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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Find a parametric description r(t) for the following curves. 5. The line segment from (1, 2, 3) to (5, 4, 0)Find a parametric description r(t) for the following curves. 6. The quarter-circle from (1,0) to (0,1) with its center at the originFind a parametric description r(t) for the following curves. 7. The segment of the parabola x = y2 + 1 from (5, 2) to (17, 4)Find an expression for the vector field F = x y, y x (in terms of t) along the unit circle r(t) = cos t, sin t.Suppose C is the curve r(t) = t,t3, for 0 t 2, and F = x,2y. Evaluate CFT ds using the following steps. a. Convert the line integral CFTds to an ordinary integral. b. Evaluate the integral in part (a).Suppose C is the circle r(l) = cos l, sin l , for 0 t 2, and F = 1, x. Evaluate C F n ds Using the following steps. a. Convert the line integral C F n ds to an ordinary integral. b. Evaluate the integral in part (a)State two other forms for the line integral CFTds given that F = f, g, h.Assume F is continuous on a region containing the smooth curve C from point A to point B and suppose Cfds=10 12. Explain the meaning of the curve C and state the value of Cfds.Assume F is continuous on a region containing the smooth curve C from point A to point B and suppose Cfds=10 13. Suppose P is a point on the curve C between A and B, where C1 is the part of the curve from A to P, and C2 is the part of the curve from P to B. Assuming C1fds=3, find the value of C2fds.How is the circulation of a vector field on a closed smooth oriented curve calculated?16EScalar line integrals Evaluate the following line integrals along the curve C. Cxy ds; C is the unit circle r(t) = cos t, sin t, for 0 t 2.Scalar line integrals Evaluate the following line integrals along the curve C. C(x22y2) ds; C is the line segment r(t) = t2,t2 for 0 t 4.Scalar line integrals Evaluate the following line integrals along the curve C. 19. C(2x+y)ds; C is the line segment r(t)=3t,4t, for 0 t 2.Scalar line integrals Evaluate the following line integrals along the curve C. 20. Cx ds; C is the curve r(t)=t3,4t, for 0 t 1.Scalar line integrals Evaluate the following line integrals along the curve C. 21. Cxy3ds; C is the quarter-circle r(t)=2cost,2sint, for 0 t /2.Scalar line integrals Evaluate the following line integrals along the curve C. 22. C3x cos y ds; C is the curve r(t)=sint,t, for 0 t /2.Scalar line integrals Evaluate the following line integrals along the curve C. 23. C(yz)ds; C is the helix r(t)=3cost,3sint,4t, for 0 t 2.Scalar line integrals in 3Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. 26.C(x+y+2z)ds; C is the circle r(t)=1,3cost,3sint, for 0 t 2.Scalar line integrals Evaluate the following line integrals along the curve C. 25. C(x2+y2)ds; C is the circle of radius 4 centered at (0, 0).Scalar line integrals Evaluate the following line integrals along the curve C. 26. C(x2+y2)ds; C is the line segment from (0, 0) to (5, 5).Scalar line integrals Evaluate the following line integrals along the curve C. 27. Cxx2+y2ds; C is the line segment from (1, 1) to (10, 10).Scalar line integrals Evaluate the following line integrals along the curve C. 28. C(xy)1/3ds; C is the curve y = x2, for 0 x 1.Scalar line integrals Evaluate the following line integrals along the curve C. 29. Cxy ds; C is a portion of the ellipse x24+y216=1 in the first quadrant, oriented counter clockwise.Scalar line integrals Evaluate the following line integrals along the curve C. 30. C(2x3y) ds; C is the line segment from (1, 0) to (0, 1) followed by the line segment from (0, 1) to (1, 0).Scalar line integrals Evaluate the following line integrals along the curve C. 31. C(x+y+z)ds; C is the semicircle r(t)=2cost,0,2sint for 0 t .Scalar line integrals in 3Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. 28.Cxyzds; C is the line segment from (1, 4, 1) to (3, 6, 3).Scalar line integrals Evaluate the following line integrals along the curve C. 33. Cxzds; C is the line segment from (0, 0, 0) to (3, 2, 6) followed by the line segment from (3, 2, 6) to (7, 9, 10)Scalar line integrals Evaluate the following line integrals along the curve C. 34. Cxeyzds; C is r(t)=t,2t,2t, for 0 t 2.Mass and density A thin wire represented by the smooth curve C with a density (units of mass per length) has a mass M = C ds. Find the mass of the following wires with the given density. 63.C: {(x, y): y = 2x2, 0 x 3}; (x, y) = 1 + xyMass and density A thin wire represented by the smooth curve C with a density (units of mass per length) has a mass M = C ds. Find the mass of the following wires with the given density. 62.C: r() = cos , sin , for 0 ; () = 2/ + 1Average values Find the average value of the following functions on the given curves. 21.f(x,y)=x+2y on the line segment from (1, 1) to (2, 5)Average values Find the average value of the following functions on the given curves. 24.f(x,y)=xey on the unit circle centered at the originLength of curves Use a scalar line integral to find the length of the following curves. 31.r(t)=20sint4,20cost4,t2, for 0 t 2Length of curves Use a scalar line integral to find the length of the following curves. 32.r(t)=30sint,40sint,50cost, for 0 t 2Line integrals of vector fields in the plane Given the following vector fields and oriented curves C, evaluate CFTds. 33.F = (x, y) on the parabola r(t) = (4t, t2), for 0 t 1Line integrals of vector fields in the plane Given the following vector fields and oriented curves C, evaluate CFTds. 34.F = (y, x) on the semicircle r(t) = (4 cos t, 4 sin t), for 0 tLine integrals of vector fields in the plane Given the following vector fields and oriented curves C, evaluate CFTds. 35.F = (y, x) on the line segment from (1, 1) to (5, 10)Line integrals of vector fields in the plane Given the following vector fields and oriented curves C, evaluate CFTds. 36.F = y, x on the parabola y = x2 from (0, 0) to (1, 1)Line integrals of vector fields in the plane Given the following vector fields and oriented curves C, evaluate CFTds. 37.F=x,y(x2+y2)3/2 on the curve r(t)=t2,3t2, for 1 t 2Line integrals of vector fields in the plane Given the following vector fields and oriented curves C, evaluate CFTds. 38.F=x,yx2+y2 on the line r(t)=t,4t, for 1 t 1047–48. Line integrals from graphs Determine whether along the paths C1 and C2 shown in the following vector fields is positive or negative. Explain your reasoning.
47.
