Solved: 4950. Flux Consider the vector fields and curves

How does a line integral differ from the single-variable integral 1 b a f 1x2 dx?

How do you evaluate the line integral 1C f ds, where C is parameterized by a parameter other than arc length?

If a curve C is given by r1t2 = 8t, t29, what is 0 r_1t2 0 ?

Given a vector field F and a parameterized curve C, explain how to evaluate the line integral 1C F # T ds.

How can 1C F # T ds be written in the alternative form 1 b a 1 f 1t2x_1t2 + g1t2y_1t2 + h1t2z_1t22 dt?

Given a vector field F and a closed smooth oriented curve C, what is the meaning of the circulation of F on C?

How is the circulation of a vector field on a closed smooth oriented curve calculated?

Given a two-dimensional vector field F and a smooth oriented curve C, what is the meaning of the flux of F across C?

How do you calculate the flux of a two-dimensional vector field across a smooth oriented curve C?

Sketch the oriented quarter circle from 11, 02 to 10, 12 and supply a parameterization for the curve. Draw the unit normal vector at several points on the curve.

1114. Scalar line integrals with arc length as parameter Evaluate the following line integrals. L C xy ds; C is the unit circle r1s2 = 8cos s, sin s9, for 0 s 2p.

1114. Scalar line integrals with arc length as parameter Evaluate the following line integrals. L C 1x + y2 ds; C is the circle of radius 1 centered at 10, 02.

1114. Scalar line integrals with arc length as parameter Evaluate the following line integrals. L C 1x2 - 2y22 ds; C is the line r1s2 = 8s>12, s>129, for 0 s 4.

1114. Scalar line integrals with arc length as parameter Evaluate the following line integrals. L C x2y ds; C is the line r1s2 = 8s>12, 1 - s>129, for 0 s 4.

1520. Scalar line integrals in the plane a. Find a parametric description for C in the form r1t2 = 8x1t2, y1t29, if it is not given. b. Evaluate  r1t2 . c. Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. L C 1x2 + y22 ds; C is the circle of radius 4 centered at 10, 02.

1520. Scalar line integrals in the plane a. Find a parametric description for C in the form r1t2 = 8x1t2, y1t29, if it is not given. b. Evaluate  r1t2 . c. Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. L C 1x2 + y22 ds; C is the line segment from 10, 02 to 15, 52.

1520. Scalar line integrals in the plane a. Find a parametric description for C in the form r1t2 = 8x1t2, y1t29, if it is not given. b. Evaluate  r1t2 . c. Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. L C x x2 + y2 ds; C is the line segment from 11, 12 to 110, 102.

1520. Scalar line integrals in the plane a. Find a parametric description for C in the form r1t2 = 8x1t2, y1t29, if it is not given. b. Evaluate  r1t2 . c. Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. L C 1xy21>3 ds; C is the curve y = x2, for 0 x 1.

1520. Scalar line integrals in the plane a. Find a parametric description for C in the form r1t2 = 8x1t2, y1t29, if it is not given. b. Evaluate  r1t2 . c. Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. L C xy ds; C is the portion of the ellipse x2 4 + y2 16 = 1 in the first quadrant, oriented counterclockwise.

1520. Scalar line integrals in the plane a. Find a parametric description for C in the form r1t2 = 8x1t2, y1t29, if it is not given. b. Evaluate  r1t2 . c. Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. L C 12x - 3y2 ds; C is the line segment from 1-1, 02 to 10, 12 followed by the line segment from 10, 12 to 11, 02.

2124. Average values Find the average value of the following functions on the given curves. f 1x, y2 = x + 2y on the line segment from 11, 12 to 12, 52

2124. Average values Find the average value of the following functions on the given curves. f 1x, y2 = x2 + 4y2 on the circle of radius 9 centered at the origin

2124. Average values Find the average value of the following functions on the given curves. f 1x, y2 = 24 + 9y2>3 on the curve y = x3>2, for 0 x 5

2124. Average values Find the average value of the following functions on the given curves. f 1x, y2 = xey on the unit circle centered at the origin

2530. Scalar line integrals in 3 Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. L C 1x + y + z2 ds; C is the circle r1t2 = 82 cos t, 0, 2 sin t9, for 0 t 2p.

2530. Scalar line integrals in 3 Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. L C 1x - y + 2z2 ds; C is the circle r1t2 = 81, 3 cos t, 3 sin t9, for 0 t 2p.

2530. Scalar line integrals in 3 Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. L C xyz ds; C is the line segment from 10, 0, 02 to 11, 2, 32.

