = y2, z2, x; C is the circle r(t) = 3 cos t, 4 cos t, 5 sin t, for 0 t 2p.Stokes Theorem for evaluating line integrals Evaluate the line integral CFdrby evaluating the surface integral in Stokes Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. 16.F = 2xy sin z, x2 sin z, x2y cos z; C is the boundary of the plane z = 8 2x 4y in the first octant.Stokes Theorem for evaluating surface integrals Evaluate the line integral in Stakes Theorem to determine the value of the surface integralS(F)ndS. Assume that n points in an upward direction. 17.F = x, y, z; S is the upper half of the ellipsoid x2/4 + y2/9 + z2 = 1.Stokes Theorem for evaluating surface integrals Evaluate the line integral in Stakes Theorem to determine the value of the surface integralS(F)ndS. Assume that n points in an upward direction. 18.F = r/|r|; S is the paraboloid x = 9 y2 z2 for 0 x 9 (excluding its base), where r = x, y, z.Stokes Theorem for evaluating surface integrals Evaluate the line integral in Stakes Theorem to determine the value of the surface integralS(F)ndS. Assume that n points in an upward direction. 19.F = 2y z, x y z; S is the cap of the sphere (excluding its base) x2 + y2 + z2 = 25, for 3 x 5.Stokes Theorem for evaluating surface integrals Evaluate the line integral in Stakes Theorem to determine the value of the surface integral S(F)ndS. Assume that n points in an upward direction. 20.F = x + y, y + z, z + x; S is the titled disk enclosed by r(t) = cos t, 2 sin t, 3cost.Stokes Theorem for evaluating surface integrals Evaluate the line integral in stokes Theorem to determine the value of the surface integral s(F)n dS. Assume n points in an upward direction. 21. F = y,zx,y;S is the part of the paraboloid z = 2 x2 2y2 that lies within the cylinder x2+y2=1.Stokes Theorem for evaluating surface integrals Evaluate the line integral in stokes Theorem to determine the value of the surface integral s(F)n dS. Assume n points in an upward direction. 22. F=4x,8z,4y; S is the part of the paraboloid z = 1 2x2 3y2 that lies within the paraboloid z = 2x2 + y2.Stokes Theorem for evaluating surface integrals Evaluate the line integral in stokes Theorem to determine the value of the surface integral s(F)n dS. Assume n points in an upward direction. 23. F=y,1,z; S is the part of the surface z=2x that lies within the cone z=x2+y2.Stokes Theorem for evaluating surface integrals Evaluate the line integral in stokes Theorem to determine the value of the surface integral s(F)n dS. Assume n points in an upward direction. 24. F=ex,1/z,y; S is the part of the surface z = 4 3y2 that lies within the paraboloid z = x2 + y2.Interpreting and graphing the curl For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. 21.v = 0, 0, yInterpreting and graphing the curl For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. 22.v = 1 z2, 0, 0Interpreting and graphing the curl For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. 23.v = 2z, 0, 1Interpreting and graphing the curl For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. 24.v = 0, z, yExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.A paddle wheel with its axis in the direction 0.1, 1 would not spin when put in the vector field F = 1, 1, 2 x, y, z. b.Stokes Theorem relates the flux of a vector field F across a surface to the values of F on the boundary of the surface. c.A vector field of the form F = a + f(x), b + g(y), c + h(z), where a, b, and c are constants, has zero circulation on a closed curve. d.If a vector field has zero circulation on all simple closed smooth curves C in a region D, then F is conservative on D.Conservative fields Use Stokes Theorem to find the circulation of the following vector fields around any simple closed smooth curve C 26.F = 2x, 2y, 2zConservative fields Use Stokes Theorem to find the circulation of the following vector fields around any simple closed smooth curve C 27.F = (x sin y ez)Conservative fields Use Stokes Theorem to find the circulation of the following vector fields around any simple closed smooth curve C 28.F = 3x2y, x3 + 2yz2, 2y2zConservative fields Use Stokes Theorem to find the circulation of the following vector fields around any simple closed smooth curve C 29.F = y2z3, 2xyz3, 3xy2z2Tilted disks Let S be the disk enclosed by the curve C: r(t) = cos cos t, sin t, sin cos t, for 0 t 2p, where 0 p/2 is a fixed angle. 30.What is the area of S? Find a vector normal to S.Tilted disks Let S be the disk enclosed by the curve C: r(t) = cos cos t, sin t, sin cos t, for 0 t 2p, where 0 p/2 is a fixed angle. 