Line integrals from graphs Determine whether c F dr along the paths C1 and C2 shown in the following vector fields is positive or negative. Explain your reasoning. a. C1Fdr b. C2Fdr 48.Work integrals Given the force field F, find the work required to move an object on the given oriented curve. 39.F = y, x on the path consisting of the line segment from (1, 2) to (0, 0) followed by the line segment from (0, 0) to (0, 4)Work integrals Given the force field F, find the work required to move an object on the given oriented curve. 40.F = x, y on the path consisting of the line segment from (1, 0) to (0, 8) followed by the line segment from (0, 8) to (2, 8)Work integrals Given the force field F, find the work required to move an object on the given oriented curve. 41.F = y, x on the parabola y = 2x2 from (0, 0) to (2, 8)Work integrals Given the force field F, find the work required to move an object on the given oriented curve. 42.F = y, x on the line y = 10 2x from (1, 8) to (3, 4)Work integrals in 3 Given the force field F, find the work required to move an object on the given oriented curve. 43. F = x, y, z) on the tilted ellipse r(t) = 4 cos t, 4 sin t, 4 cos t, for 0 t 2Work integrals in 3 Given the force field F, find the work required to move an object on the given oriented curve. 44. F = y, x, z) on the helix r(t) = 2 cos t, 2 sin t, t/2, for 0 t 2Work integrals in 3 Given the force field F, find the work required to move an object on the given oriented curve. 45. F=x,y,z(x2+y2+z2)3/2on the line segment from (1, 1, 1) to (10, 10, 10)Work integrals in 3 Given the force field F, find the work required to move an object on the given oriented curve. 46. F=x,y,zx2+y2+z2 on the line segment from (1, 1, 1) to (8, 4, 2)Circulation Consider the following vector fields F and closed oriented curves C in the plane (see figures). a. Based on the picture make a conjecture about whether the circulation of F on C is positive, negative, or zero. b. Compute the circulation and interpret the result. 47.F = y x, x); C: r(t) = 2 cos t, 2 sin t), for 0 t 2Circulation Consider the following vector fields F and closed oriented curves C in the plane (see figures). a. Based on the picture, make a conjecture about whether the circulation of F on C is positive, negative, or zero. b. Compute the circulation and interpret the result. 58. F=y,2x4x2+y2; C : r(t)=2cost,4sint, for 0 t 2Flux Consider the vector fields and curves in Exercises 5758. a. Based on the picture, make a conjecture about whether the outward flux of F across C is positive, negative, or zero. b. Compute the flux for the vector fields and curves. 59. F and C given in Exercise 57 57. F=yx,x; C : r(t)=2cost,2sint, for 0 t 2Flux Consider the vector fields and curves in Exercises 5758. a. Based on the picture, make a conjecture about whether the outward flux of F across C is positive, negative, or zero. b. Compute the flux for the vector fields and curves. 60. F and C given in Exercise 58 58. F=y,2x4x2+y2; C : r(t)=2cost,4sint, for 0 t 2Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If a curve has a parametric description r(t) = x(t), y(t), z(t), where t is the arc length, then |r(t)|=1. b. The vector field F = y, x has both zero circulation along and zero flux across the unit circle centered at the origin. c. If at all points of a path a force acts in a direction orthogonal to the path, then no work is done in moving an object along the path. d. The flux of a vector field across a curve in 2 can be computed using a line integral.Flying into a headwind An airplane flies in the xz-plane, where x increases in the eastward direction and z 0 represents vertical distance above the ground. A wind blows horizontally out of the west, producing a force F = 150, 0. On which path between the points (100, 50) and (100, 50) is the most work done overcoming the wind? a. The straight line r(t) = x(t), z(t) = t, 50, for 100 t 100 b. The arc of a circle r(t) = 100 cos t, 50 + 100 sin t), for 0 tFlying into a headwind
How does the result of Exercise 62 change if the force due to the wind is F = ⟨141, 50⟩ (approximately the same magnitude, but a different direction)?
How does the result of Exercise 62 change if the force due to the wind is F = ⟨141, −50⟩ (approximately the same magnitude, but a different direction)?
62. Flying into a headwind An airplane flies in the xz-plane, where x increases in the eastward direction and z ≥ 0 represents vertical distance above the ground. A wind blows horizontally out of the west, producing a force F = ⟨150, 0⟩. On which path between the points (100, 50) and (−100, 50) is more work done overcoming the wind?
The line segment r(t) = ⟨x(t), z(t)⟩ = ⟨−t, 50⟩, for −100 ≤ t ≤ 100
The are of the circle r(t) = ⟨100 cos t, 50 + 100 sin t⟩, for 0 ≤ t ≤ π
64EChanging orientation Let f(x, y) = x and let C be the segment of the parabola y = x2 joining O(0, 0) and P(1, 1). a.Find a parameterization of C in the direction from O to P. Evaluate Cf ds. b.Find a parameterization of C in the direction from P to O. Evaluate Cf ds. c.Compare the results of (a) and (b).Work in a rotation field Consider the rotation field F = y, x and the three paths shown in the figure. Compute the work done in the presence of the force field F on each of the three paths. Does it appear that the line integral C F T ds is independent of the path, where C is any path from (1, 0) to (0, 1)?Work in a hyperbolic field Consider the hyperbolic force field F = ⟨y, x⟩ (the streamlines are hyperbolas) and the three paths shown in the figure for Exercise 66. Compute the work done in the presence of F on each of the three paths Does it appear that the line integral is independent of the path, where C is any path from (1, 0) to (0, 1)?
66. Work in a rotation field Consider the rotation field F = ⟨−y, x⟩ and the three paths shown in the figure. Compute the work done on each of the three paths. Does it appear that the line integral is independent of the path, where C is any path from (1, 0) to (0, 1)?
Assorted line integrals Evaluate each line integral using the given curve C. 68. Cx2dx+dy+ydz; C is the curve r(t)=t,2t,t2, for 0 t 3.Assorted line integrals Evaluate each line integral using the given curve C. 69. Cx3ydx+xzdy+(x+y)2dz; C is the helix r(t)=2t,sint,cost, for 0 t 4.Assorted line integrals Evaluate each line integral using the given curve C. 71. Cyx2+y2dxxx2+y2dy; C is a quarter-circle from (0, 4) to (4, 0).Assorted line integrals Evaluate each line integral using given curve C. 72. C(x+y)dx+(xy)dy+xdz;C is the line segment from (1, 2, 4) to (3, 8, 13).Flux across curves in a vector field Consider the vector Field F = y, x shown in the figure. a.Compute the outward flux across the quarter circle C: r(t) = 2 cos t, 2 sin t), for 0 t /2. b.Compute the outward flux across the quarter circle C: r(t) = 2 cos t, 2 sin t), for /2 t . c.Explain why the flux across the quarter circle in the third quadrant equals the flux computed in part (a). d.Explain why the flux across the quarter circle in the fourth quadrant equals the flux computed in part (b). e.What is the outward flux across the full circle?74EZero circulation fields 57.Consider the vector field F = ax + by, cx + dy. Show that F has zero circulation on any oriented circle centered at the origin, for any a, b, c, and d, provided b = c.Zero flux fields 58.For what values of a and d does the vector field F = ax, dy have zero flux across the unit circle centered at the origin and oriented counterclockwise?Zero flux fields 59.Consider the vector field F = ax + by, cx + dy. Show that F has zero flux across any oriented circle centered at the origin, for any a, b, c, and d, provided a = d.Heat flux in a plate A square plate R = {(x, y): 0 x 1, 0 y 1} has a temperature distribution T(x, y) = 100 50x 25y. a.Sketch two level curves of the temperature in the plate. b.Find the gradient of the temperature T(x, y). c.Assume that the flow of heat is given by the vector field F = T(x, y). Compute F. d.Find the outward heat flux across the boundary {(x, y): x= 1, 0 y 1}. e.Find the outward heat flux across the boundary {(x, y): 0 x 1, y = 1}.79ELine integrals with respect to dx and dy Given a vector field F = ⟨f, 0⟩ and curve C with parameterization r(t) = ⟨x(t), y(t)⟩ for a ≤ t ≤ b, we see that the line integral simplifies to .
