Review 2: Many True/False

11. If ∇ × F = 0 then F is conservative:

FALSE. This was true as long as F is defined on all of R 3 .

10. If F is conservative then ∇ · F = 0:

FALSE. This would imply that ∂ 2 f ∂x2 + ∂ 2 f ∂y2 = 0 (or analogously in any number of dimensions), but lots of functions (e.g. f(x, y) = x 2 + y 2 ) violate this equality.

5. If F = hP, Qiand Py = Qx in an open region D then F is conservative.

FALSE: D must be simply connected.

17. If R C f ds = 0 then C is a closed curve.

FALSE: for example, the integral of sin x as x goes from 0 to 2π.

20. If a force points only in the x direction then the work done by the force on a particle depends only on the particle's starting and ending x-positions.

FALSE: the force's strength could depend on the y-position of the particle (Imagine swimming up a river versus walking along the bank. The river's current does different amounts of work.) 1

54. If a function f has a single global maximum at (a, b) then ∇f(x, y) points along the line segment from (x, y) to (a, b).

FALSE: the gradient points in the direction of steepest ascent, which is not necessarily directly toward the global maximum. (For example, most points on a non-circular ellipse)

24. For any x, f(x − ∇f(x)) ≤ f(x).

FALSE: the negative gradient is a descent direction, so what's true is that if ∇f 6= 0 then there exists λ > 0 (possibly very small) such that f(x − λ∇f(x)) < f(x).

63. For any integrable function f, Za x=0 Za y=x f(x, y) dx dy = Za y=0 Za x=y f(x, y) dx dy.

FALSE: the two integrals describe different triangular domains.

8. There is a vector field F such that ∇ × Fhx, y, zi.

FALSE: this function has non-zero divergence, but an earlier true/false implies that the divergence of the curl of any smooth function is zero.

4. If f has continuous partial derivatives on R 3 and C is any circle then R C ∇f · dr = 0.

FALSE: this is necessarily true only if the curl of f is zero.

35. A region D simply connected if any two points in D can be joined by a curve that stays inside D.

FALSE: this only describes a connected region (mostly). D must also not have any "holes," so a doughnut shape would be a counterexample.

27. If f(x, y) = g(x)h(y), then RR D f(x, y) dA = RR D g(x) dA RR D h(y) dA .

FALSE: you can split the integral as a product of two single-variable integrals if the integral is over a rectangle.

65. If f(x, y) = ln y then ∇f(x, y) = 1/y.

FALSE: ∇f(x, y) = h0, 1/yi.

56. If fx and fy exist and are continuous in a neighborhood around (a, b) then f is differentiable at (a, b).

TRUE

62. If f(x, y) = f(y, x) for all x, y ∈ R then Za x=0 Zb y=0 f(x, y) dy dx = Zb x=0 Za y=0 f(x, y) dy dx.

TRUE

66. If f has a local minimum at (a, b) and f is differentiable at (a, b) then ∇f(a, b) = 0.

TRUE

9. If F is conservative then ∇ × F = 0:

TRUE

31. If fxx > 0 and fyy < 0 at a point (x, y) then (x, y) is a saddle point of f.

TRUE, IF (x, y) is a critical point. Otherwise it's definitely false.

13. curl(div(F)) is not a meaningful expression.

TRUE, since curl must take a 3-D vector field as its argument, but div(F) is a scalar.

41. The kinetic energy of an object plus its potential energy due to gravity is always constant:

depends on whether you assume that gravity is the only thing doing work on an object. If yes, then true, but I would say in general FALSE.

74. If f(x, y) is continuous and we define g0(y) = f(0, y), then g is also continuous:

true.

23. If f is differentiable along every straight line going through a point (x, y) then f is differentiable at (x, y).

FALSE: Still might not even be continuous!

52. Z1 0 Zx 0 p x + y 2 dy dx = Zx 0 Z1 0 p x + y 2 dx dy

FALSE: The second integral isn't even well-defined on account of the Rx 0 term! You have to be more careful when changing the limits of integration and make sure that your new limits specify the same geometric domain as the old ones.