31.What is the length of C?Tilted disks Let S be the disk enclosed by the curve C: r(t) = cos cos t, sin t, sin cos t, for 0 t 2p, where 0 p/2 is a fixed angle. 32.Use Stokes Theorem and a surface integral to find the circulation on C of the vector field F = y, x, 0 as a function of . For what value of is the circulation a maximum ?Tilted disks Let S be the disk enclosed by the curve C: r(t) = cos cos t, sin t, sin cos t, for 0 t 2p, where 0 p/2 is a fixed angle. 33.What is the circulation on C of the vector field F = y, z, x as a function of ? For what value of is the circulation a maximum?38ECirculation in a plane A circle C in the plane x + y + z = 8 has a radius of 4 and center (2, 3, 3). Evaluate CFdr for F = 0, z, 2y where C has a counterclockwise orientation when viewed from above. Does the circulation depend on the radius of the circle? Does it depend on the location of the center of the circle?No integrals Let F = (2z, z, 2y + x) and let S be the hemisphere of radius a with its base in the xy-plane and center at the origin. a.Evaluate S(F)ndS by computing F and appealing to symmetry. b.Evaluate the line integral using Stokes Theorem to check part (a).Compound surface and boundary Begin with the paraboloid z = x2 + y2, for 0 z 4, and slice it with the plane y = 0. Let S be the surface that remains for y 0 (including the planar surface in the xzplane) (see figure). Let C be the semicircle and line segment that bound the cap of .S in the plane z = 4 with counterclockwise orientation. Let F = 2z + y, 2x + z, 2y + x. a.Describe the direction of the vectors normal to the surface that are consistent with the orientation of C. b.Evaluate S(F)ndS c.Evaluate CFdrand check for agreement with part (b).Ampres Law The French physicist AndrMarie Ampre (17751836) discovered that an electrical current I in a wire produces a magnetic field B. A special case of Ampres Law relates the current to the magnetic field through the equation CBdr=I, where C is any closed curve through which the wire passes and is a physical constant. Assume that the current I is given in terms of the current density J as I=SJndS, where S is an oriented surface with C as a boundary. Use Stokes Theorem to show that an equivalent form of Ampres Law is B = J.Maximum surface integral Let S be the paraboloid z = a(1 x2 y2), for z 0, where a 0 is a real number. Let F = (x y, y + z, z x). For what value(s) of a (if any) does S(F)ndShave its maximum value?Area of a region in a plane Let R be a region in a plane that has a unit normal vector n = a, b, c and boundary C. Let F = bz, cx, ay. a.Show that F = n b.Use Stokes Theorem to show that areaofR=CFdr. c.Consider the curve C given by r = 5 sin t, 13 cos t, 12 sin t, for 0 t 2p. Prove that C lies in a plane by showing that rr is constant for all t. d.Use part (b) to find the area of the region enclosed by C in part (c). (Hint: Find the unit normal vector that is consistent with the orientation of C.)Choosing a more convenient surface The goal is to evaluateA=S(F)ndS, where F = yz, xz, xy and S is the surface of the upper half of the ellipsoid x2 + y2 + 8z2 = 1(z 0). a.Evaluate a surface integral over a more convenient surface to find the value of A. b.Evaluate A using a line integral.Radial fields and zero circulation Consider the radial vector fields F = r/|r|p, where p is a real number and r = x, y, z. Let C be any circle in the xy-plane centered at the origin. a.Evaluate a line integral to show that the field has zero circulation on C. b.For what values of p does Stokes Theorem apply? For those values of p, use the surface integral in Stokes Theorem to show that the field has zero circulation on C.Zero curl Consider the vector field F=yx2+y2i+xx2+y2j+zk. a.Show that F = 0. b.Show that CFdr is not zero on a circle C in the xy-plane enclosing the origin. c.Explain why Stokes Theorem does not apply in this case.Average circulation Let S be a small circular disk of radius R centered at the point P with a unit normal vector n. Let C be the boundary of S. a.Express the average circulation of the vector field F on S as a surface integral of F. b.Argue that for small R, the average circulation approaches ( F)|pn (the component of F in the direction of n evaluated at P) with the approximation improving as R0.Proof of Stokes Theorem Confirm the following step in the proof of Stokes Theorem. If z =s(x, y) and f, g, and h are functions of x, y, and z, with M = f + hzr and N = g + hzy, then My = fy + fzzy + hzxy + zx(hy + hzzy)and Ny = gy + gzzx + hzyx + zy(hx + hzzx).Stokes Theorem on closed surfaces Prove that if F satisfies the conditions of Stokes