Show that .
Use the vector field F = ⟨0, g⟩ to show that .
Evaluate , where C is the line segment from (0, 0) to (5, 12).
Evaluate , where C is a segment of the parabola x = y2 from (1, −1) to (1, 1).
Looking ahead: Area from line integrals The area of a region R in the plane, whose boundary is the closed curve C, may be computed using line integrals with the formula areaofR=Cxdy=Cydx. These ideas reappear later in the chapter. 67.Let R be the rectangle with vertices (0, 0), (a, 0), (0, b), and (a, b), and let C be the boundary of R oriented counterclockwise. Use the formula A = C x dy to verify that the area of the rectangle is ab.82EIs a figure-8 curve simple? Closed? Is a torus connected? Simply connected?Explain why a potential function for a conservative vector field is determined up to an additive constant.Verify by differentiation that the potential functions found in Example 2 produce the corresponding vector fields.Explain why the vector field (xy + xz yz) is conservative.What does it mean for a function to have an absolute extreme value at a point c of an interval [a, b]?What are local maximum and minimum values of a function?What conditions must be met to ensure that a function has an absolute maximum value and an absolute minimum value on an interval?How do you determine whether a vector field in 3 is conservative?Briefly describe how to find a potential function for a conservative vector field F = f, g.If F is a conservative vector field on a region R, how do you evaluate C F dr. where C is a path between two points A and B in R?If F is a conservative vector field on a region R, what is the value of CFdr, where C is a simple closed piecewise-smooth oriented curve in R?Give three equivalent properties of conservative vector fields.How do you determine the absolute maximum and minimum values of a continuous function on a closed interval?Explain how a function can have an absolute minimum value at an endpoint of an interval.Testing for conservative vector fields Determine whether the following vector fields are conservative (in 2 or 3). 11. F = y, xTesting for conservative vector fields Determine whether the following vector fields are conservative on 2. 12.F = y, x + yTesting for conservative vector fields Determine whether the following vector fields are conservative on 2. 13.F = ex cos y, ex sin yTesting for conservative vector fields Determine whether the following vector fields are conservative on 2. 14.F = 2x3 + xy2, 2y3 + x2yTesting for conservative vector fields Determine whether the following vector fields are conservative (in 2 or 3). 15. F = yz cos xz, sin xz, xy cos xzTesting for conservative vector fields Determine whether the following vector fields are coservative (in 2 or 3). 16. F = yexz, exz, yexzFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R and D be open regions of 2 and 3 respectively, that do not include the origin. 15.F = x, y on 2Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R and D be open regions of 2 and 3 respectively, that do not include the origin. 16.F = y, x on 2Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R and D be open regions of 2 and 3 respectively, that do not include the origin. 17.F=x3xy,x22+yon2Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R and D be open regions of 2 and 3 respectively, that do not include the origin. 18.F=x,yx2+y2onRDesigning a function Sketch a graph of a function f continuous on [0, 4] satisfying the given properties. 19.f(x) = 0 for x = 1 and 2; f has an absolute maximum at x = 4; f has an absolute minimum at x = 0: and f has a local minimum at x = 2.Finding Potential functions Determine Whether the following vector fields are conservative on the specified region. If so, determine a potertial function. Let R and D be open regions of 2 and 3, respectively, that do not include the origin. 22. F= y, x, x y on 3Designing a function Sketch a graph of a function f continuous on [0, 4] satisfying the given properties. 21.f(1) and f(3) are undefined; f(2) = 0; f has a local maximum at x = 1; f has a local minimum at x = 2; f has an absolute maximum at x = 3; and f has an absolute minimum at x = 4.Designing a function Sketch a graph of a function f continuous on [0, 4] satisfying the given properties. 22.f(x) = 0 at x = 1 and 3; f(2) is undefined; f has an absolute maximum at x = 2; f has neither a local maximum nor a local minimum at x = 1; and f has an absolute minimum at x = 3.Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R and D be open regions 2 and 3, respectively, that do not include the origin. 25. F = ez, ez, ez(x y)on 3Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R and D be open regions of 2 and 3, respectively, that do not include the origin. 26. F = 1, z, y on 3Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R and D be open regions of 2 and 3 respectively, that do not include the origin. 23.F = y + z, x + z, x + y on 3Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R and D be open regions of 2 and 3 respectively, that do not include the origin. 24.F=x,y,zx2+y2+z2onDFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R and D be open regions of 2 and 3 respectively, that do not include the origin. 25.F=x,y,zx2+y2+z2onDFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R and D be open regions of 2 and 3 respectively, that do not include the origin. 26.F = x3, 2y, z3 on 3Evaluating line integrals Evaluate the line integral Cdr for the following functions and oriented curves C in two ways. a.Use a parametric description of C to evaluate the integral directly. b.Use the Fundamental Theorem for line integrals. 27.(x, y) = xy; C: r(t) = cos t, sin t, for 0 tEvaluating line integrals Evaluate the line integral Cdr for the following functions and oriented curves C in two ways. a.Use a parametric description of C to evaluate the integral directly. b.Use the Fundamental Theorem for line integrals. 29.(x, y) = x + 3y; C: r(t) = 2 t, t, for 0 t 2Evaluating line integrals Evaluate the line integral Cdr for the following functions and oriented curves C in two ways. a.Use a parametric description of C to evaluate the integral directly. b.Use the Fundamental Theorem for line integrals. 31.(x, y, z) = (x2 + y2 + z2)/2; C: r(t) = cos t, sin t, t/, for 0 t 2Evaluating line integrals Evaluate the line integral Cdr for the following functions and oriented curves C in two ways. a.Use a parametric description of C to evaluate the integral directly. b.Use the Fundamental Theorem for line integrals. 32.