Problem sets and quizzes

Here are the quizzes and their associated problem sets (Quiz 1, Quiz 2, Quiz 3, Quiz 4, Quiz 5, Quiz 6, Quiz 7); but anything whose assigned date is in the future is subject to change!

1. Vectors and curves:
   • Date taken: April 10 Wednesday.
   • Practice Exercises from §11 (pages 638&639): 19, 25, 29, 31, 35, 37, 43, 50.
   • Exercises from §15.1 (page 826): 1–8.
   • Exercises from §12.1 (pages 648–650): 1, 4, 6, 7, 11, 14, 15, 17, 19, 20, 23.
   • Exercises from §12.2 (pages 654–656): 1, 4, 6, 12, 15, 17, 21, 22.
   • Exercises from §12.3 (page 660): 1, 5, 8, 9, 11, 15, 18.
   • Exercises from §13.1 (pages 682–684): 3, 4, 7, 9, 10, 15, 17, 18, 19, 23, 24, 30, 31–36, 39, 40, 42, 52, 54, 59, 62.
   • Additional extra-credit exercise: Acceleration (change in velocity) can be viewed as a combination of change in speed and change in direction. Writing v = vT (where v is velocity, v =|v| is speed, and T = v̂ = v/|v| is the unit tangent vector, indicating direction), the Product Rule tells us that dv/dt = (dv/dt)T + v(dT/dt); note that dv/dt is the vector acceleration a, while dv/dt is the scalar acceleration a. As for dT/dt, it can be further broken down into its own magnitude and direction; these are ω = |dT/dt|, the angular speed (in radians per unit time), and N = (dT/dt)ˆ = (dT/dt)⁠/|dT/dt|, the unit normal vector (indicating the direction of curvature); that is, dT/dt = ωN. So in summary, a = aT + ωvN. (Some related material is in Sections 12.4&12.5 of the textbook; use ω = κv to convert between my notation and the textbook's.) Based on this, if an object has an instantaneous velocity of 6.00 metres per second due east, is speeding up by 3.00 m/s², and is changing direction towards the north by 10.0 degrees per second (remembering that a degree is π/180 radian), then what are the east/west and north/south components of its acceleration vector, to the nearest cm/s²? Give a final answer something like ‹3.12 m/s² to the east and 1.24 m/s² to the north› (although the correct answer is different) and show at least what numerical calculations you make to get your answer.
2. Functions of several variables:
   • Date taken: April 18 Thursday.
   • Exercises from §13.2 (pages 690–693): 2, 6, 11, 18, 23, 28, 31, 32, 36, 39, 43, 46, 55.
   • Exercises from §15.2 (pages 838–840): 5, 6, 39, 41, 43.
   • Additional exercises (you must attempt these to get full credit for the problem set):
     1. Given α = 3x dx + 4x²y dy, evaluate α at (x, y) = (2, 6) along ⟨dx, dy⟩ = ⟨0.003, 0.005⟩. (Answer.)
     2. Given α = 2xy dx + 2yz dy + 2xz dz, evaluate α at (x, y, z) = (−1, 3, 2) along ⟨dx, dy, dz⟩ = ⟨0.01, 0.02, −0.01⟩.
     3. Given β = x² dx + xy dy + xz dz, evaluate β at (x, y, z) = (4, 3, −2). (Answer.)
     4. Given β = 5x² dx − 3xy dy, evaluate β at (x, y) = (1, 2).
   • Exercises from §13.3 (pages 702–704): 3, 4, 10, 12, 24, 26, 30, 39, 43, 46, 55, 57, 75, 82, 91.
   • Exercises from §13.5 (pages 720&721): 2, 3, 7, 8, 14, 15, 16, 20, 23.
   • Exercises from §15.2 again (page 838): 1, 4.
   • Additional extra-credit exercise: Prove that the two definitions of continuity in Section 2.3 on page 26 of my notes are equivalent. Actually, just do the direction that is not incredibly difficult: Given a function f of several variables and a point P₀, show that if the conditions in the second definition of the continuity of f at P₀ (the one in terms of ε and δ in the third paragraph on page 26) are met, then the conditions in the first definition (the one in terms of the continuity of composite functions in the second paragraph on page 26) must also be met (but don't try to prove it the other way around). To provide the link between these definitions, you will need to refer to the ε-δ definition of continuity for an ordinary function of one variable; see the definition about two-thirds of the way down page 8 in my notes from Calculus 1 if you don't know precisely how that goes.
3. Applications of differentiation:
   • Date taken: April 29 Monday.
   • Exercises from §13.4 (pages 711&712):
     • Use any method (including differentials, gradients, or matrices): 2, 4, 7, 10;
     • Try to use at least two methods (differentials, matrices, or branch diagrams) for each: 13, 15, 17, 19;
     • 27, 28, 41;
     • Extra credit: 44. (The answer to part (a) is often written ∂⁠/∂r = cos θ ∂⁠/∂x + sin θ ∂⁠/∂y and ∂⁠/∂θ = −r sin θ ∂⁠/∂x + r cos θ ∂⁠/∂y, and you can write the answer to part (b) similarly. Expressions such as these are differential operators.)