Theorem, then S(F)ndS=0, where S is a smooth surface that encloses a region.Rotated Greens Theorem Use Stokes Theorem to write the circulation form of Greens Theorem in the yz-plane.Interpret the Divergence Theorem in the case that F=a,b,c is a constant vector field and D is a ball.In Example 3. does the vector field have negative components anywhere in the cube D? Is the divergence negative anywhere in D?Draw unit cube D = {(x, y, z) : 0 x 1, 0 y 1, 0 z 1} and sketch the vector field F=x,y,2z on the six faces of the cube. Compute and interpret div F.Review Questions 1.Explain the meaning of the surface integral in the Divergence Theorem.Interpret the volume integral in the Divergence Theorem.Explain the meaning of the Divergence Theorem.What is the net outward flux of the rotation field F = 2z + y, x, 2x across the surface that encloses any region?What is the net outward flux of the radial field F = x, y, z across the sphere of radius 2 centered at the origin?What is the divergence of an inverse square vector field?Suppose div F = 0 in a region enclosed by two concentric spheres. What is the relationship between the outward fluxes across the two spheres?If div F 0 in a region enclosed by a small cube, is the net flux of the field into or out of the cube?Verifying the Divergence Theorem Evaluate both integrals of the Divergence Theorem for the following vector fields and regions. Check for agreement. 9.F = 2x, 3y, 4z; D = {(x, y, z): x2 + y2 + z2 4}F = x, y, z; D = {(x, y, z): |x| 1, |y| 1, |z| 1}Basic Skills 912.Verifying the Divergence Theorem Evaluate both integrals of the Divergence Theorem for the following vector fields and regions. Check for agreement. 11.F = z y, x, x; D = {(x, y, z): x2/4 + y2/8 + z2/12 1}F = x2, y2, z2; D = {(x, y, z): |x| 1, |y| 2, |z| 3}Rotation fields 13.Find the net outward flux of the field F = 2z y, x, 2x across the sphere of radius 1 centered at the origin.Rotation fields 14.Find the net outward flux of the field F = z y, x z, y x across the boundary of the cube {(x, y, z): |x| 1, |y| 1,Find the net outward flux of the field F = bz cy, cx az, ay bx across any smooth closed surface in 3, where a, b, and c are constants.Rotation fields 16.Find the net outward flux of F = a r across any smooth closed surface in 3 where a is a constant nonzero vector and r = x, y, z.Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. 17.F = x, 2y, 3z; S is the sphere {(x, y, z): x2 + y2 + z2 = 6}.Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. 18.F = x2, 2xz, y2; S is the surface of the cube cut from the first octant by the planes x = 1, y = 1, and z = 1.F = x, 2y, z; S is the boundary of the tetrahedron in the first octant formed by the plane x + y + z = 1.Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. 20.F = x2, y2, z2; S is the sphere {(x, y, z): x2 + y2 + z2 = 25}.F = y 2x, x3 y, y2 z; S is the sphere {(x, y, z): x2 + y2 + z2 = 4}.Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. 22.F = y + z, x + z, x + y; S consists of the faces of the cube {(x, y, z): |x| 1, |y| 1, |z| 1}.Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. 23.F = x, y, z; S is the surface of the paraboloid z = 4 x2 y2, for z 0, plus its base in the xy-plane.Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. 24.F = x, y, z; S is the surface of the cone z2 = x2 + y2, for 0 z 4, plus its top surface in the plane z = 4.Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. 25.F = z x, x y, 2y z; D is the region between the spheres of radius 2 and 4 centered at the origin.Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. 26.F = r |r| = x, y, z x2+y2+z2; D is the region between the spheres of radius 1 and 2 centered at the origin.Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. 27.F=r|r|=x,y,zx2+y2+z2; D is the region between the spheres of radius 1 and 2 centered at the origin.Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. 28.F = z y, x z, 2y x; D is the region between two cubes: {(x, y, z): 1 |x| 3, 1 |y| 3, 1 |z| 3}.F = x2, y2, z2); D is the region in the first octant between the planes z = 4 x y and z = 2 x y.Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. 30.F = x, 2y, 3z); D is the region between the cylinders x2 + y2 = 1 and x 2 + y2 = 4, for 0 z 8.