(x, y, z) = xy + xz + yz; C: r(t) = t, 2t, 3t, for 0 t 4Applying the Fundamental Theorem of Line integrals Suppose the vector field F is continuous on 2, F = f, g = , (1, 2) = 7, (3, 6) = 10 and (6, 4) = 20. Evaluate the following integrals for the given curve C, if possible. 35. CFdr;C: r(t) = 2t 1, t2 + t, for 1 t 2Applying the Fundamental Theorem of Line integrals Suppose the vector field F is continuous on 2, F = f, g = , (1, 2) = 7, (3, 6) = 10 and (6, 4) = 20. Evaluate the following integrals for the given curve C, if possible. 36. CFTds;C is a smooth curve from (1, 2) to (6, 4).Applying the Fundamental Theorem of Line integrals Suppose the vector field F is continuous on 2, F = (f, g) = , (1, 2) =7, (3, 6) = 10, and (6, 4) = 20. Evaluate the following integral for the given curve C, if possible. 37. Cfdx+gdy;C is the path consisting of the line segment from A(6, 4) to B(1, 2) followed by the line segment from B(1, 2) to C(3, 6).Applying the Fundamental Theorem of Line integrals Suppose the vector field F is continuous on 2, F = f, g = , (1, 2) = 7, (3, 6) = 10 and (6, 4) = 20. Evaluate the following integrals for the given curve C, if possible. 38.CFdr;C is a circle, oriented clockwise, starting and ending at the point A(6, 4)Using the Fundamental Theorem for line integrals Verify that the Fundamental Theorem for line integrals can be used to evaluate the given integral, and then evaluate the integral. 39.C2x,2ydr,where C is a smooth curve from (0, 1) to (3, 4)Using the Fundamental Theorem for line integrals Verify that the Fundamental Theorem for line integrals can be used to evaluate the given integral, and then evaluate the integral. 40.C1,1,1dr,where C is a smooth curve from (1, 1, 2) to (3, 0, 7)Using the Fundamental Theorem for line integrals Verify that the Fundamental Theorem for line integrals can be used to evaluate the given integral, and then evaluate the integral. 41. C(excosy)dr,where C is the line segment from (0, 0) to (ln 2, 2)Using the Fundamental Theorem for line integrals verify that the Fundamental Theorem for line integrals can be used to evaluate the given integral, and then evaluate the integral. 42. C(1+x2yz)dr, where C is the helix r(t) = cos2t,sin2t,t, for 0 t 4Using the Fundamental Theorem for line integrals verify that the Fundamental Theorem for line integrals can be used to evaluate the given integral, and then evaluate the integral. 43. Ccos (2y z) dx 2x sin (2y z) dy + x sin (2y z) dz, where C is the curve r(t) = t2,t,t, for 0 tUsing the Fundamental Theorem for line integral verify that the fundamental Theorem for line integrals can be used to evaluate the given integral, and then evaluate the integral. 44. C exy dx + ex dy, where C is the parabola r(t) t+1,t2, for 1 t 3Line integrals of vector fields on closed curves Evaluate CFdr for the following vector fields and closed oriented curves C by parameterizing C. If the integral is not zero, give an explanation. 33.F = x, y; C is the circle of radius 4 centered at the origin oriented counterclockwise.Line integrals of vector fields on closed curves Evaluate CFdr for the following vector fields and closed oriented curves C by parameterizing C. If the integral is not zero, give an explanation. 34.F = y, x; C is the circle of radius 8 centered at the origin oriented counterclockwise.Line integrals of vector fields on closed curves Evaluate CFdr for the following vector fields and closed oriented curves C by parameterizing C. If the integral is not zero, give an explanation. 35.F = x, y; C is the triangle with vertices (0, 1) and (1, 0) oriented counterclockwise.Line integrals of vector fields on closed curves Evaluate CFdr for the following vector fields and closed oriented curves C by parameterizing C. If the integral is not zero, give an explanation. 36.F = y, x; C is the circle of radius 3 centered at the origin oriented counterclockwise.Line integrals of vector fields on closed curves Evaluate CFdr for the following vector fields and closed oriented curves C by parameterizing C. If the integral is not zero, give an explanation. 37.F = x, y, z; C: r(t) = cos t, sin t, 2, for 0 t 2Line integrals of vector fields on closed curves Evaluate CFdr for the following vector fields and closed oriented curves C by parameterizing C. If the integral is not zero, give an explanation. 38.F = y z, z x, xy; C: r(t) = cos t, sin t, cos t, for 0 t 2Evaluating line integral using level curves Suppose the vector field F, whose potential function is , is continuous on 2. Use the curves C1 and C2 and level curves of (see figure) to evaluate the following line integrals. 51.C1 F drEvaluating line integral using level curves Suppose the vector field F, whose potential function is , is continuous on 2. Use the curves C1 and C2 and level curves of (see figure) to evaluate the following line integrals. 52.C2 F drLine integrals Evaluate each line integral using a method of your choice. 43.CFdr, where F = 2xy + z2, x2, 2xz and C is the circle r(t) = 3 cos t, 4 cos t, 5 sin t, for 0 t 2Line integrals Evaluate each line integral using a method of your choice. 42.Cex(cosydx+sinydy), where C is the square with vertices (1, 1) oriented counterclockwiseLine integrals Evaluate the following line integrals using a method of your choice. 55. C (sin xy) dr, where C is the line segment from (0,0) to (2, /4)Line integrals Evaluate the following line integrals using a method of your choice. 56. C x3 dx + y3dy, where C is the curve r(t) 1+sint,cos2t, for 0 t /257E58EWork in force fields Find the work required to move an object in the following force fields along a line segment between the given points. Check to see whether the force is conservative. 45.F = x, 2 from A(0, 0) to B(2, 4)Work in force fields Find the work required to move an object in the following force fields along a line segment between the given points. Check to see whether the force is conservative. 46.F = x, y from A(1, 1) to B(3, 6)Work in force fields Find the work required to move an object in the following force fields along a line segment between the given points. Check to see whether the force is conservative. 47.F = x, y, z from A(1, 2, 1) to B(2, 4, 6)Work in force fields Find the work required to move an object in the following force fields along a line segment between the given points. Check to see whether the force is conservative. 48.F = ex+y 1, 1, z from A(0, 0, 0) to B(1, 2, 4)Suppose C is a circle centered at the origin in a vector field F (see figure). a. If C is oriented counterclockwise, isC F dr positive, negative, or zero? b. If C is oriented clockwise, isC F dr positive, negative, or zero? c. Is F conservative in 2? Explain.A vector field that is continuous in R2 is given (see figure). Is it conservative?
