9 Chapter in Review Answers to selected odd-numbered problems begin on page ANS-26.

Answer Problems 1–20 without referring back to the text. Fill in the blank or answer true/false. Where appropriate, assume continuity of P, Q, and their first partial derivatives.

  1. A particle whose position vector is r(t) = cos ti + cos tj + sin tk moves with constant speed.
  2. The path of a moving particle whose position vector is r(t) = (t2 + 1)i + 4j + t4k lies in a plane.
  3. The binormal vector is perpendicular to the osculating plane.
  4. If r(t) is the position vector of a moving particle, then the velocity vector v(t) = r′(t) and the acceleration vector a(t) = r″(t) are orthogonal.
  5. ∇z is perpendicular to the graph of z = f(x, y).
  6. If ∇f = 0, then f = constant.
  7. The integral (x2 + y2) dx + 2xy dy, where C is given by y = x3 from (0, 0) to (1, 1), has the same value on the curve y = x6 from (0, 0) to (1, 1).
  8. The value of the integral 2xy dx − x2 dy between two points A and B depends on the path C.
  9. If C1 and C2 are two smooth curves such that dx + Q dy = dx + Q dy, then dx + Q dy is independent of the path.
  10. If the work F · dr depends on the curve C, then F is nonconservative.
  11. If ∂P/∂x = ∂Q/∂y, then P dx + Q dy is independent of the path.
  12. In a conservative force field F, the work done by F around a simple closed curve is zero.
  13. Assuming continuity of all partial derivatives, ∇ × ∇f = 0.
  14. The surface integral of the normal component of the curl of a conservative vector field F over a surface S is equal to zero.
  15. Work done by a force F along a curve C is due entirely to the tangential component of F.
  16. For a two-dimensional vector field F in the plane z = 0, Stokes’ theorem is the same as Green’s theorem.
  17. If F is a conservative force field, then the sum of the potential and kinetic energies of an object is constant.
  18. If P dx + Q dy is independent of the path, then F = Pi + Qj is the gradient of some function ϕ.
  19. If ϕ = is a potential function for a conservative force field F, then F = .
  20. If F = f(x)i + g(y)j + h(z)k, then curl F = .
  21. Find the velocity and acceleration of a particle whose position vector is r(t) = 6ti + tj + t2k as it passes through the plane –x + y + z = –4.
  22. The velocity of a moving particle is v(t) = –10ti + (3t2 – 4t)j + k. If the particle starts at t = 0 at (1, 2, 3), what is its position at t = 2?
  23. The acceleration of a moving particle is a(t) = sin ti + cos tj. Given that the velocity and position of the particle at t = π/4 are v(π/4) = –i + j + k and r(π/4) = i + 2j + (π/4)k, respectively, what was the position of the particle at t = 3π/4?
  24. Given that r(t) = t2i + t3j – t2k is the position vector of a moving particle, find the tangential and normal components of the acceleration at any t. Find the curvature.
  25. Sketch the curve traced by r(t) = cosh ti + sinh tj + tk.
  26. Suppose that the vector function of Problem 25 is the position vector of a moving particle. Find the vectors T, N, and B at t = 1. Find the curvature at that point.

In Problems 27 and 28, find the directional derivative of the given function in the indicated direction.

  1. f(x, y) = x2y – y2x; Du f in the direction of 2i + 6j
  2. F(x, y, z) = 1n(x2 + y2 + z2); DuF in the direction of –2i +j + 2k
  3. Consider the function f(x, y) = x2y4. At (1, 1) what is
    1. The rate of change of f in the direction of i?
    2. The rate of change of f in the direction of i – j?
    3. The rate of change of f in the direction of j?
  4. Let w =
    1. If x = 3 sin 2t, y = 4 cos 2t, and z = 5t3, find dw/dt.
    2. If x = 3 sin 2 , y = 4 cos 2 , and z = 5t3r3, find ∂w/∂t.
  5. Find an equation of the tangent plane to the graph of
    z = sin (xy) at .
  6. Determine whether there are any points on the surface z2 + xy – 2x – y2 = 1 at which the tangent plane is parallel to z = 2.
  7. Express the volume of the solid shown in FIGURE 9.R.1 as one or more iterated integrals using the order of integration (a) dy dx and (b) dx dy. Choose either part (a) or part (b) to find the volume.
    A graph. A solid is graphed on a three dimensional x y z coordinate system. The solid, graphed in the first octant, is bound by the plane z = 0 at the bottom, the vertical plane that follows the line labeled y = x on the x y plane on the left, the vertical plane that follows the line y = 2 x on the x y plane on the right, and the surface labeled z = sqrt(1 minus x^2) in the front.

    FIGURE 9.R.1 Solid for Problem 33

  8. A lamina has the shape of the region in the first quadrant bounded by the graphs of y = x2 and y = x3. Find the center of mass if the density at a point P is directly proportional to the square of the distance from the origin.
  9. Find the moment of inertia of the lamina described in Problem 34 about the y-axis.
  10. Find the volume of the sphere x2 + y2 + z2 = a2 using a triple integral in (a) rectangular coordinates, (b) cylindrical coordinates, and (c) spherical coordinates.
  11. Find the volume of the solid that is bounded between the cones z = and z = and the plane z = 3.
  12. Find the volume of the solid shown in FIGURE 9.R.2.
    A graph. A solid shaped like an inverted truncated cone, is graphed on a three dimensional x y z coordinate system. The solid is bound above by the surface labeled z = sqrt(4 minus x^2 minus y^2), and the surface labeled z = sqrt(1 minus x^2 minus y^2) below. The lateral surface of the solid is labeled z = sqrt(3 x^2 + 3 y^2).

