1. Let C be the line segment from (0,0) to (0,1). In each part, evaluate the line integral along C by inspection, and explain your reasoning.

Transcription

1 MA112: Prepared b Dr. Archara Pacheenburawana 1 Eercise hapter 7 Eercise onfirm that φ is a potential function for F(r) on some region. 1. (a) φ(,) = tan 1 F(,) = 1+ 2 i j 2 (b) φ(,,z) = z 2 F(,,z) = 2i 6j+8zk 2. (a) φ(,) = F(,) = (6 3 )i+( )j (b) φ(,,z) = sinz +sin+zsin F(,,z) = (sinz +cos)i+(sin+zcos)j+(sin +cosz)k 3 8 Find divf and curlf. 3. F(,,z) = 2 i 2j+zk 4. F(,,z) = z 3 i j+5z 2 k 5. F(,,z) = 7 3 z 2 i 8 2 z 5 j 3 4 k 6. F(,,z) = e i cosj+sin 2 zk 7. F(,,z) = z 2(i+j+zk) 8. F(,,z) = lni+e z j+tan 1 (z/)k 9 10 Find (F G). 9. F(,,z) = 2i+j+4k G(,,z) = i+j zk 10. F(,,z) = zi+zj+k G(,,z) = j+zk Find ( F). 11. F(,,z) = sini+cos( )j+zk 12. F(,,z) = e z i+3e j e z k Find ( F). 13. F(,,z) = j+zk 14. F(,,z) = 2 i 3zj+k

2 MA112: Prepared b Dr. Archara Pacheenburawana 2 Eercise Let be the line segment from (0,0) to (0,1). In each part, evaluate the line integral along b inspection, and eplain our reasoning. (a) ds (b) sind 2. Let be the line segment from (0,2) to (0,4). In each part, evaluate the line integral along b inspection, and eplain our reasoning. (a) ds (b) e d 3. Let be the curve represented b the equations = 2t, = 3t 2 (0 t 1) In each part, evaluate the line integral along. (a) ( )ds (b) ( )d (c) ( )d 4. Let be the curve represented b the equations = t, = 3t 2, z = 6t 3 (0 t 1) In each part, evaluate the line integral along. (a) z 2 ds (b) z 2 d (c) z 2 d (d) z 2 dz 5. In each part, evaluate the integral (3+2)d+(2 )d along the stated curve. (a) The line segment from (0,0) to (1,1). (b) The parabolic arc = 2 from (0,0) to (1,1). (c) The curve = sin(π/2) from (0,0) to (1,1). (d) The curve = 3 from (0,0) to (1,1). 6. In each part, evaluate the integral d+zd dz along the stated curve.

3 MA112: Prepared b Dr. Archara Pacheenburawana 3 7. (a) The line segment from (0,0,0) to (1,1,1). (b) The twisted cubic = t, = t 2, z = t 3 from (0,0,0) to (1,1,1). (c) The heli = cospit, = sinπt, z = t from (1,0,0) to ( 1,0,1) Evaluate the line integral with respect to s along the curve ds : r(t) = ti+ 2 3 t3/2 j (0 t 3) ds 2 : = 1+2t, = t (0 t 1) zds : = t, = t 2, z = 2 3 t3 (0 t 1) e z ds : r(t) = 2costi+2sintj+tk (0 t 2π) Evaluate the line integral along the curve. (+2)d+( )d : = 2cost, = 4sint (0 t π/4) 12. ( 2 2 )d+d : = t 2/3, = t ( 1 t 1) 13. d+d : 2 = 3 from (3,3) to (0,0). 14. ( )d+ 2 d : 2 = 3 from (1, 1) to (1,1). 15. ( )d d : = 1, counterclockwise from (1,0) to (0,1).

4 MA112: Prepared b Dr. Archara Pacheenburawana ( )d+d : the line segment from (3,4) to (2,1). zd zd +dz : = e t, = e 3t, z = e t (0 t 1) 2 d+d +z 2 dz : = sint, = cost, z = t 2 (0 t π/2) Evaluate d d along the curve shown in the figure. 19. (a) (0,1) (1,0) (b) (1,1) (1,0) 20. (a) (1,1) (2,0)

5 MA112: Prepared b Dr. Archara Pacheenburawana 5 (b) (0,5) ( 5, 0) (5,0) Evaluate F dr along the curve. 21. F(,) = 2 i+j : r(t) = 2costi+2sintj (0 t π) 22. F(,) = 2 i+4j : r(t) = e t i+e t j (0 t 1) 23. F(,) = ( ) 3/2 (i+j) : r(t) = e t sinti+e t costj (0 t 1) 24. F(,,z) = zi+j+k : r(t) = sinti+3sintj+sin 2 tk (0 t π/2) Find the work done b the force field F on a particle that moves along the curve. 25. F(,) = i+ 2 j : = 2 from (0,0) to (1,1). 26. F(,) = ( 2 +)i+( 2 )j : = t, = 1/t (1 t 3) 27. F(,,z) = i+zj+zk : r(t) = ti+t 2 j+t 3 k (0 t 1) 28. F(,,z) = (+)i+j z 2 k : along line segments from (0,0,0) to (1,3,1) to (2, 1,4) Find the work done b the force field 1 F(,) = 2 + i j 2 on a particle that moves along the curve show in the figure.

