* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Chapter 17
Maxwell's equations wikipedia , lookup
Vector space wikipedia , lookup
Electrostatics wikipedia , lookup
Electromagnet wikipedia , lookup
Feynman diagram wikipedia , lookup
Electromagnetism wikipedia , lookup
Weightlessness wikipedia , lookup
Four-vector wikipedia , lookup
Speed of gravity wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Metric tensor wikipedia , lookup
Time in physics wikipedia , lookup
Navier–Stokes equations wikipedia , lookup
History of quantum field theory wikipedia , lookup
Kaluza–Klein theory wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Equations of motion wikipedia , lookup
Noether's theorem wikipedia , lookup
Mathematical formulation of the Standard Model wikipedia , lookup
Path integral formulation wikipedia , lookup
Classical central-force problem wikipedia , lookup
Field (physics) wikipedia , lookup
Chapter 17 (x 2 3xy y )dx , where c is the arc 2 y 2x2 , 0 x 2 . 1. Evaluate the line integral 2. Consider the vector field F( x, y) i xj . If a particle starts at the point (10, 4) in the velocity field given by F, find an equation of the path it follows. 3. c If a wire with linear density 10 lies along a space curve C, its moment of inertia about the z-axis is given by I z 10 ( x 2 y 2 ) ds . Find the moment of inertia for the wire in the shape of the helix C x 2 sin(t ), y cos(t ), z 3t, 0 t 2 4. Find a function f such that F f , and use it to evaluate F dr along the given curve C. C F e2 y i (1 2xe2 y ) j , C : r(t ) tet i (1 t ) j, 0 t 1 2r where r = x i + y j + z k. Find the | r |3 work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d 2 from these points to the origin. 5. Suppose that F is an inverse square force field, that is, F 6. Use Green's Theorem to find the work done by the force F( x, y) x( x 5 y)i (4xy 2 ) j in moving a particle from the origin along the x-axis to (3, 0) then along the line segment to (0, 3) and then back to the origin along the y-axis. Select the correct answer. a. 7. 49.5 b. -1.5 c. 4.5 d. -4.5 e. -49.5 Let r = xi + yj + zk and r = | r |. Find (9 r r) . 8. Which of the equations below is an equation of a plane? Select the correct answer. a. r (u, v) (5 10u)i (u 9v) j (2 6u 4v)k b. r (u, v) u cosvi u sin vj u k 2 9. Find a parametric representation for the part of the sphere x 2 y 2 z 2 4 that lies above the cone z x2 y 2 . 10. Find the area of the part of paraboloid x y 2 z 2 that lies inside the cylinder z 2 y 2 25 . 11. Find the area of the part of the surface y 4x z 2 that lies between the planes x = 0, x = 7, z = 0, and z = 3. Select the correct answer. a. 121.339 b. 113.269 12. Evaluate the surface integral. c. 107.759 d. 110.234 e. 111.239 4( x 2 y z 2 ) dS S S is the part of the cylinder x 2 y 2 9 between the planes z = 0 and z = 4. 13. Evaluate the surface integral F dS , where F( x, y, z) x , and S is the part of the plane S x y z 1 in the first octant. 14. Find parametric equations for C, if C is the curve of intersection of the hyperbolic paraboloid z y 2 x 2 and the cylinder x 2 y 2 16 oriented counterclockwise as viewed from above. 15. Find the moment of inertia about the z-axis of a thin funnel in the shape of a cone z 5 4 x 2 4 y 2 , 1 z 4 , if its density function is ( x, y, z) 3 z . 2 2 2 16. Use Gauss's Law to find the charge contained in the solid hemisphere x y z 81, z 0 , if the electric field is E( x, y, z ) xi yj 2 zk . 