69. If ∇f(a, b) = 0, fxx(a, b) > 0 and fyy(a, b) > 0, then f has a local minimum at (a, b).

FALSE: if fxy is sufficiently large then the point could still be a saddle point. But (a, b) is definitely not a local maximum.

28. If fxx > 0 and fyy > 0 at a point (x, y) then the point (x, y) is a local minimum of the function f.

FALSE: if fxy, fyx are large then it could be a saddle point.

51. Every point in R 3 is uniquely represented by a set of spherical coordinates (ρ, θ, φ).

FALSE: in particular, when ρ = 0 the point will be the origin regardless of θ and φ.

61. If ∇f(x, y) = λ∇g(x, y) for some λ then x is an extreme value of f on the set {(a, b) : g(a, b) = g(x, y)}.

FALSE: it could be a saddle point (for example, if g(x, y) = 0 for all (x,y) then this is just the same as finding a critical point of f.)

30. If ∇f(x, y) = 0 then (x, y) is a local minimum or maximum of f.

FALSE: it could be a saddle point.

38. If there exists a closed curve C in D such that R C F · dr = 0 then F is conservative on D.

FALSE: the condition is that this holds for all curves in D.

1. If F is a vector field then ∇ · F is a vector field.

FALSE: the divergence is a scalar function.

32. If ∇f is never zero then the minimum and maximum of f on a closed and bounded domain D must occur on the boundary.

TRUE, assuming f is continuous and differentiable.

67. If f(x, y) = sin x+ sin y then − √ 2 ≤ Duf(x, y) ≤ √ 2 for all unit vectors u.

TRUE, since |Duf(x, y)| = |∇f(x, y) · u| ≤ |∇f(x, y)| · |u| = |∇f(x, y)|.

59. fy(a, b) = lim y→b f(a, y) − f(a, b) y − b .

TRUE.

84. For any vectors u and v in R^n, u + v = v + u.

True.

76. For any u, v, w ∈ R 3 , u · (v × w) = (u × v) · w.

Visual intuition: up to sign, yes, because (with the zero vector) the triple product describes the volume of a parallelepiped.

80. The intersection of two non-parallel planes is always a line.

Yup.

43. If a particle travels in a closed loop then the total work done on the particle over the loop is zero.

...FALSE: the force may not be conservative (a racecar can accelerate on a circular track. . . non-zero work done)

49. If a particle has fixed coordinates φ and θ but moves with dρ/dt constant, then the speed of the particle is constant.

...TRUE, since the particle is moving on a straight line from the origin it speed is equal to |dρ/dt|.

94. If the force on a particle is always parallel to the particle's velocity then the particle will never change direction:

Depends on whether you count stopping and reversing direction as being in the "same" direction. It's still parallel to the original velocity vector, but I would say FALSE.

46. If a particle is moving in a constant force field then the work done on the particle is proportional to the distance the object travels.

FALSE

14. The integral Zπ/2 φ=0 Zπ/2 θ=0 Z1 ρ=0 ρ 2 sin θ dρ dθ dφ gives the volume of 1/4 of a sphere.

FALSE for two reasons: the sin θ should be sin φ, and even with this change it only gives the volume of 1/8 of a sphere.

48. If a particle moves around on the surface of a sphere with dφ/dt and dθ/dt constant, then the speed of the particle is constant.

FALSE, since dθ/dt being constant means that the particle will move faster when it is near the equator than when it is near one of the poles.

58. There exists a function f with continuous second-order partial derivatives such that fx(x, y) = x + y 2 and fy(x, y) = x − y 2 .

FALSE, since we would have fxy(x, y) = 2y but fyx(x, y) = 1.

33. If f has a critical point in the interior of a closed and bounded domain D then the minimum and maximum of f on D occur in the interior of D.