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.If F = 0 at all points of a region D. then F n = 0 at all points of the boundary of D. b.If SFndS=0 on all closed surfaces in 3, then F is constant. c.If |F| 1, then |DFdV| is less than the area of the surface of D.Flux across a sphere Consider the radial field F = x, y, z and let S be the sphere of radius a centered at the origin. Compute the outward flux of F across S using the representation z=a2x2y2 for the sphere (either symmetry or two surfaces must be used).Flux integrals Compute the outward flux of the following vector fields across the given surfaces S You should decide which integral of the Divergence Theorem to use. 33.F = x2ey cos z, 4xey cos z, 2xey sin z; S is the boundary of the ellipsoid x2/4 + y2 + z2 = 1.Flux integrals Compute the outward flux of the following vector fields across the given surfaces S. You should decide which integral of the Divergence Theorem to use. 34.F = yz, xz, 1); S is the boundary of the ellipsoid x2/4 + y2/4 + z2 = 1.Flux integrals Compute the outward flux of the following vector fields across the given surfaces S You should decide which integral of the Divergence Theorem to use. 35.F = x sin y, cos y, z sin y; S is the boundary of the region bounded by the planes x = 1, y = 0, y = p/2, z = 0, and z = x.Radial fields Consider the radial vector field F=r|r|p=x,y,z(x2+y2+z2)p2. Let S be the sphere of radius a centered at the origin. a. Use a surface integral to show that the outward flux of F across S is 4a3 p. Recall that the unit normal to the sphere is r/|r|. b. For what values of p does F satisfy the conditions of the Divergence Theorem? For these values of p, use the fact (Theorem 17.10) that F=3p|r|p to compute the flux across S using the Divergence Theorem.Singular radial field Consider the radial field F=r|r|=x,y,z(x2+y2+z2)1/2. a.Evaluate a surface integral to show thatSFndS=4a2, where S is the surface of a sphere of radius a centered at the origin. b.Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a) is finite. Evaluate the triple integral of the Divergence Theorem as an improper integral as follows. Integrate div F over the region between two spheres of radius a and 0 g a. Then let g 0+ to obtain the flux computed in part (a).Logarithmic potential Consider the potential function (x,y,z)=12ln(x2+y2+z2)=ln|r|, where r = x, y, z. a.Show that the gradient field associated with is F=r|r|2=x,y,zx2+y2+z2. b.Show that SFndS=4a, where S is the surface of a sphere of radius a centered at the origin. c.Compute div F. d.Note that F is undefined at the origin, so the Divergence Theorem does not apply directly. Evaluate the volume integral as described in Exercise 37.Gauss Law for electric fields The electric field due to a point charge Q is E=Q40r|r|3, where r = x, y, z, and 0 is a constant. a.Show that the flux of the field across a sphere of radius a centered at the origin is SEndS=Q0. b.Let S be the boundary of the region between two spheres centered at the origin of radius a and b with a b. Use the Divergence Theorem to show that the net outward flux across S is zero. c.Suppose there is a distribution of charge within a region D Let q(x, y, z) be the charge density (charge per unit volume). Interpret the statement that SEndS=10Dq(x,y,z)dV. d.Assuming E satisfies the conditions of the Divergence Theorem on D. conclude from part (c) that E=q0. e.Because the electric force is conservative, it has a potential function . From part (d). conclude that 2==q0.Gauss Law for gravitation The gravitational force due to a point mass M at the origin is proportional to F = GMr/|r|3, where r = x, y, z and G is the gravitational constant. a.Show that the flux of the force field across a sphere of radius a centered at the origin is SFndS=4GM b.Let S be the boundary of the region between two spheres centered at the origin of radius a and b with a b. Use the Divergence Theorem to show that the net outward flux across S is zero. c.Suppose there is a distribution of mass within a region D Let (x, y, z) be the mass density (mass per unit volume). Interpret the statement that SFndS=4GD(x,y,z)dV. d.Assuming F satisfies the conditions of the Divergence Theorem on D. conclude from part (c) that F = 4pG. e.Because the gravitational force is conservative, it has a potential function . From part (d). conclude that 2 = 4pGp.Heat transfer Fouriers Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = kT. which means that heat energy flows from hot regions to cold regions. The constant k 0 is called the conductivity, which has metric units of J/m-s-K. A temperature function for a region D is given. Find the net outward heat flux SFndS=kSTndSacross the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1. 