6 MA112: Prepared b Dr. Archara Pacheenburawana (0,4) (4,0) 29. (6,3) Eercise Determine whether F is a conservative vector field. If so, find a potential function for it. 1. F(,) = i+j 2. F(,) = 3 2 i+6j 3. F(,) = 2 i+5 2 j 4. F(,) = e cosi e sinj 5. F(,) = lni+lnj 6. F(,) = (cos +cos)i+(sin sin)j 7. (a) Show that the line integral 2 d+2d is independent of the path. (b) Evaluate the integral in part (a) along the line segment from ( 1,2) to (1,3). (c) Evaluate the integral (1,3) ( 1,2) 2 d+2d using Theorem16.1, and confirm that the value is the same as that obtained in part (b). 8. (a) Show that the line integral sind cosd is independent of the path. (b) Evaluate the integral in part (a) along the line segment from (0,1) to (π, 1). (c) Evaluate the integral (π, 1) (0,1) sind cosd using Theorem16.1, and confirm that the value is the same as that obtained in part (b) Show that the line integral is independent of the path, and use Theorem16.1 to find its value.

7 MA112: Prepared b Dr. Archara Pacheenburawana (4,0) (1,2) (3,2) (0,0) 3d+3d 10. 2e d+ 2 e d 12. (1,π/2) (0,0) ( 1,0) (2, 2) e sind+e cosd 2 3 d d (0,1) ( 1,2) (3,3) (1,1) (3 +1)d (+4 +2)d ) ( ) (e ln e e d+ e ln d, where and are positive onfirm that the force field F is conservative in some open connected region containing the points P and Q, and then find the work done b the force field on a particle moving along an arbitrar smooth curve in the region from P to Q. 15. F(,) = 2 i+ 2 j; P(1,1), Q(0,0) 16. F(,) = 2 3 i j; P( 3,0), Q(4,1) 17. F(,) = e i+e j; P( 1,1), Q(2,0) 18. F(,) = e cosi e sinj; P(π/2,1), Q( π/2,0) Find the eact value of F dr using an method. 19. F(,) = (e +e )i+(e +e )j : r(t) = sin(πt/2)i+lntj (1 t 2) 20. F(,) = 2i+( 2 +cos)j : r(t) = ti+tcos(t/3)j (0 t π) Eercise Evaluate the line integral using Green s Theorem and check the answer b evaluating it directl. 2 d + 2 d, where is the square with vertices (0,0), (1,0), (1,1), and (0,1) oriented counterclockwise. 2. d+d, where is the unit circle oriented counterclockwise Use Green s Theorem to evaluate the integral. In each eercise, assume that the curve is oriented counterclockwise.

8 MA112: Prepared b Dr. Archara Pacheenburawana d+2d, where is the rectangle bounded b = 2, = 4, = 1, and = 2. ( 2 2 )d+d, where is the circle = 9. cosd sind,where isthesquarewithvertices(0,0), (π/2,0),(π/2,π/2), and (0, π/2). tan 2 d+tand, where is the circle 2 +( +1) 2 = 1. ( 2 )d+d, where is the circle = 4. (e + 2 )d+(e + 2 )d, where is the boundar of the region between = 2 and =. ln(1+)d d, where is the trianglewith vertices (0,0), (2,0), and (0,4) d 2 d, where is the boundar of the region in the first quadrant, enclosed between the coordinate aes and the circle = 16. tan 1 d 2 d, where is the square with vertices (0,0), (1,0), (1,1), and 1+2 (0,1). cossind+sincosd, where is the triangle with vertices (0,0), (3,3), and (0,3). 2 d+( + 2 )d, where is the boundar of the region enclosed b = 2 and = Use a line integral to find the area enclosed b the astroid = acos 3 φ, = asin 3 φ (0 φ 2π) 15. Use a line integral to find the area of the triangle with vertices (0,0), (a,0), and (0,b), where a > 0 and b > Use Green s Theorem to find the work done b the force field F on a particle that moves along the stated path.

9 MA112: Prepared b Dr. Archara Pacheenburawana F(,) = i+ ( ) j; theparticlestartsat(5,0), traversestheuppersemicircle = 25, and returns to its starting point along the -ais. 17. F(,) = i+ j; theparticlemoves counterclockwise onetimearoundtheclosed curve given b the equations = 0, = 2, and = 3 /4. Etra Problems 1. onsider the vector field F = 2 i+zj+z 3 k. (a) ompute div F. (b) ompute curl F. 2. Find the divergence and the curl of the vector field F(,,z) = zcosi+sinj+z 2 k. 3. Suppose that F(,,z) = e i cosj+sin 2 zk. Find divf and curlf. 4. Evaluate the line integral f(,,z)ds where f(,,z) = z and is the heli r(t) = 4costi+4sintj+3tk for 0 t 2π. 5. Evaluate the line integral ( 2 +)d+(+1)d where is the curve starting at (0,0), traveling along a line segment to (1,2) and then traveling along a second line segment to (0,3). 6. Evaluate d d along the curve shown in the figure. (1,1) (2,0) 7. onfirm that the force field F is conservative in some open connected region containing thepointsp andq, andthenfindtheworkdonebtheforcefieldonaparticlemoving along an arbitrar smooth curve in the region from P to Q. F(,) = 2 3 i j; P( 3,0), Q(4,1) 8. Let F(,) = cos( )i+[sin cos( )]j be a force field on -plane. (a) Show that F is a conservative vector field.

10 MA112: Prepared b Dr. Archara Pacheenburawana 10 (b) Find a potential function φ b first integrating φ. (c) omputetheworkdoneinmoving aparticleintheforcefieldfalongthestraight line segment from the origin to the point P(π/2,π/4). 9. Use Green s Theorem to evaluate the line integral (tan 1 +2)d+( 2 +ln 2 )d along the closed path bounding region R. = 4 = 0 = 10. Use Green s Theorem to evaluate the integral the triangle with vertices (0,0), (2,0), and (0,4). 4 ln(1+)d d, where is 1+ 2