FALSE. (e.g. f(x, y) = x 2 − y 2 on the unit circle)

71. If f has a single global minimum at (a, b), then the minimum of f on the unit circle occurs at the point on the circle closest to (a, b).

FALSE. 4

68. If f is differentiable at (a, b) and ∇f(a, b) = 0 then f has a local maximum or minimum at (a, b).

FALSE. It could be a saddle point.

42. R −C f ds = − R C f ds: very

FALSE. Not to be confused with the work done on a particle moving one direction along a curve versus moving in the opposite direction.

6. If F and G are vector fields and ∇ × F = ∇ × G then F = G.

FALSE: F can be G plus any function whose curl is zero.

70. If f(x, y) has two local maxima then f must have a local minimum.

FALSE: It could just have a saddle point in between the maxima (imagine a mountain with two peaks: it doesn't have a local minimum elevation).

22. If fx = fy = 0 at a point (x, y) then f is differentiable at (x, y).

FALSE: it might not even be continuous! (come up with examples)

44. If we have a region S in (u,v)-space and form a region R by the transformation x = 2u + v, y = u − 2v, then the area of R is 1/5 the area of S.

FALSE: it's 5 times the area.

12. Green's Theorem is just the Divergence Theorem in two dimensions.

FALSE: it's Stokes' Theorem in two dimensions.

40. If two triangles share an edge then the work a force field does on a particle traveling along the two triangles one at a time is the same as if the particle traveled along the quadrilateral boundary of the union of the two triangles.

FALSE: only if the particle travels along both triangles in the same direction, in which case it's true (the integrals over the shared edge have to cancel out).

47. If a particle is moving in a constant force field then the work done on the particle is proportional to the particle's distance from its starting position.

FALSE: only it's displacement in the direction parallel to the force matters, no movement normal to the force matters.

29. If (x, y) is a local minimum of a function f then f is differentiable at (x, y) and ∇f(x, y) = 0.

FALSE: say, f(x, y) = |x| + |y|.

57. If f has a unique global maximum at a point a then the maximum value of f on a domain D occurs at the point in D closest to a.

FALSE: say. f(x, y) = −10x 2 − y 2 where D is the line y = 1 − x.

37. If P and Q are continuously differentiable and ∂P ∂y = ∂Q ∂x throughout D then F is conservative on D.

FALSE: the domain must be simply connected.

34. If x is a minimum of f given the constraints g(x) = h(x) = 0 then ∇f(x) = λ∇g(x) and ∇f(x) = µ∇h(x) for some scalars λ and µ.

FALSE: ∇f(x) = λ∇g(x) + µ∇h(x) for some λ, µ.

64. If fx(a, b) and fy(a, b) both exist then f is differentiable at (a, b):

FALSE— just because two directional derivatives exists doesn't mean the function is differentiable. (recall the function f(x, y) = xy2/(x 2 + y 4 ))

85. For any vectors u and v in R n, |u + v| = |u| + |v|.

False, unless the vectors are pointing in the same direction.

72. If f(x, y) → L as (x, y) → (a, b) along every straight line through (a, b) then lim(x,y)→(a,b) f(x, y) = L.

False.

82. If x = f(t) and y = g(t) are twice differentiable, then d 2y dx2 = d2y/dt2 / d2x/dx2 .

False.

83. The distance traveled by an object is equal to the integral of its velocity over time.

False. . . it's the integral of speed over time.

89. If dy/dt = 0 at some point on a curve then the tangent line at that point is horizontal.

False: if dx/dt is also zero then the tangent line is not necessarily horizontal. 5

91. If f(θ) = f(−θ) for all θ, then the curve defined by r = f(θ) will have a vertical axis of symmetry.

False: it will have a horizontal axis of symmetry.

86. The set of points {x, y, z|x^2 + y^2 = 1} is a circle.

False: it's an infinite cylinder.

73. If f is a function then lim(x,y)→(2,5)f(x,y) = f(2, 5):

False: this is true if f is continuous.