41.T(x, y, z) = 100 + x + 2y + z; D = {(x, y, z): 0 x 1, 0 1, 0 z 1}Heat transfer Fouriers Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = kT. which means that heat energy flows from hot regions to cold regions. The constant k 0 is called the conductivity, which has metric units of J/m-s-K. A temperature function for a region D is given. Find the net outward heat flux SFndS=kSTndSacross the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1. 42.T(x, y, z) = 100 + x2 + y2 + z2; D = {(x, y, z): 0 x 1, 0 1, 0 z 1}Heat transfer Fouriers Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = kT. which means that heat energy flows from hot regions to cold regions. The constant k 0 is called the conductivity, which has metric units of J/m-s-K. A temperature function for a region D is given. Find the net outward heat flux SFndS=kSTndSacross the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1. 43.T(x, y, z) = 100 + e-z; D = {(x, y, z): 0 x 1, 0 1, 0 z 1}Heat transfer Fouriers Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = kT. which means that heat energy flows from hot regions to cold regions. The constant k 0 is called the conductivity, which has metric units of J/m-s-K. A temperature function for a region D is given. Find the net outward heat flux SFndS=kSTndSacross the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1. 44.T(x, y, z) = 100 + x2 + y2 + z2; D is the unit sphere centered at the origin.Heat transfer Fouriers Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = kT. which means that heat energy flows from hot regions to cold regions. The constant k 0 is called the conductivity, which has metric units of J/m-s-K. A temperature function for a region D is given. Find the net outward heat flux SFndS=kSTndSacross the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1. 45.T(x,y,z)=100ex2y2z2 D is the sphere of radius a centered at the origin.Inverse square fields are special Let F be a radial field F = r/|r|p, where p is a real number and r = x, y, z. With p = 3, F is an inverse square field. a.Show that the net flux across a sphere centered at the origin is independent of the radius of the sphere only for p = 3. b.Explain the observation in part (a) by finding the flux of F = r/|r|p across the boundaries of a spherical box {(, , ): a b, 1 2, 1 2} for various values of p.A beautiful flux integral Consider the potential function (x, y, z) = G(p), where G is any twice differentiable function and =x2+y2+z2; therefore, G depends only on the distance from the origin. a.Show that the gradient vector field associated with is F==G()r, where r = x, y, z and = |r|. b.Let S be the sphere of radius a centered at the origin and let D be the region enclosed by S. Show that the flux of F across S is sFndS=4a2G(a). c.Show thatF==2G()+G(). d.Use part (c) to show that the flux across S (as given in part (b)) is also obtained by the volume integral DFdV. (Hint: use spherical coordinates and integrate by parts.)Integration by parts (Gauss' Formula) Recall the Product Rule of Theorem 14.11: (uF) = uF + u(F). a.Integrate both sides of this identity over a solid region D with a closed boundary S and use the Divergence Theorem to prove an integration by parts rule: Du(F)dV=SuFndSDuFdV. b.Explain the correspondence between this rule and the integration by parts rule for single-variable functions. c.Use integration by parts to evaluate D(x2y+y2z+z2x)dV, where D is the cube in the first octant cut by the planes x = 1, y = 1, and z = 1.49E50EGreens Second Identity Prose Greens Second Identity for scalar-valued functions u and v defined on a region D: D(u2vv2u)dV=S(uvvu)ndS. (Hint: Reverse the roles of u and v in Greens First Identity.)52E53E54EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.The rotational field F = y, x has zero curl and zero divergence. b. = 0 c.Two vector fields with the same curl differ by a constant vector field. d.Two vector fields with the same divergence differ by a constant vector field. e.If F = x, y, z) and S encloses a region D, then SFndS is three times the volume of D.Matching vector fields Match vector fields a-f with the graphs A-F. Let r = x, y. a.F = x, y b.F = 2y, 2x c.F = r/|r| d.F = y x, x e.F = ey, ex f.F = sin px, sin pyGradient fields in 2 Find the vector field F = for the following potential functions. Sketch a few level curves of and sketch the general appearance of F in relation to the level curves. 3.(x, y) = x2 + 4y2, for |x| 5, |y| 5Gradient fields in 2 Find the vector field F = for the following potential functions. Sketch a few level curves of and sketch the general appearance of F in relation to the level curves. 4.(x, y) = (x2 + y2)/2, for |x| 2, |y| 2Gradient fields in 3 Find the vector field F = for the following potential functions. 