65EConservation of energy Suppose an object with mass m moves in a region R in a conservative force field given by F = , where is a potential function in a region R. The motion of the object is governed by Newtons Second Law of Motion, F = ma, where a is the acceleration. Suppose the object moves from point A to point B in R. a.Show that the equation of motion is mdvdt=. b.Show that dvdtv=12ddt(vv). c.Take the dot product of both sides of the equation in part (a) with v(t) = r(t) and integrate along a curve between A and B. Use part (b) and the fact that F is conservative to show that the total energy (kinetic plus potential) 12m|v|2+ is the same at A and B. Conclude that because A and B are arbitrary, energy is conserved in R.Gravitational potential The gravitational force between two point masses M and m is F=GMmr|r|3=GMmx,y,z(x2+y2+z2)3/2, where G is the gravitational constant. a.Verify that this force field is conservative on any region excluding the origin. b.Find a potential function for this force field such that F = . c.Suppose the object with mass m is moved from a point A to a point B, where A is a distance r1 from M and B is a distance r2, from M. Show that the work done in moving the object is GMm(1r21r1). d.Does the work depend on the path between A and B? Explain.Radial Fields in 3 are conservative Prove that the radial field F=r|r|p, where r = x, y, z and p is a real number, is conservative on any region not containing the origin. For what values of p is F conservative on a region that contains the origin?55.Rotation fields are usually not conservative a.Prove that the rotation field F=y,x|r|p, where r = x, y, is not conservative for p 2. b.For p = 2, show that F is conservative on any region not containing the origin. c.Find a potential function for F when p = 2.Linear and quadratic vector fields a.For what values of a, b, c and d is the field F = ax + by, cx + dy conservative? b.For what values of a, b, and c is the field F = ax2 by2, cxy conservative?71E72E73E74E75ECompute gxfy for the radial vector field F=x,y.What does this tell you about the circulation on a simple closed curve?Compute fxgy for the radial vector field F=y,x.What does this tell you about the outward flux of F across a simple closed curve?3QCExplain why Greens Theorem proves that if gx = fy, then the vector field F=f,g is conservative.Explain why the two forms of Greens Theorem are analogs of the Fundamental Theorem of Calculus.Referring to both forms of Greens Theorem, match each idea in Column 1 to an idea in Column 2: Line integral for flux Double integral of the curl Line integral for circulation Double integral of the divergence3EWhy does a two-dimensional vector field with zero curl on a region have zero circulation on a closed curve that bounds the region?Why does a two-dimensional vector field with zero divergence on a region have zero outward flux across a closed curve that bounds the region?Sketch a two-dimensional vector field that has zero curl everywhere in the plane.Sketch a two-dimensional vector field that has zero divergence everywhere in the plane.Discuss one of the parallels between a conservative vector field and a source-free vector field.Assume C is a circle centered at the origin, oriented counter clockwise, that encloses disk R in the plane. Complete the following steps for each vector field F. a. Calculate the two-dimensional curl of F. b. Calculate the two-dimensional divergence of F. c. Is F irrotational on R? d. Is F source free on R? 9. F=x,yAssume C is a circle centered at the origin, oriented counter clockwise, that encloses disk R in the plane. Complete the following steps for each vector field F. a. Calculate the two-dimensional curl of F. b. Calculate the two-dimensional divergence of F. c. Is F irrotational on R? d. Is F source free on R? 10. F=y,xAssume C is a circle centered at the origin, oriented counterclockwise, that encloses disk R in the plane. Complete the following steps for each vector field F. a. Calculate the two-dimensional curl of F. b. Calculate the two-dimensional divergence of F. c. is F irrotational on R? d. is F source free on R? 11. F=y,3x12EAssume C is a circle centered at the origin, oriented counter clockwise, that encloses disk R in the plane. Complete the following steps for each vector field F. a. Calculate the two-dimensional curl of F. b. Calculate the two-dimensional divergence of F. C. Is F irrotational on R? d. Is F source free on R? 13. F = 4x2y,xy2+x4Assume C is a circle centered at the origin, oriented counter clockwise, that encloses disk R in the plane. Complete the following steps for each vector field F. a. Calculate the two-dimensional curl of F. b. Calculate the two-dimensional divergence of F. c. Is F irrotational on R? d. Is F source free on R? 14. F = 4x3+y,12xySuppose C is the boundary of region R = {(x, y):x2 y 1},oriented counter clockwise (see figure); F = 1,x. a. Compute the two-dimensional curl of F and determine whether F is irrotational. b. Find parameterizations r1(t) and r2(t) for C1 and C2, respectively. c. Evaluate both e line integral and the double integral in the circulation form of Greens Theorem and check for consistency. d. Compute the two-dimensional divergence of F and use the flux form of Greens Theorem to explain why the outward flux is 0.Suppose C is the boundary of region R = {(x, y): 2x2 – 2x ≤ y ≤ 0}, oriented counterclockwise (see figure); let F = .
Compute the two-dimensional curl of F and use the circulation form of Green’s Theorem to explain why the circulation is 0.
Compute the two-dimensional divergence of F and determine whether F is source free.
Find parameterizations r1(t) and r2(t) for C1 and C2, respectively.
Evaluate both the line integral and the double integral in the flux form of Green’s Theorem and check for consistency.