75. If f(x, y) has no global maximum or minimum and g(x) = f(0, x), then g(x) also has no global maximum or minimum:

False: try f(x, y) = x 2 − y 2 .

87. The curve defined by any set of parametric equations (x, y) = (f(t), g(t)) can also be defined by an equation of the form y = h(x).

False: x = cos θ, y = sin θ defines a circle, and y cannot be expressed as a function of x.

79. If u × v = 0 then u = 0 or v = 0.

Nope. They can be parallel.

78. If u · v = 0 then u = 0 or v = 0.

Nope. They can be perpendicular.

77. For any u, v, w ∈ R 3 , u × (v × w) = (u × v) × w.

Nope. i × (i × j) = i × k = −j, but (i × i) × j = 0 × j = 0.

25. If f(x, y) = 1 then RR D f(x, y) dA is equal to the area of the domain D.

TRUE

26. For any a, b ∈ R and continuous function f, Ra x=0 Rb y=0 f(x, y) dy dx = Rb y=0 Ra x=0 f(x, y) dx dy.

TRUE

45. If we instead apply the transformation x = 2u + v, y = 4u + 2v then the region of R is zero.

TRUE

55. For any unit vector u and any point a, Df−u(a) = −Dfu(a).

TRUE, since ∇f · (−u) = −∇f · u at any point for any vector.

15. Z1 r=−1 Z1 θ=0 e r 2+θ 2 dθ dr = Z1 r=−1 e r 2 dr Z1 θ=0 e θ 2 dθ :

TRUE.

18. If the work done by a force F on an object moving along a curve is W, then if the object moves along the curve in the opposite direction the work done by F will be −W.

TRUE.

19. If a particle moves along a curve C, the total work done by a force F on the object is independent of how quickly the particle moves.

TRUE.

2. If F is a vector field then ∇ × F is a vector field.

TRUE.

3. If f has continuous partial derivatives on R 3 then ∇ · (∇ × f) = 0.

TRUE.

36. If F is conservative on D and F = Pi + Qj (where P and Q are continuously differentiable) then ∂P ∂y = ∂Q ∂x throughout D.

TRUE.

60. If f and g are both differentiable, then ∇(fg) = f∇g + g∇f.

TRUE.

7. The work done by a conservative force field in moving a particle around a closed path is zero.

TRUE.

39. If F is conservative on a region D then there is some function f on D such that ∇f = F.

TRUE. 2

53. When f(x, y, z) = 1, the integral RRR V f(x, y, z) dx dy dz gives the volume of the region V .

TRUE. 3

21. If ∇f exists everywhere then f is continuous everywhere.

TRUE: if a function is differentiable it must be continuous (but not the other way round!)

50. Z4 y=1 Z1 x=0 (x 2 + √ y) sin(x 2 y 2 ) dx dy ≤ 9.

TRUE: in general if f(x, y) ≤ K and a domain D has area A, RR D f(x, y) dx dy ≤ K · A. Here, the domain is a rectangle with area 3, so the trick is to show that (x 2 + √y) sin(x 2y 2 ) ≤ 3 for all (x, y) in the rectangle.

93. If the force on a particle is always perpendicular to the particle's velocity then the particle will never change speed.

TRUE: the normal component of acceleration (which here is all of the acceleration) only affects the curvature, not the speed.

90. If a circle is parametrized as (x, y) = (cost,sin t), then for any t the angle between (x(t), y(t)) and the positive x-axis will be equal to t.

True for circles, but not for ellipses.

81. The polar curves r = 1 − sin 2θ, r = sin 2θ − 1 have the same graph.

True.

92. If f(θ) = f(θ + π) for all θ, then the curve defined by r = f(θ) will be unchanged when it is rotated by 180 degrees about the origin.

True.

88. The curve defined by any equation of the form y = h(x) can also be defined by a set of parametric equations (x, y) = (f(t), g(t)).

True: set x = t and g(t) = h(x).

16. If C is a closed curve then R C f ds = 0 for any function f:

Very FALSE.