Determine whether or not F is a conservative vector field. If it is, find a function f such that F = enter NONE.

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1 Ch17 Practice Test Sketch the vector field F. F(x, y) = (x - y)i + xj Evaluate the line integral, where C is the given curve. C xy 4 ds. C is the right half of the circle x 2 + y 2 = 4 oriented counterclockwise. Evaluate the line integral C xyz 2 ds. C is the line segment from (-2, 6, 0) to (0, 7, 5). Evaluate the line integral C F dr, where C is given by r(t), 0 t 1. F(x, y) = e x-1 i + xy j, r(t) = t 4 i + t 5 j Evaluate y 2 dx + x dy along the following paths. (a) C = C 1 is the line segment from (-13, -7) to (0, 6) (b) C = C 2 is the arc of the parabola x = 36 - y 2 from (-13, -7) to (0, 6). Determine whether or not F is a conservative vector field. If it is, find a function f such that F = enter NONE. f. If it is not, F(x, y) = (e x cos(y))i + (e x sin(y))j F(x, y) = (sin(xy) + xcos(xy)y)i + (x 2 cos(xy))j Consider F and C below. F(x, y, z) = yz i + xz j + (xy + 10z)k: C is the line segment from (2, 0, -1) to (6, 4, 3) (a) Find a function f such that F = f. (b) Use part (a) to evaluate C f dr along the given curve C.

2 Find the work done by the force field F in moving an object from P to Q. F(x, y) = e -y i - xe -y j, P(0, 4), Q(4, 0) Consider the force field and circle defined below. F(x, y) = x 2 i + xy j, x 2 + y 2 = 144 Find the work done by the force field on a particle that moves once around the circle oriented in the clockwise direction. Find the work done by the force field F(x, y) = x i + (y + 7)j in moving an object along an arch of the cycloid r(t) = (t sin t)i + (2 cos t)j, 0 t 2π. Find the work done by the force field F(x, y) = xsin(y)i + y j on a particle that moves along the parabola y = x 2 from (1, 1) to (2, 4). Consider F and C below. F(x, y, z) = e y i + xe y j + (z+1)e z k, C: r(t) = t i + t 2 j + t 3 k, 0 t 1 (a) Find a function f such that F = f. (b) Use part (a) to evaluate C f dr along the given curve C. Evaluate the line integral by the two following methods. (x y)dx + (x + y)dy, C is counterclockwise around the circle with center the origin and radius 3. Evaluate the line integral by the two following methods. xy dx + x 2 y 3 dy, C is counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 3). Evaluate the line integral by the two following methods: x dx + y dy, C consists of the line segments from (0, 1) to (0, 0) and from (0, 0) to (1, 0) and the parabola y = 1 - x 2 from (1, 0) to (0, 1). Use Green's Theorem to evaluate the line integral along the given positively oriented curve. C xy 2 dx + 4x 2 y dy, C is the triangle with vertices (0, 0), (2, 2), and (2, 4) Use Green's Theorem to evaluate the line integral along the given positively oriented curve. C y 3 dx - x 3 dy, C is the circle x 2 + y 2 = 4

3 Use Green's Theorem to evaluate C F dr. (Check the orientation of the curve before applying the theorem.) F(x, y) = y 2 cos(x), x 2 + 2ysin(x), C is the triangle from (0, 0) to (1, 3) to (1, 0) to (0, 0). Evaluate 2y 2 dx + 6xy dy, where C is the boundary of the semiannular region D in the upper half-plane between the circles x 2 + y 2 = 16 and x 2 + y 2 = 25. If F(x, y) = (-yi + xj)/(x 2 + y 2 ), show that F dr = 2π for every positively oriented simple closed path that encloses the origin. Evaluate the surface integral: S z ds S is the surface x = y + 5z 2, 0 y 1, 0 z 2. Evaluate the surface integral: S y ds S is the part of the paraboloid y = x 2 + z 2 that lies inside the cylinder x 2 + z 2 = 4. Evaluate the surface integral: S y 2 ds S is the part of the sphere x 2 + y 2 + z 2 = 100 that lies inside the cylinder x 2 + y 2 = 25 Evaluate the surface integral S F ds for the given vector field F and the oriented surface S. In other words, find the flux of F across S. F(x, y, z) = xze y i xze y j + z k. S is the part of the plane x + y + z = 9 in the first octant and has downward orientation. F(x, y, z) = x i - z j + y k. S is the part of the sphere x 2 + y 2 + z 2 = 25 in the first octant, with orientation toward the origin. F(x, y, z) = xz i + x j + y k. S is the hemisphere x 2 + y 2 + z 2 = 25, y 0, oriented in the direction of the positive y- axis. F(x, y, z) = x i + y j + 7 k. S is the boundary of the region enclosed by the cylinder x 2 + z 2 = 1 and the planes y=0 and x+y = 5.

4 F(x, y, z) = x 2 i + y 2 j + z 2 k. S is the boundary of the solid half-cylinder 0 z, 0 x 2. F(x,y,z) = 6y cos z i + e x sin z j + xe y k. S is the hemisphere x 2 + y 2 + z 2 = 9, z 0, oriented upward. F(x,y,z) = x 2 z 2 i + y 2 z 2 j + xyz k S is the part of the paraboloid z = x 2 +y 2 that lies inside the cylinder x 2 +y 2 = 1, oriented upward. F(x,y,z) = x 2 y 3 z i + sin(xyz) j + xyz k. S is the part of the cone y 2 = x 2 +z 2 that lies between the planes y = 0 and y = 1, oriented in the direction of the positive y-axis. F(x,y,z) = e xy cos(z) i + x 2 z j + xy k. S is the hemisphere, oriented in the direction of the positive x-axis. Use Stokes' Theorem to evaluate C F dr. C is oriented counterclockwise as viewed from above. F(x,y,z) = e -x i + e x j + e z k C is the boundary of the part of the plane 3x+y+3z = 3 in the first octant. Use Stokes' Theorem to evaluate C F dr. C is oriented counterclockwise as viewed from above. F(x,y,z) = xy i + 2z j + 4y k. C is the curve of intersection of the plane x+z = 7 and the cylinder x 2 +y 2 = 81. Evaluate F dr, where F(x, y, z) = -y 2 i + 2xj + 3z 2 k and C is the curve of the intersection of the plane y + z = 3 and the cylinder x 2 + y 2 = 16. (Orient C to be counterclockwise when viewed from above.) Use the Divergence Theorem to calculate the surface integral S F ds; that is, calculate the flux of F across S. F(x,y,z) = 3xy 2 i + xe z j + z 3 k. S is the surface of the solid bounded by the cylinder y 2 + z 2 = 9 and the planes x = -1 and x = 3. Use the Divergence Theorem to calculate the surface integral S F ds; that is, calculate the flux of F across S. F(x,y,z) = x 4 i - x 3 z 2 j + 4xy 2 z k. S is the surface of the solid bounded by the cylinder x 2 +y 2 = 4 and the planes z = 0 and z = x+2.

5 Evaluate F ds, where F(x, y, z) = xyi + (y 2 + e z )j + cos(xy)k and S is the surface of the region E bounded by the parabolic cylinder z = 4 - x 2 and the planes z = 0, y = 0, and y + z = 6.