Greens Theorem, circulation form Consider the following regions R and vector fields F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. 17. F = 2y,2x ; R is the region bounded by y = sin x and y = 0, for 0 xGreens Theorem, circulation form Consider the following regions R and vector fields F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. 18. F = 3y,3x ; R is the triangle with vertices (0, 0), (1, 0), and (0, 2).Greens Theorem, circulation form Consider the following regions R and vector fields F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. 19. F = 2xy,x2 ; R is the region bounded by y = x(2 x) and y=0.Greens Theorem, circulation form Consider the following regions R and vector fields F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. 20. F = 0,x2+y2; R = {(x, y) : x2 + y2 1}Area of regions Use a line integral on the boundary to find the area of the following regions. 17.A disk of radius 5Area of regions Use a line integral on the boundary to find the area of the following regions. 18.A region bounded by an ellipse with major and minor axes of length 12 and 8, respectively.Area of regions Use a line integral on the boundary to find the area of the following regions. 19.{(x, y): x2 + y2 16}Area of regions Use a line integral on the boundary to find the area of the following regions. 20.The region shown in the figureArea of regions Use a line integral on the boundary to find the area of the following regions. 21.The region bounded by the parabolas r(t) = t, 2t2 and r(t) = t, 12 t2, for 2 t 2Area of regions Use a line integral on the boundary to find the area of the following regions. 22.The region bounded by the curve r(t) = t(1 t2), 1 t2, for 1 t 1 (Hint: Plot the curve.)Greens Theorem, flux form Consider the following regions R and vector fields F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. 27. F = x,y ; R = {(x, y) : x2 + y2 4}Greens Theorem, flux form Consider the following regions R and vector fields F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. 28. F = x,3y ; R is the triangle with vertices (0, 0), (1, 2), and (0, 2).Greens Theorem, flux form Consider the following regions R and vector fields F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. 29. F = 2xy,x2 ; R = {(x, y) : 0 y x(2 x)}Greens theorem, flux form Consider the following regions R and vector field F. a. Compute the two dimensional divergence of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. 30. F = x2+y2,0; R = {(x, y) : x2 + y2 1}Line integrals Use Greens Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. 31. C3y+1,4x2+3dr, where C is the boundary of the rectangle with vertices (0, 0) (4, 0) (4, 2) and (0, 2)Line integrals Use Greens Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. 32. Csiny,xdr, where C is the boundary of the triangle with vertices (0, 0) (2,0), 0) and (2,2)Line integrals Use Greens Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. 33. Cxeydx+xdy, where C is the boundary of the region bounded by the curves y = x2, x = 2, and the x-axis3140. Line integrals Use Greens Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. C11+y2dx+ydy, where C is the boundary of the triangle with vertices (0, 0), (1, 0), and (1, 1)Line integrals Use Greens Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. 35. C(2x+ey2)dx(4y2+ex2)dx, where C is the boundary of the rectangle with vertices (0, 0) (1, 0) (1, 1) and (0, 1)Line integrals Use Greens Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. 36.C(2x3y)dy(3x+4y), where C is the unit of circleLine integrals Use Greens Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. 37. Cfdygdx where f,g=0,xy and C is the triangle with vertices (0, 0) (2, 0) and (0, 4)Line integrals Use Greens Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. 38.cfdygdx where f,g=x and C upper half of the unit circle and the line segment -1 x 1, oriented clockwiseLine integrals use Greens Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. 39. The circulation line integral of F = x2+y2,4x+y3 where C is boundary of {(x, y) : 0 y sin x, 0 x }Line integrals Use Greens Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. 40. The flux line integral of F = exy,eyx where C is boundary of {(x, y) : 0 y x, 0 x 1}General regions For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. 35.F = (x, y); R is the half-annulus {(r, ): 1 r 2, 0 }.General regions For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. 36.F = (y, x); R is the annulus {(r, ): 1 r 3, 0 2}.General regions For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. 37.F = (2x + y, x 4y); R is the quarter-annulus {(r, ): 1 r 4, 0 /2}.General regions For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. 38.F = (x y, x + 2y); R is the parallelogram {(x, y): 1 x y 3 x, 0 x 1}.Circulation and flux For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region, Assume boundary curves have counterclockwise orientation. 41.F=(x2+y2), where R is the half annulus {(r,):1r3,0}Circulation and flux for the following vector fields, compute (a) the circulation on, and (b) the outward flux across, the boundary of the given region. Assume boundary curves are oriented counterclockwise.
F = ; R is the eighth-annulus {(r, θ): 1 ≤ r ≤ 2, 0 ≤ θ ≤ π/4}.
Circulation and flux For the following vector fields, compute (a) the circulation on, and (b) the outward flux across, the boundary of the given region. Assume boundary curves are oriented counterclockwise. 47. F=x+y2,x2y;R={(x,y):y2x2y2}.Circulation and flux For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region, Assume boundary curves have counterclockwise orientation. 42.F=(ycosx,sinx), where R is the square {(x,y):0x/2,0y/2}Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.The work required to move an object around a closed curve C in the presence of a vector force field is the circulation of the force field on the curve. b.If a vector field has zero divergence throughout a region (on which the conditions of Greens Theorem are met), then the circulation on the boundary of that region is zero. c.If the two-dimensional curl of a vector field is positive throughout a region (on which the conditions of Greens Theorem are met), then the circulation on the boundary of that region is positive (assuming counterclockwise orientation).Special line integrals Prove the following identities, where C is a simple closed smooth oriented curve. 44.Cdx=Cdy=0Special line integrals Prove the following identities, where C is a simple closed smooth oriented curve. 45.Cf(x)dx+g(y)dy=0, where f and g have continuous derivatives on the region enclosed by C52EArea line integral Show that the value of Cxy2dx+(x2y+2x)dy depends only on the area of the region enclosed by C.Area line integral In terms of the parameters a and b, how is the value of Caydx+bxdy related to the area of the region enclosed by C, assuming counterclockwise orientation of C?Stream function Recall that if the vector field F = (f, g) is source free (zero divergence), then a stream function exists such that f = y and g = x. a.Verify that the given vector field has zero divergence. b.Integrate the relations f = y and g= x to find a stream function for the field. 49.F = (4, 2)Stream function Recall that if the vector field F = (f, g) is source free (zero divergence), then a stream function exists such that f = y and g = x. a.Verify that the given vector field has zero divergence. b.Integrate the relations f = y and g = x to find a stream function for the field. 50.F = (y2, x2)Stream function Recall that if the vector field F = (f, g) is source free (zero divergence), then a stream function exists such that f = y and g = x. a.Verify that the given vector field has zero divergence. b.Integrate the relations f = y and g = x to find a stream function for the field. 51.F = (ex sin y, ex cos y)Stream function Recall that if the vector field F = (f, g) is source free (zero divergence), then a stream function exists such that f = y and g = x. a.Verify that the given vector field has zero divergence. b.Integrate the relations f = y and g = x to find a stream function for the field. 52.F = (x2, 2xy)Applications 5356. Ideal flow A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region (excluding the origin if necessary). a.Verify that the curl and divergence of the given field is zero. b.Find a potential function and a stream function for the field. c.Verify that and satisfy Laplaces equation xx+yy=xx+yy=0. 53.F = (ex cos y, ex sin y)Applications 5356. Ideal flow A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region (excluding the origin if necessary). a.Verify that the curl and divergence of the given field is zero. b.Find a potential function and a stream function for the field. c.Verify that and satisfy Laplaces equation xx+yy=xx+yy=0. 54.F = (x3 3xy2, y3 3x2y)Applications 5356. Ideal flow A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region (excluding the origin if necessary). a.Verify that the curl and divergence of the given field is zero. b.Find a potential function and a stream function for the field. c.Verify that and satisfy Laplaces equation xx+yy=xx+yy=0. 55.F=tan1yx,12ln(x2+y2)Applications 5356. Ideal flow A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region (excluding the origin if necessary). a.Verify that the curl and divergence of the given field is zero. b.Find a potential function and a stream function for the field. c.Verify that and satisfy Laplaces equation xx+yy=xx+yy=0. 56.F=(x,y)x2+y263EGreens Theorem as a Fundamental Theorem of Calculus Show that if the circulation form of Greens Theorem is applied to the vector field 0,f(x)c and R={(x,y):axb,0yc}, then the result is the Fundamental Theorem of Calculus, abdfdxdx=f(b)f(a).Greens Theorem as a Fundamental Theorem of Calculus Show that if the flux form of Greens Theorem is applied to the vector field f(x)c,0 and R={(x,y):axb,0yc}, then the result is the Fundamental Theorem of Calculus, abdfdxdx=f(b)f(a).Whats wrong? Consider the rotation field F=(y,x)x2+y2. a.Verify that the two-dimensional curl of F is zero, which suggests that the double integral in the circulation form of Greens Theorem is zero. b.Use a line integral to verify that the circulation on the unit circle of the vector field is 2. c.Explain why the results of parts (a) and (b) do not agree.Whats wrong? Consider the radial field F=(x,y)x2+y2. a.Verify that the divergence of F is zero, which suggests that the double integral in the flux form of Greens Theorem is zero. b.Use a line integral to verify that the outward flux across the unit circle of the vector field is 2. c.Explain why the results of parts (a) and (b) do not agree.68EFlux integrals Assume the vector field F = (f, g) is source free (zero divergence) with stream function . Let C be any smooth simple curve from A to the distinct point B. Show that the flux integral CFnds is independent of path; that is, CFnds=(B)(A).Streamlines are tangent to the vector field Assume that the vector field F = (f, g) is related to the stream function by y = f and x = g on a region R Prove that at all points of R, the vector field is tangent to the streamlines (the level curves of the stream function).Streamlines and equipotential lines Assume that on 2, the vector field F = {f, g) has a potential function such that f = x and g = y, and it has a stream function such that f = y and g = x. Show that the equipotential curves (level curves of ) and the streamlines (level curves of ) are everywhere orthogonal.Channel flow The flow in a long shallow channel is modeled by the velocity field F = (0, 1 x2), where R = {(x, y): |x| 1 and |y| = 5}. a.Sketch R and several streamlines of F. b.Evaluate the curl of F on the lines x = 0, x = 14, x = 12, and x = 1. c.Compute the circulation on the boundary of R. d.How do you explain the fact that the curl of F is nonzero at points of R, but the circulation is zero?Show that is a vector field has the form F=f(y,z),g(x,z),h(x,y), then div F = 0.Verify the claim made in Example 3d by showing that the net outward flux of F across C is positive. (Hint: If you use Green’s Theorem to evaluate the integral , convert to polar coordinates.)