2530. Scalar line integrals in 3 Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. L C xy z ds; C is the line segment from 11, 4, 12 to 13, 6, 32.

2530. Scalar line integrals in 3 Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. L C 1y - z2 ds; C is the helix r1t2 = 83 cos t, 3 sin t, t9, for 0 t 2p.

2530. Scalar line integrals in 3 Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. L C xeyz ds; C is r1t2 = 8t, 2t, -4t9, for 1 t 2.

3132. Length of curves Use a scalar line integral to find the length of the following curves. r1t2 = h 20 sin t 4 , 20 cos t 4 , t 2 i, for 0 t 2

3132. Length of curves Use a scalar line integral to find the length of the following curves. r1t2 = 830 sin t, 40 sin t, 50 cos t9, for 0 t 2p

3338. Line integrals of vector fields in the plane Given the following vector fields and oriented curves C, evaluate 1C F # T ds. F = 8x, y9 on the parabola r1t2 = 84t, t29, for 0 t 1

3338. Line integrals of vector fields in the plane Given the following vector fields and oriented curves C, evaluate 1C F # T ds. F = 8 -y, x9 on the semicircle r1t2 = 84 cos t, 4 sin t9, for 0 t p

3338. Line integrals of vector fields in the plane Given the following vector fields and oriented curves C, evaluate 1C F # T ds. F = 8y, x9 on the line segment from 11, 12 to 15, 102

3338. Line integrals of vector fields in the plane Given the following vector fields and oriented curves C, evaluate 1C F # T ds. F = 8 -y, x9 on the parabola y = x2 from 10, 02 to 11, 12

3338. Line integrals of vector fields in the plane Given the following vector fields and oriented curves C, evaluate 1C F # T ds. F = 8x, y9 1x2 + y223>2 on the curve r1t2 = 8t2, 3t29, for 1 t 2

3338. Line integrals of vector fields in the plane Given the following vector fields and oriented curves C, evaluate 1C F # T ds. F = 8x, y9 x2 + y2 on the line r1t2 = 8t, 4t9, for 1 t 10

3942. Work integrals Given the force field F, find the work required to move an object on the given oriented curve. F = 8y, -x9 on the path consisting of the line segment from 11, 22 to 10, 02 followed by the line segment from 10, 02 to 10, 42

3942. Work integrals Given the force field F, find the work required to move an object on the given oriented curve. F = 8x, y9 on the path consisting of the line segment from 1-1, 02 to 10, 82 followed by the line segment from 10, 82 to 12, 82

3942. Work integrals Given the force field F, find the work required to move an object on the given oriented curve. F = 8y, x9 on the parabola y = 2x2 from 10, 02 to 12, 82

3942. Work integrals Given the force field F, find the work required to move an object on the given oriented curve. F = 8y, -x9 on the line y = 10 - 2x from 11, 82 to 13, 42

4346. Work integrals in 3 Given the force field F, find the work required to move an object on the given oriented curve. F = 8x, y, z9 on the tilted ellipse r1t2 = 84 cos t, 4 sin t, 4 cos t9, for 0 t 2p

4346. Work integrals in 3 Given the force field F, find the work required to move an object on the given oriented curve. F = 8 -y, x, z9 on the helix r1t2 = 82 cos t, 2 sin t, t>2p9, for 0 t 2p

4346. Work integrals in 3 Given the force field F, find the work required to move an object on the given oriented curve. F = 8x, y, z9 1x2 + y2 + z223>2 on the line segment from 11, 1, 12 to 110, 10, 102

4346. Work integrals in 3 Given the force field F, find the work required to move an object on the given oriented curve. F = 8x, y, z9 x2 + y2 + z2 on the line segment from 11, 1, 12 to 18, 4, 22

4748. 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. F = 8y - x, x9; C: r1t2 = 82 cos t, 2 sin t9, for 0 t 2p

4748. 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. F = 8x, y9 1x2 + y221>2 ; C is the boundary of the square with vertices 1{2, {22, traversed counterclockwise.

4950. Flux Consider the vector fields and curves in Exercises 4748. 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. 49. F and C given in Exercise 47

4950. Flux Consider the vector fields and curves in Exercises 4748. 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. 49. F and C given in Exercise 48

Explain 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 r1t2 = 8x1t2, y1t2, z1t29, where t is the arc length, then _ r_1t2 _ = 1. b. The vector field F = 8y, x9 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 = 8150, 09. On which path between the points 1100, 502 and 1-100, 502 is the most work done overcoming the wind? a. The straight line r1t2 = 8x1t2, z1t29 = 8 -t, 509, for -100 t 100 b. The arc of a circle r1t2 = 8100 cos t, 50 + 100 sin t9, for 0 t p

Flying into a headwind a. How does the result of Exercise 52 change if the force due to the wind is F = 8141, 509 (approximately the same magnitude, but different direction)? b. How does the result of Exercise 52 change if the force due to the wind is F = 8141, -509 (approximately the same magnitude, but different direction)?