5.(x, y, z) = 1/|r|, where r = x, y, zGradient fields in 3 Find the vector field F = for the following potential functions. 6.(x,y,z)=12ex2y2z2Normal component Let C be the circle of radius 2 centered at the origin with counterclockwise orientation. Give the unit outward normal vector at any point (x, y) on C.Line integrals Evaluate the following line integrals. 8.C(x22xy+y2)ds;C is the upper half of the circle r(t) = 5 cos t, 5 sin t, for 0 t .Line integrals Evaluate the following line integrals. 9. Cyexzds; C is the path r(t)=2t,3t,6t, for 0 t 2.Line integrals Evaluate the following line integrals. 10.C(xzy2)ds; C is the line segment from (0, 1, 2) to (3, 7, 1).Two parameterizations Verify that C(x2y+3z)ds has the same value when C is given by r(t) = 2 cos t, 2 sin t, 0, for 0 t 2, and by r(t) = 2 cos t2, 2 sin t2, 0, for 0t2.Work integral Find the work done in moving an object from P(1, 0, 0) to Q(0, 1, 0) in the presence of the force F = 1, 2y, 4z along the following paths. a.The line segment from P to Q b.The line segment from P to O(0, 0, 0) followed by the line segment from O to Q c.The arc of die quarter circle from P to Q d.Is the work independent of the path?Work integrals in R3 Given the following force fields, find the work required to move an object on the given curve. 13. F = y, z, x on the path consisting of the line segment from (0, 0, 0) to (0, 1, 0) followed by the line segment from (0, 1, 0) to (0, 1, 4)Work integrals in 3 Given the following force fields, find the work required to move an object on the given curve. 14.F=x,y,z(x2+y2+z2)3/2 on the path r(t) = t2, 3t2, t2, for 1 t 2Circulation and flux Find the circulation and the outward flux of the following vector fields for the curve r(t) = 2 cos t, 2 sin t, for 0 t 2. 15.F = y x, yCirculation and flux Find the circulation and the outward flux of the following vector fields for the curve r(t) = 2 cos t, 2 sin t, for 0 t 2. 16.F = x, yCirculation and flux Find the circulation and the outward flux of the following vector fields for the curve r(t) = 2 cos t, 2 sin t, for 0 t 2. 17.F = r/|r|2, where r = x, yCirculation and flux Find the circulation and the outward flux of the following vector fields for the curve r(t) = 2 cos t, 2 sin t, for 0 t 2. 18.F = x y, xFlux in channel flow Consider the flow of water in a channel whose boundaries are the planes y = L and z=12. The velocity field in the channel is v = v0(L2 y2), 0, 0. Find the flux across the cross section of the channel at x = 0 in terms of v0 and L.Conservative vector fields and potentials Determine whether the following vector fields are conservative on their domains. If so, find a potential function. 20.F = y2, 2xyConservative vector fields and potentials Determine whether the following vector fields are conservative on their domains. If so, find a potential function. 21.F = y, x + z2, 2yzConservative vector fields and potentials Determine whether the following vector fields are conservative on their domains. If so, find a potential function. 22.F = ex cos y, ex sin yConservative vector fields and potentials Determine whether the following vector fields are conservative on their domains. If so, find a potential function. 23.F = ezy, x, xyEvaluating line integrals Evaluate the line integral CFdr for the following vector fields F and curves C in two ways. a.By parameterizing C b.By using the Fundamental Theorem for line integrals, if possible 24.F=(x2y);C:r(t)=9t2,t, for 0 t 3Evaluating line integrals Evaluate the line integral CFdr for the following vector fields F and curves C in two ways. a.By parameterizing C b.By using the Fundamental Theorem for line integrals, if possible 25.F=(xyz);C:r(t)=cost,sint,t/, for 0 tEvaluating line integrals Evaluate the line integral CFdr for the following vector fields F and curves C in two ways. a.By parameterizing C b.By using the Fundamental Theorem for line integrals, if possible 26.F = x, y; C is the square with vertices (1, 1) with counterclockwise orientation.Evaluating line integrals Evaluate the line integral CFdr for the following vector fields F and curves C in two ways. a.By parameterizing C b.By using the Fundamental Theorem for line integrals, if possible 27.F = y, z, x; C: r(t) = cos t, sin t, 4, for 0 t 2Radial fields in R2 are conservative Prove that the radial field F=r|r|p, where r = x, y and p is a real number, is conservative on R2 with the origin removed. For what value of p is F conservative on R2 (including the origin)?Greens Theorem for line integrals Use either form of Greens Theorem to evaluate the following line integrals. 