Show that is a vector field has the form F=f(x),g(y),h(z), then F = 0.Is (uF) a vector function or a scalar function?Explain how to compute the divergence of the vector field F = (f, g, h).Interpret the divergence of a vector field.What does it mean if the divergence of a vector field is zero throughout a region?Explain how to compute the curl of the vector field F = (f, g, h).Interpret the curl of a general rotation vector field.What does it mean if the curl of a vector field is zero throughout a region?What is the value of ( F)?What is the value of u?Divergence of vector fields Find the divergence of the following vector fields. 9.F = (2x, 4y, 3z)Divergence of vector fields Find the divergence of the following vector fields. 10.F = (2y, 3x, z)Divergence of vector fields Find the divergence of the following vector fields. 11.F = (12x, 6y, 6z)Divergence of vector fields Find the divergence of the following vector fields. 12.F = (x2yz, xy2z, xyz2)Divergence of vector fields Find the divergence of the following vector fields. 13.F = (x2 y2, y2 z2, z2 x2)Divergence of vector fields Find the divergence of the following vector fields. 14.F = (ex + y, ey + z, ez + x)Divergence of vector fields Find the divergence of the following vector fields. 15.F=(x,y,z)1+x2+y2Divergence of vector fields Find the divergence of the following vector fields. 16.F = (yz sin x, xz cos y, xy cos z)Divergence of radial fields Calculate the divergence of the following radial fields. Express the result in terms of the position vector r and its length |r|. Check for agreement with Theorem 17.10 17. F=x,y,zx2+y2+z2=r|r|2Divergence of radial fields Calculate the divergence of the following radial fields. Express the result in terms of the position vector r and its length |r|. Check for agreement with Theorem 17.10 18. F=x,y,z(x2+y2+z2)3/2=r|r|3Divergence of radial fields Calculate the divergence of the following radial fields. Express the result in terms of the position vector r and its length |r|. Check for agreement with Theorem 17.10 19. F=x,y,z(x2+y2+z2)2=r|r|4Divergence of radial fields Calculate the divergence of the following radial fields. Express the result in terms of the position vector r and its length |r|. Check for agreement with Theorem 17.10 20. F=x,y,z(x2+y2+z2)=r|r|2Divergence and flux from graphs Consider the following vector fields, the circle C, and two points P and Q. a.Without computing the divergence, does the graph suggest that the divergence is positive or negative at P and Q? Justify your answer. b.Compute the divergence and confirm your conjecture in part (a). c.On what part of C is the flux outward? Inward? d.Is the net outward flux across C positive or negative? 21.F = (x, x + y)Divergence and flux from graphs Consider the following vector fields, the circle C, and two points P and Q. a.Without computing the divergence, does the graph suggest that the divergence is positive or negative at P and Q? Justify your answer. b.Compute the divergence and confirm your conjecture in part (a). c.On what part of C is the flux outward? Inward? d.Is the net outward flux across C positive or negative? 22.F = (x, y2)Curl of a rotational field Consider the following vector fields, where r = (x, y, z). a.Compute the curl of the field and verify that it has the same direction as the axis of rotation. b.Compute the magnitude of the curt of the field. 23.F = (1, 0, 0) rCurl of a rotational field Consider the following vector fields, where r = (x, y, z). a.Compute the curl of the field and verify that it has the same direction as the axis of rotation. b.Compute the magnitude of the curt of the field. 24.F = (1, 1, 0) rCurl of a rotational field Consider the following vector fields, where r = (x, y, z). a.Compute the curl of the field and verify that it has the same direction as the axis of rotation. b.Compute the magnitude of the curt of the field. 25.F = (1, 1, 1) rCurl of a rotational field Consider the following vector fields, where r = (x, y, z). a.Compute the curl of the field and verify that it has the same direction as the axis of rotation. b.Compute the magnitude of the curt of the field. 26.F = (1, 2, 3) rCurl of a vector field Compute the curl of the following vector fields. 27.F = (x2 y2, xy, z)Curl of a vector field Compute the curl of the following vector fields. 28.F = (0, z2 y2, yz)Curl of a vector field Compute the curl of the following vector fields. 29.F = (x2 z2, 1, 2xz)Curl of a vector field Compute the curl of the following vector fields. 30.F = r = (x, y, z)Curl of a vector field Compute the curl of the following vector fields. 31.F=x,y,z(x2+y2+z2)3/2=r|r|3Curl of a vector field Compute the curl of the following vector fields. 32.F=x,y,z(x2+y2+z2)1/2=r|r|Curl of a vector field Compute the curl of the following vector fields. 33.F = (z2 sin y, xz2 cos y, 2xz sin y)Curl of a vector field Compute the curl of the following vector fields. 34.F=(3xz3ey2,2xz3ey2,3xz2ey2)Derivative rules Prove the following identities. Use Theorem 17.13 (Product Rule) whenever possible. 35.(1|r|3)=3r|r|5 (used in Example 5) Example 5 more properties of radical field Letr=x,y,zandlet=1|r|=(x2+y2+z2)1/2 be a potential function. a. Find the associate gradient field F=(1|r|). b. Compute F.Derivative rules Prove the following identities. Use Theorem 17.13(Product Rule) whenever possible. 36. (1|r|2)=2r|r|4Derivative rules Prove the following identities. Use Theorem 17.13 (Product Rule) whenever possible. 37. (1|r|2)=2|r|4 (Hint: Use Exercise 36.)Derivative rules Prove the following identities. Use Theorem 17.13 (Product Rule) whenever possible. 38. (ln|r|)=r|r|2Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.For a function f of a single variable, if f(x) = 0 for all x in the domain, then f is a constant function. If F = 0 for all points in the domain, then F is constant. b.If F = 0, then F is constant. c.A vector field consisting of parallel vectors has zero curl. d.A vector field consisting of parallel vectors has zero divergence. e.curl F is orthogonal to F.Another derivative combination Let F = (f, g, h) and let u be a differentiable scalar-valued function. a.Take the dot product of F and the del operator; then apply the result to u to show that (F)u=(fx+gy+hz)u=fux+guy+huz. b.Evaluate (F)(xy2z3) at (1, 1, 1), where F = (1, 1, 1).Does it make sense? Are the following expressions defined? If so, state whether the result is a scalar or a vector. Assume F is a sufficiently differentiable vector field and is a sufficiently differentiable scalar-valued function. a. b.F c. d. () e.