Changing orientation Let f 1x, y2 = x + 2y and let C be the unit circle. a. Find a parameterization of C with counterclockwise orientation and evaluate 1C f ds. b. Find a parameterization of C with clockwise orientation and evaluate 1C f ds. c. Compare the results of (a) and (b).

Changing orientation Let f 1x, y2 = x and let C be the segment of the parabola y = x2 joining O10, 02 and P11, 12. a. Find a parameterization of C in the direction from O to P. Evaluate 1C f ds. b. Find a parameterization of C in the direction from P to O. Evaluate 1C f ds. c. Compare the results of (a) and (b).

For what values of b and c does the vector field F = 8by, cx9 have zero circulation on the unit circle centered at the origin and oriented counterclockwise?

Consider the vector field F = 8ax + by, cx + dy9. Show that F has zero circulation on any oriented circle centered at the origin, for any a, b, c, and d, provided b = c.

For what values of a and d does the vector field F = 8ax, dy9 have zero flux across the unit circle centered at the origin and oriented counterclockwise?

Consider the vector field F = 8ax + by, cx + dy9. Show that F has zero flux across any oriented circle centered at the origin, for any a, b, c, and d, provided a = -d.

Work in a rotation field Consider the rotation field F = 8 -y, x9 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 1C F # T ds is independent of the path, where C is any path from 11, 02 to 10, 12?

Work in a hyperbolic field Consider the hyperbolic force field F = 8y, x9 (the streamlines are hyperbolas) and the three paths shown in the figure for Exercise 60. Compute the work done in the presence of F on each of the three paths. Does it appear that the line integral 1C F # T ds is independent of the path, where C is any path from 11, 02 to 10, 12?

6263. Mass and density A thin wire represented by the smooth curve C with a density r (units of mass per length) has a mass M = 1C r ds. Find the mass of the following wires with the given density. C: r1u2 = 8cos u, sin u9, for 0 u p; r1u2 = 2u>p + 1

6263. Mass and density A thin wire represented by the smooth curve C with a density r (units of mass per length) has a mass M = 1C r ds. Find the mass of the following wires with the given density. C: 51x, y2: y = 2x2, 0 x 36; r1x, y2 = 1 + xy

Heat flux in a plate A square plate R = 51x, y2: 0 x 1, 0 y 16 has a temperature distribution T1x, y2 = 100 - 50x - 25y. a. Sketch two level curves of the temperature in the plate. b. Find the gradient of the temperature _ T1x, y2. c. Assume that the flow of heat is given by the vector field F = -_ T1x, y2. Compute F. d. Find the outward heat flux across the boundary 51x, y2: x = 1, 0 y 16. e. Find the outward heat flux across the boundary 51x, y2: 0 x 1, y = 16.

Inverse force fields Consider the radial field F = r _ r _p = 8x, y, z9 _ r _p , where p 7 1 (the inverse square law corresponds to p = 3). Let C be the line from 11, 1, 12 to 1a, a, a2, where a 7 1, given by r1t2 = 8t, t, t9, for 1 t a. a. Find the work done in moving an object along C with p = 2. b. If aS _ in part (a), is the work finite? c. Find the work done in moving an object moving along C with p = 4. d. If aS _ in part (c), is the work finite? e. Find the work done in moving an object moving along C for any p 7 1. f. If aS _ in part (e), for what values of p is the work finite?

Flux across curves in a vector field Consider the vector field F = 8y, x9 shown in the figure. a. Compute the outward flux across the quarter circle C: r1t2 = 82 cos t, 2 sin t9, for 0 t p>2. b. Compute the outward flux across the quarter circle C: r1t2 = 82 cos t, 2 sin t9, for p>2 t p. 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?

Let R be the rectangle with vertices 10, 02, 1a, 02, 10, b2, and 1a, b2, and let C be the boundary of R oriented counterclockwise. Use the formula A = 1C x dy to verify that the area of the rectangle is ab.

Let R = 51r, u2: 0 r a, 0 u 2p6 be the disk of radius a centered at the origin and let C be the boundary of R oriented counterclockwise. Use the formula A = - 1C y dx to verify that the area of the disk is pr2.