29.Cxy2dx+x2ydy;C is the triangle with vertices (0, 0), (2, 0), and (0, 2) with counterclockwise orientation.Greens Theorem for line integrals Use either form of Greens Theorem to evaluate the following line integrals. 30.C(3y+x3/2)dx+(xy2/3)dy; C is the boundary of the half disk {(x, y): x2 + y2 2, y 0} with counterclockwise orientation.Greens Theorem for line integrals Use either form of Greens Theorem to evaluate the following line integrals. 31.C(x3+xy)dy+(2y22x2y)dx; C is the square with vertices (1, 1) with counterclockwise orientation.Greens Theorem for line integrals Use either form of Greens Theorem to evaluate the following line integrals. 32.C3x3dy3y3dx; C is the circle of radius 4 centered at the origin with clockwise orientation.Areas of plane regions Find the area of the following regions using a line integral. 33.The region enclosed by the ellipse x2 + 4y2 = 16Areas of plane regions Find the area of the following regions using a line integral. 34.The region bounded by the hypocycloid r(t) = cos3 t, sin3 t, for 0 t 2Circulation and flux Consider the following vector fields. a.Compute the circulation on the boundary of the region R (with counterclockwise orientation). b.Compute the outward flux across the boundary of R. 35.F = r/|r|, where r = x, y and R is the half-annulus {(r, ): 1 r 3, 0 }Circulation and flux Consider the following vector fields. a.Compute the circulation on the boundary of the region R (with counterclockwise orientation). b.Compute the outward flux across the boundary of R. 36.F = sin y, x cos y, where R is the square {(x, y): 0 x /2, 0 y /2}Parameters Let F = ax + by, cx + dy, where a, b, c, and d are constants. a.For what values of a, b, c, and d is F conservative? b.For what values of a, b, c, and d is F source free? c.For what values of a, b, c, and d is F conservative and source free?Divergence and curl Compute the divergence and curl of the following vector fields. State whether the field is source free or irrotational. 38.F = yz, xz, xyDivergence and curl Compute the divergence and curl of the following vector fields. State whether the field is source free or irrotational. 39.F=r|r|=x,y,zx2+y2+z2Divergence and curl Compute the divergence and curl of the following vector fields. State whether the field is source free or irrotational. 40.F = sin xy, cos yz, sin xzDivergence and curl Compute the divergence and curl of the following vector fields. State whether the field is source free or irrotational. 41.F = 2xy + z4, x2, 4xz3Identities Prove that (1|r|4)=4r|r|6 and use the result to prove that (1|r|4)=12|r|6.Maximum curl Let F=z,x,y. a. What is the scalar component of curl F in the direction of n=1,0,0? b. What is the scalar component of curl F in the direction of n=0,1/2,1/2? c. In the direction of what unit vector n is the scalar component of curl F a maximum?Paddle wheel in a vector field Let F = 0, 2x, 0 and let n be a unit vector aligned with the axis of a paddle wheel located on the y-axis. a.If the axis of the paddle wheel is aligned with n = 1, 0, 0, how fast does it spin? b.If the axis of the paddle wheel is aligned with n = 0, 0, 1, how fast does it spin? c.For what direction n does the paddle wheel spin fastest?Surface areas Use a surface integral to find the area of the following surfaces. 45.The hemisphere x2 + y2 + z2 = 9, for z 0 (excluding the base)Surface areas Use a surface integral to find the area of the following surfaces. 46.The frustum of the cone z2 = x2 + y2, for 2 z 4 (excluding the bases)Surface areas Use a surface integral to find the area of the following surfaces. 47.The plane z = 6 x y above the square |x| 1, |y| 1Surface areas Use a surface integral to find the area of the following surfaces. 48.The surface f(x,y)=2xy above the region {(r,):0r2,02}Surface integrals Evaluate the following surface integrals. 49.S(1+yz)dS; S is the plane x + y + z = 2 in the first octant.Surface integrals Evaluate the following surface integrals. 50.S0,y,zndS; S is the curved surface of the cylinder y2+z2=a2,|x|8 with outward normal vectors.Surface integrals Evaluate the following surface integrals. 51.S(xy+z)dS; S is the entire surface including the base of the hemisphere x2 + y2 + z2 = 4, for z 0.Flux integrals Find the flux of the following vector fields across the given surface. Assume the vectors normal to the surface point outward. 52.F = x, y, z across the curved surface of the cylinder x2 + y2 = 1, for |z| 8Flux integrals Find the flux of the following vector fields across the given surface. Assume the vectors normal to the surface point outward. 