( ) f. (F) g. h. (F) i. ( F)Zero divergence of the rotation field Show that the general rotation field F = a r, where a is a nonzero constant vector and r = (x, y, z), has zero divergence.General rotation fields a.Let a = (0, 1, 0), r = (x, y, z), and consider the rotation field F = a r. Use the right-hand rule for cross products to find the direction of F at the points (0, 1, 1), (1, 1, 0), (0, 1, 1), and (1, 1, 0). b.With a = (0, 1, 0), explain why the rotation field F = a r circles the y-axis in the counterclockwise direction looking along a from head to tail (that is, in the negative y-direction).44ECurl of the rotation field For the general rotation field F = a r, where a is a nonzero constant vector and r = (x, y, z), show that curl F = 2a.Inward to outward Find the exact points on the circle x2 + y2 = 2 at which the field F = (f, g) = (x2, y) switches from pointing inward to outward on the circle, or vice versa.Maximum divergence Within the cube {(x, y, z): |x| 1, |y| 1, |z| 1}, where does div F have the greatest magnitude when F = (x2 y, xy2z, 2xz)?Maximum curl Let F =
Find the scalar component of curl F in the direction of the unit vector n =
Find the scalar component of curl F in the direction of the unit vector n =
Find the unit vector n that maximizes scaln and state the value of scaln in the direction.
Zero component of the curl For what vectors n is (curl F)n = 0 when F = (y, 2z, x)?50EFind a vector Field Find a vector field F with the given curl. In each case, is the vector field you found unique? 51.curl F = (0, z, y)52EPaddle wheel in a vector field Let F = z, 0, 0 and let n be a unit vector aligned with the axis of a paddle wheel located on the x-axis (see figure). a.If the paddle wheel is oriented with n = (1, 0, 0), in what direction (if any) does the wheel spin? b.If the paddle wheel is oriented with n = (0, 1, 0), in what direction (if any) does the wheel spin? c.If the paddle wheel is oriented with n = (0, 0, 1), in what direction (if any) does the wheel spin?Angular speed Consider the rotational velocity field v = 2y, 2z, 0. a.If a paddle wheel is placed in the xy-plane with its axis normal to this plane, what is its angular speed? b.If a paddle wheel is placed in the xz-plane with its axis normal to this plane, what is its angular speed? c.If a paddle wheel is placed in the yz-plane with its axis normal to this plane, what is its angular speed?Angular speed Consider the rotational velocity field v = 0, 10z, 10y. If a paddle wheel is placed in the plane x + y + z = 1 with its axis normal to this plane, how fast does the paddle wheel spin (revolutions per unit time)?Heat flux Suppose a solid object in 3 has a temperature distribution given by T(x, y, z). The heat flow vector field in the object is F = kT, where the conductivity k 0 is a property of the material. Note that the heat flow vector points in the direction opposite that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is F = k T = k2T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions. 56.T(x,y,z)=100ex2+y2+z2Heat flux Suppose a solid object in 3 has a temperature distribution given by T(x, y, z). The heat flow vector field in the object is F = kT, where the conductivity k 0 is a property of the material. Note that the heat flow vector points in the direction opposite that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is F = k T = k2T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions. 57.T(x,y,z)=100ex2+y2+z258EGravitational potential The potential function for the gravitational force field due to a mass M at the origin acting on a mass m is =GMm/|r|, where r = x, y, z is the position vector of the mass m and G is the gravitational constant. a.Compute the gravitational force field F = b.Show that the field is irrotational; that is. F = 0.Electric potential The potential function for the force field due to a charge q at the origin is =140q|r|, where r = x, y, z is the position vector of a point in the field and 0 is the permittivity of free space. a.Compute the force field F = b.Show that the field is irrotational; that is F = 0.Navier-Stokes equation The Navier-Stokes equation is the fundamental equation of fluid dynamics that models the flow in everything from bathtubs to oceans. In one of its many forms (incompressible, viscous flow), the equation is (Vt+(V)V)=p+()V. In this notation, V = (u, v, w) is the three-dimensional velocity field, p is the (scalar) pressure, is the constant density of the fluid, and is the constant viscosity. Write out the three component equations of this vector equation. (See Exercise 40 for an interpretation of the operations.)Stream function and vorticity The rotation of a three-dimensional velocity field V = u, v, w is measured by the vorticity = V. If = 0 at all points in the domain, the flow is irrotational. a.Which of the following velocity fields is irrotational: V = 2, 3y, 5z or V = y, x z, y? b.Recall that for a two-dimensional source-free flow V = (u, v, 0), a stream function (x, y) may be defined such that u = y and v = r. For such a two-dimensional flow, let = k V be the k-component of the vorticity. Show that 2 = = . c.Consider the stream function (x, y) = sin x sin y on the square region R = {(x, y): 0 x , 0 y }. Find the velocity components u and v; then sketch the velocity field. d.For the stream function in part (c), find the vorticity function as defined in part (b). Plot several level curves of the vorticity function. Where on R is it a maximum? A minimum?Amperes Law One of Maxwells equations for electromagnetic waves is B=CEt, where E is the electric field, B is the magnetic field, and C is a constant. a.Show that the fields E(z,t)=Asin(kzt)iB(z,t)=Asin(kzt)j satisfy the equation for constants A. k, and , provided =k/C. b.Make a rough sketch showing the directions of E and B64EProperties of div and curl Prove the following properties of the divergence and curl. Assume F and G are differentiable vector fields and c is a real number. a.(F+G)=F+G b.(F+G)=(F)+(G) c.(cF)=c(F) d.(cF)=c(F)