MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 1 Etra Problems hapter 7 1. onsider the vector field F = i+z j +z 3 k. a) ompute div F. b) ompute curl F. Solution a) div F = +z +3z b) curl F = i k. Find the divergence and the curl of the vector field F,,z) = zcos i+sin j +z k. Solution div F = zsin+cos+z, curl F = z i z cos) j+sin k 3. Suppose that F,,z) = e i cos j +sin z k. Find div F and curl F. Solution div F = e +sin +sinzcosz, curl F = e k 4. Given F,) = zcos,sin,z. a) What is div F? b) What is curl F? 5. Given F,,z) = e cos i+e sin j +z k. a) What is div F? b) What is curl F? Solution a) div F = e cos +1 b) curl F = e sin k 6. Given F,,z) = i+ +αz) j+ k. Find α such that divf) = curlf) i. Solution α = 0 7. Evaluate the line integral heli rt) = 4cost i+4sint j +3t k for 0 t π. Solution 10π 8. Given F = i+ j. Evaluate rt) = t+sin f,,z)ds where f,,z) = z + and is the )) πt i+ F d r, where the curve is parametrized b t+cos )) πt j for t [0,1].
MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 9. Evaluate 10. Evaluate 11. a) Evaluate 3 ds, where is defined as the following parametric equations: = 4sint, = 4cost, z = 3t, t [0,π]. ds, where is the parabola =, 4. +3z)ds, where is the line segment shown in Figure below z 0,0,0) 1,,1) b) Let be the oriented curve given b rt) = cost i + sint j + t k, 0 t 3π. Evaluate F d r where F,,z) = i j +z k. Solution a) 5 6 b) 6 3 +9π3 1. Evaluate ) d+4 +)d, where is the circle of radius 4 centered at the point, 1). 13. Evaluate the line integral +)d++1)d where is the curve starting at 0,0), traveling along a line segment to 1,) and then traveling along a second line segment to 0,3). Solution 4 14. Evaluate d d along the curve shown in the figure. 1,1) 1 3,0) Solution
MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 3 15. 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. F,) = 3 i+3 j; P 3,0), Q4,1) Solution W = 16 16. Find the work done b the force field F = 3 i+ j on a particle moving along the smooth curve = 3 in the region from 1,3) to,1). Solution W = 85 4 17. Given three vector fields F 1 = e i+e j F = +5 ) i+5+) j F 3 = i+ +e z ) j +3e 3z k Answer the following questions: a) Which vector fields are conservative? b) Evaluate F d r, when : t) = t, t) = t+1, t [0,1]. 18. Evaluate 1,,0) to 3,0,1). F d r where F,,z) = z i z k and is the line segment from Solution 3 19. Evaluate F d r where F,,z) = + z) i +z +) j + +) k and is the line segment from 1,1,0) to,0, 1). Solution 3 0. Determine whether the given field F is conservative. If so, find a potential function φ such that F = φ. a) F = sin +4e ) i+ cos j b) F = 45 4 6 6 +3) i+18 5 1 5 +7) j Solution a) F is conservative. φ,) = sin +4e + b) F is not conservative.
MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 4 1. Let F,) = cos ) i+[sin cos )] j be a force field on -plane. a) Show that F is a conservative vector field. b) Find a potential function φ b first integrating φ. c) ompute the work done in moving a particle in the force field F along the straight line segment from the origin to the point Pπ/,π/4). Solution b) φ = sin ) cos +K c) W = 1. onsider the vector field F,) =, +. a) Show that F is conservative vector field. b) Find a potential function φ such that F = φ. c) Use the Fundamental Theorem of Line Integrals to evaluate an path from 1,) to 3,0). F d r along Solution b) φ = + + c) 0 3. onsider the vector field F,) = +1) i+ +) j. a) Show that F is conservative vector field. b) Find a potential function φ such that F = φ. c) Use the Fundamental Theorem of Line Integrals to evaluate an path from 1,0) to 1,0). F d r along Solution b) φ = ++ +K c) 4. onsider the vector field F,) = α e, 3 e where α is a constant. a) Find the value for α that makes F a conservative vector field. b) With α as in a) find a potential function φ such that F = φ. c) Use the Fundamental Theorem of Line Integrals to evaluate F d r along an path from 1,1) to,0). Solution a) α = 3 b) φ = 3 e + c) 1+e 5. onsider the vector field F,) = e +,e +α where α is a constant. a) Find the value for α that makes F a conservative vector field. b) With α as in a) find a potential function φ such that F = φ.
MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 5 c) Use the Fundamental Theorem of Line Integrals to evaluate an path from 1,) to 1,0). F d r along Solution a) α = b) φ = e + + c) 3 e 6. A force field is given b F,) = cos) i+cos) ) j. a) Eamine whether this field is conservative or not. If it is conservative find a potential function. b) ompute the work done b the force in moving an object from point 0,0) to point π,1) along a straight line. Solution a) F is conservative. φ,) = sin) + b) 0 7. Evaluate )d+d where is a piecewise smooth curve consisting of the following curves: 1 the line segment from 0,0) to 1,0), the segment of the unit circle + = 1 from 1,0) to 0,1), and 3 the line segment from 0,1) to 0,0) b a) evaluating it directl WITHOUT using Green s Theorem, b) evaluating it using Green s Theorem. 8. Use Green s Theorem to evaluate the line integral along the closed path bounding region R. = 4 tan 1 +)d+ +ln )d = 0 = Solution 16 3 9. Evaluate F d r, where F = e +e ) i+e +e ) j is a conservative vector field ) πt and rt) = sin i+lnt j, for t [1,].
MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 6 30. Use Green s Theorem to evaluate the integral F d r, when F = i+ 3 j and is the rectangle with vertices 0,0),,0),,3), and 0,3). 31. Use Green s Theorem to evaluate the integral ln1+)d d, where is 1+ the triangle with vertices 0,0),,0), and 0,4). 4 Solution 4 3. Use Green s Theorem to evaluate F d r, when F = 1+e ) i+ 5 +cos ) j and is the square with vertices 0,0),1,0),1,1), and 0,1) oriented counterclockwise. Solution 1 6 33. Use Green s Theorem to evaluate the integral e + ) d + 4+sin ) ) d, where traces the triangle with vertices 0,0),,), 0,) traversed in this order. 0,),) 0,0) Solution 6 34. Use Green s Theorem to evaluate the integral d+ 3 d, where traces the triangle with vertices 0,0), 1,0), 1,) traversed in this order.
MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 7 1,) 1 Solution 3 35. Use Green s Theorem to evaluate the integral 1+3 d+d, where is the boundar of the region R and oriented counterclockwise. = R = 3 Solution 4 1 36. Evaluate the surface integral +z)ds where σ is the part of the plane σ +z = 3 above the triangle R sketched in Figure below. 1,1) R 1,0) Solution 9 6 8 37. Evaluate the surface integral σ ) z ds where σ is the part of the clinder = that lies in the first octant between the planes z = 0, z = 5, = 1, and = 4 see Figure below).
MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 8 z 0,1,5) 0,4,5) 0,1,0) 0,4,0) Solution 5 [ 65 3/ 5 3/] 4 = 38. Let the surface σ be the portion of the plane z = 3 that lies inside the clinder + = 1. Evaluate the surface integral zds. 39. Use the Divergence Theorem to find the outward flu of the vector field F,,z) = i+z j +z 3 k σ across the surface of the rectangular defined b the inequalities 0 1, 0, 0 z 3. Solution 60 40. Let G be a solid bounded b the graphs of + = 4, z = 0, and z = 3. Let σ denote the surface of G. Use the Divergence Theorem to find the outward flu of the vector field F,,z) = 3 i+ 3 j +z k across the surface σ. Solution 108π 41. Find the outward flu of the vector field F = z j + z k across the surface of the region that is enclosed b +z = 1 and the planes = 0, = 1, and z = 0. 4. Let σ be the surface of the cone z = + with 0 z 1 with upward orientation. Let be the curve oriented counter clockwise) of intersection of the cone z = + and the plane z = 1 which forms the boundar of σ in the plane z = 1. Use Stokes Theorem to evaluate F d r where F = sin 3 3,cos + 3 3,z. 43. Use Stokes Theorem to find the work performed b the force field F,,z) = i+z j + k on a particle that traverses the oriented triangle ling in the plane z = 6 shown in Figure below.
MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 9 z 6 z = 6 Solution 9 3 3 44. Let σ be the portion of the paraboloid z = 9 for which z 0 with upward orientation, and is the positivel oriented circle that form the boundar of σ in the -plane. Use the Stokes Theorem to evaluate F d r where F = 3z i+4 j + k. Solution 36π