17. Use Stokes' Theorem to evaluate curl F dS . S F( x, y, z) 2xyzi 2xyj 2x 2 yzk S consists of the top and the four sides (but not the bottom) of the cube with vertices (3, 3, 3) oriented outward. 18. Use Stokes’ theorem to calculate the surface integral F dS . S F ( x, y , z ) (e x sin y )i (e x cos y z ) j yk , and S is the part of the cone 1 z 2 . 19. Use Stokes' Theorem to evaluate z 2 x 2 y 2 for which F dr . C F( x, y, z) 0.16xi 5 yj 6( y 2 x 2 )k C is the boundary of the part of the paraboloid z 0.16 x 2 y 2 in the first octant. C is oriented counterclockwise as viewed from above. xz 20. Find the div F if F ( x, y, z ) e (cos yz i sin yz j k ) . 1. Find the gradient vector field of f ( x, y) ln( x 2 y) . 2. Find the gradient vector field of f ( x, y, z ) x cos 3. Consider the vector field F(x, y) = i + xj. If a particle starts at the point (10, 4) in the velocity field given by F, find an equation of the path it follows. 4. Evaluate 2y . 5z xy 4 ds , where C is the right half of the circle x 2 y 2 25 . C 5. Evaluate xy dx ( x y ) dy , where C consists of line segments from (0, 0) to (3, 0) and from (3, C 0) to (2, 5). 6. Evaluate yz dy xy dz , where C is given by x 10 t , y 3t , z 10t 2 , 0 t 1 . C Select the correct answer. a. 193.93 225.25 b. 20.82 c. 208.23 d. 1,939.29 e. 7. A thin wire is bent into the shape of a semicircle x 2 y 2 4, x 0 . If the linear density is 7, find the exact mass of the wire. 8. Find the curl of the vector field. F( x, y, z) 5e x sin( y)i 3e x cos(y) j 8zk 9. If a wire with linear density 6 lies along a space curve C, its moment of inertia about the z-axis is given by I z 6( x 2 y 2 ) ds . Find the moment of inertia for the wire in the shape of the helix C x 2sin(t ), y 2 cos(t ), z 3t, 0 t 2 . Select the correct answer. a. e. 48 13 b. 48 13 2 c. 12 13 d. 24 11 12 11 10. Find the work done by the force field F(x, y) = 3xi + (3y + 10)j in moving an object along an arch of the cycloid r (t ) (t sin(t )) i (1 cos(t )) j , 0 t 2 . 11. Find the work done by the force field F(x, y) =xsin(y)i + yj on a particle that moves along the parabola y x 2 from (-2, 4) to (1, 1). 12. Find the work done by the force field F( x, y, z) xzi yxj zyk on a particle that moves along the curve r (t ) t 2 i t 3 j t 4k , 1 t 0 . 13. Determine whether or not F is a conservative vector field. If it is, find a function f such that F f . F (4 2xy ln x)i x 2 j 14. Determine whether or not F is a conservative vector field. If it is, find a function f such that F f . F (9 ye9 x sin y)i (e9 x x cos y) j 15. Find a function f such that F f and use it to evaluate F dr along the given curve C. C F yi ( x 2 y) j C is the upper semicircle that starts at (1, 2) and ends at (5, 2). 16. Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F f . F( x, y, z) 10 xi 10 yj 10 zk 17. Find the work done by the force field F in moving an object from P to Q. F( x, y) x3 y 4i x 4 y3 j ; P(0, 0), Q(3, 2) 18. Let F f , where f ( x, y) sin( x 8 y) . Which of the following equations does the line segment from (0, 0) to (0, ) satisfy? Select the correct answer. a. F dr 0 C b. F dr 1 C c. none of these 9r is an inverse square force field, where r = x i + y j + z k. | r |3 Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d 2 from these points to the origin. 19. Suppose that F 2 20. Use Green's Theorem to find the work done by the force F( x, y) x( x 4 y)i 4xy j in moving a particle from the origin along the x-axis to (1, 0) then along the line segment to (0, 1) and then back to the origin along the y-axis.