53.F = r/|r| across the sphere of radius a centered at the origin, where r = x, y, zThree methods Find the surface area of the paraboloid z = x2 + y2, for 0 z 4, in three ways. a.Use an explicit description of the surface. b.Use the parametric description r = v cos u, v sin u, v2). c.Use the parametric description r=vcosu,vsinu,v.Flux across hemispheres and paraboloids Let S be the hemisphere x2 + y2 + z2 = a2, for z 0, and let T be the paraboloid z = a (x2 + y2)/a, for z 2= 0, where a 0. Assume the surfaces have outward normal vectors. a.Verify that S and T have the same base (x2 + y2 a2) and the same high point (0, 0, a). b.Which surface has the greater area? c.Show that the flux of the radial field F = x, y, z across S is 2a3. d.Show that the flux of the radial field F = x, y, z across T is 3a3/2.Surface area of an ellipsoid Consider the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1, where a, b, and c are positive real numbers. a.Show that the surface is described by the parametric equations r(u, v) = a cos u sin v, b sin u sin v, c cos v for 0 u 2, 0 v . b.Write an integral for the surface area of the ellipsoid.Stokes Theorem for line integrals Evaluate the line integral CFdr using Stokes Theorem. Assume C has counterclockwise orientation. 57.F = xz, yz, xy; C is the circle x2 + y2 = 4 in the xy-plane.Stokes Theorem for line integrals Evaluate the line integral CFdr using Stokes Theorem. Assume C has counterclockwise orientation. 58.F=x2y2,x,2yz; C is the boundary of the plane z = 6 2x y in the first octant.Stokes Theorem for surface integrals Use Stokes Theorem to evaluate the surface integral S(F)ndS. Assume that n is the outward normal. 59.F = z, x, y, where S is the hyperboloid z=101+x2+y2, for z 0Stokes Theorem for surface integrals Use Stokes Theorem to evaluate the surface integral S(F)ndS. Assume that n is the outward normal. 60.F = x2 z2, y2, xz, where S is the hemisphere x2 + y2 + z2 = 4, for y 0Conservative fields Use Stokes Theorem to find the circulation of the vector field F = (10 x2+ y2 + z2) around any smooth closed curs C with counterclockwise orientation.Computing fluxes Use the Divergence Theorem to compute the outward flux of the following vector fields across the given surfaces S. 62.F = x, x y, x z; S is the surface of the cube cut from the first octant by the planes x = 1, y = 1, and z = 1.Computing fluxes Use the Divergence Theorem to compute the outward flux of the following vector fields across the given surfaces S. 63.F = x3, y3, z3/3; S is the sphere {(x, y, z): x2 + y2 + z2 = 9}Computing fluxes Use the Divergence Theorem to compute the outward flux of the following vector fields across the given surfaces S. 64.F = x2, y2, z2; S is the cylinder {(x, y, z): x2 + y2 = 4, 0 4, 0 z 8}.General regions Use the Divergence Theorem to compute the outward flux of the following vector fields across the boundary of the given regions D. 65.F = (x3, y3, 10); D is the region between the hemispheres of radius 1 and 2 centered at the origin with bases in the xy-plane.General regions Use the Divergence Theorem to compute the outward flux of the following vector fields across the boundary of the given regions D. 66.F=r|r|3=x,y,z(x2+y2+z2)3/2; D is the region between two spheres with radii 1 and 2 centered at (5, 5, 5).Flux integrals Compute the outward flux of the field F = x2 + x sin y, y2 + 2 cos y, z2 + z sin y across the surface S that is the boundary of the prism bounded by the planes y = 1 x, x = 0, y = 0, z = 0, and z = 4.Stokes Theorem on a compound surface Consider the surface S consisting of the quarter-sphere x2 + y2 + z2 = a2, for z 0 and x 0, and the half-disk in the yz-plane y2 + z2 a2, for z 0. The boundary of S in the xy-plane is C, which consists of the semicircle x2 + y2 = a2, for x 0, and the line segment [a, a] on the y-axis, with a counterclockwise orientation. Let F = 2z y, x z, y 2x. a.Describe the direction in which the normal vectors point on S. b.Evaluate CFdr. c.Evaluate S(F)ndS and check for agreement with part (b).State the meaning of {x: 4 x 10}. Express the set {x: 4 x 10} using interval notation and draw it on a number line.Write the interval (, 2) in set notation and draw it on a number line.3E4E5EWrite an equation of the set of all points that are a distance 5 units from the point (2, 3).Explain how to find the distance between two points whose coordinates are known.8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28ESolving inequalities Solve the following inequalities and draw the solution on a number line. 29.x29x+20x6030E