    FIGURE 9.R.2 Solid for Problem 38

In Problems 39–42, find the indicated expression for the vector field F = x2yi + xy2j + 2xyzk.

  1. ∇ · F
  2. ∇ × F
  3. ∇ · (∇ × F)
  4. ∇(∇ · F)
  5. Evaluate , where C is given by
    x = cos 2t, y = sin 2t, z = 2t, π ≤ t ≤ 2π.
  6. Evaluate (xy + 4x) ds, where C is given by 2x + y = 2 from (1, 0) to (0, 2).
  7. Evaluate 3x2y2dx + (2x3y – 3y2) dy, where C is given by y = 5x4 + 7x2 – 14x from (0, 0) to (1, –2).
  8. Show that , where C is the circle x2 + y2 = a2.
  9. Evaluate y sin πz dx + x2ey dy + 3xyz dz, where C is given by x = t, y = t2, z = t3 from (0, 0, 0) to (1, 1, 1).
  10. If F = 4yi + 6xj and C is given by x2 + y2 = 1, evaluate F · dr in two different ways.
  11. Find the work done by the force F = x sin yi + y sin xj acting along the line segments from (0, 0) to (π/2, 0) and from (π/2, 0) to (π/2, π).
  12. Find the work done by F = i + j from (–, ) to (1, ) acting on the path shown in FIGURE 9.R.3.
    A graph. A path, consisting of 3 line segments, is graphed on an x y coordinate plane. The first line segment begins at the marked point (negative 1 over 2, 1 over 2), and ends at the marked point (negative 1, 1). The second line segment begins at the marked point (negative 1, 1), and ends at the marked point (1, 1). The third line segment begins at the marked point (1, 1), and ends at the marked point (1, sqrt(3)).

    FIGURE 9.R.3 Path for Problem 50

  13. Evaluate (z/xy) dS, where S is that portion of the cylinder z = x2 in the first octant that is bounded by y = 1, y = 3, z = 1, z = 4.
  14. If F = i + 2j + 3k, find the flux of F through the square defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, z = 2.
  15. If F = c∇(1/r), where c is constant and ||r|| = r, r = xi + yj + zk, find the flux of F through the sphere x2 + y2 + z2 = a2.
  16. Explain why the divergence theorem is not applicable in Problem 53.
  17. Find the flux of F = c∇(1/r) through any surface S that forms the boundary of a closed bounded region of space not containing the origin.
  18. If F = 6xi + 7zj + 8yk, use Stokes’ theorem to evaluate (curl F · n) dS, where S is that portion of the paraboloid z = 9 – x2 – y2 within the cylinder x2 + y2 = 4.
  19. Use Stokes’ theorem to evaluate –2y dx + 3x dy + 10z dz, where C is the circle (x – 1)2 + (y – 3)2 = 25, z = 3.
  20. Find the work F · dr done by the force F = x2i + y2j + z2k around the curve C that is formed by the intersection of the plane z = 2 – y and the sphere x2 + y2 + z2 = 4z.
  21. If F = xi + yj + zk, use the divergence theorem to evaluate (F · n) dS, where S is the surface of the region bounded by x2 + y2 = 1, z = 0, z = 1.
  22. Repeat Problem 59 for F = x3i + y3j + z3k.
  23. If F = (x2 – ey tan−1z)i + (x + y)2j – (2yz + x10)k, use the divergence theorem to evaluate (F · n) dS, where S is the surface of the region in the first octant bounded by z = 1 – x2, z = 0, z = 2 – y, y = 0.
  24. Suppose F = xi + yj + (z2 + 1)k and S is the surface of the region bounded by x2 + y2 = a2, z = 0, z = c. Evaluate eeS (F · n) dS without the aid of the divergence theorem. [Hint: The lateral surface area of the cylinder is 2πac.]
  25. Evaluate the integral (x2 + y2) dA, where R is the region bounded by the graphs of x = 0, x = 1, y = 0, and y = 1 by means of the change of variables u = 2xy, v = x2 – y2.
  26. Evaluate the integral

    where R is the region bounded by the graphs of y = x, x = 2, and y = 0 by means of the change of variables x = u + uv, y = v + uv.

  27. As shown in FIGURE 9.R.4, a sphere of radius 1 has its center on the surface of a sphere of radius a > 1. Find the surface area of that portion of the larger sphere cut out by the smaller sphere.
    The center of a smaller sphere is on the surface of a larger sphere. The intersection of the 2 spheres is represented by a dashed circle.

    FIGURE 9.R.4 Spheres in Problem 65

  28. On the surface of a globe or, more precisely, on the surface of the Earth, the boundaries of the states of Colorado and Wyoming are both “spherical rectangles.” (In this problem we assume that the Earth is a perfect sphere.) Colorado is bounded by the lines of longitude 102°W and 109°W and the lines of latitude 37°N and 41°N. Wyoming is bounded by longitudes 104°W and 111°W and latitudes 41°N and 45°N. See FIGURE 9.R.5.
    1. Without explicitly computing their areas, determine which state is larger and explain why.
    2. By what percentage is Wyoming larger (or smaller) than Colorado? [Hint: Suppose the radius of the Earth is R. Project a spherical rectangle in the Northern Hemisphere that is determined by latitudes θ1 and θ2 and longitudes ϕ1 and ϕ2 onto the xy-plane.]
    3. One reference book gives the areas of the two states as 104,247 and 97,914 mi2. How does this answer compare with the answer in part (b)?
    A map of the United States. The States of Wyoming, labeled W Y, and Colorado, labeled C O, are highlighted.

    FIGURE 9.R.5 States WY and CO are spherical rectangles in Problem 66