   • Exercise from §13.5 (pages 720&721): 28.
   • Exercises from §13.6 (pages 727–730): 3, 6, 10, 13, 14.
   • Exercises from §13.7 (pages 737–739): 2, 7, 9, 15, 27, 32, 34, 37, 43, 52, 57.
   • Exercises from §13.8 (pages 746–748): 1, 5, 10, 11, 16, 23, 29.
4. Integration on curves:
   • Date taken: May 8 Wednesday.
   • Exercises from §13.6 (pages 727–730): 19, 21, 29, 30, 33, 35, 39, 50, 54.
   • Exercises from §15.2 (pages 838–840): 10, 11, 14, 16, 17.A&B, 19, 22, 23, 24, 29.
   • Exercises from §15.1 (pages 826–828): 10, 13, 16, 22, 30.
   • Exercises from §14.1 (pages 759&760): 3, 7, 10, 17, 22, 27.
   • Exercises from §14.2 (pages 767–769):
     • 19, 23;
     • Extra credit: 80.
5. Multiple integrals:
   • Date taken: May 15 Wednesday.
   • Exercises from §14.2 (pages 767–769): 1, 2, 7, 9, 12, 14, 17, 35, 41, 47, 51, 57, 61, 82.
   • Exercises from §14.5 (pages 785–788): 3, 6, 9, 15, 21, 25, 29, 34, 37.
   • Exercises from §14.3 (page 772): 1, 4, 7, 12, 13, 14, 17, 20, 21.
   • Exercises from §14.4 (pages 777–786): 1, 3, 5, 7, 9, 17, 20, 23, 24, 28, 29, 34, 37.
   • Exercises from §14.7 (pages 803–806): 1, 2, 8, 12, 14, 23, 37, 43, 46, 57.
   • Additional extra-credit exercise: Suppose that α and β are real numbers with 0 ≤ α ≤ β ≤ 2π and that f and g are continuous functions defined at least on [α, β] with 0 ≤ f(θ) ≤ g(θ) at least whenever α ≤ θ ≤ β. Consider the region in the plane given in polar coordinates by α ≤ θ ≤ β and f(θ) ≤ r ≤ g(θ). Use the methods of §14.4 of the textbook to show that the area of this region is ½ ∫^β_α (g(θ)² − f(θ)²) dθ.
6. More higher-dimensional integration:
   • Date taken: May 28 Tuesday.
   • Exercises from §14.8 (pages 814–816): 1, 3, 7, 9, 15.
   • Exercises from §15.5 (pages 872–874): 2, 3, 6, 9, 13, 20, 23.
   • Exercises from §15.6 (pages 883–884): 1, 5, 8, 11, 16, 17, 19, 23, 25, 34, 35, 37, 41, 45.
   • Exercises from §14.6 (pages 793–795): 3, 14, 19, 25, 29.
   • Exercise from §15.1 (pages 826–828): 35.
   • Exercises from §14.7 (pages 803–806): 77.
   • Additional extra-credit exercise: Suppose that a and b are real numbers with a ≤ b, and that f is a continuously differentiable function defined at least on [a, b] with f(z) ≥ 0 at least whenever a ≤ z ≤ b. Consider the surface given in cylindrical coordinates by a ≤ z ≤ b and r = f(z). Use the methods of §6.6 of my notes or §15.5 of the textbook to show that the area of this surface is 2π ∫^b_a f(z) √(f′(z)² + 1) dz.
7. The Stokes theorems:
   • Date taken: June 5 Wednesday.
   • Exercises from §15.3 (pages 849–851): 1, 3, 6, 7, 8, 11, 14, 17, 21, 25.
   • Exercises from §15.4 (pages 861–863): 1, 4, 7, 9, 12, 15, 21, 24, 26, 33.
   • Exercises from §15.7 (pages 895–897): 1, 3, 5, 6, 9, 14, 17, 21, 28.
   • Exercises from §15.8 (pages 906–908): 1, 2, 6, 7, 8, 13, 17.
   • Additional extra-credit exercise: Suppose that F is a conservative vector field defined on all of 3-dimensional space; then there exists a scalar field f such that F = ∇f. Let U = −f. In physics, if F is a force field, then we call U a potential energy field for F. Recall that, if an object travels along a curve C in the force field F, then the work done on that object by that force field, or in other words the energy transferred to that object by that force field, is the integral ∫_P∈CF(P) ⋅ dP (or ∫_C F ⋅ dr for short). If the curve C begins at the point P₁ and ends at the point P₂, then use that F = −∇U to express the value of this work using values of the scalar field U at those points. If you imagine that U(P) is the amount of ‘potential’ energy held by an object at P by virtue of its position within this force field, then check that the amount of energy transferred to the object by the field (the work) is the opposite of the change in the object's potential energy. (In other words, we have conservation of energy: the total change in energy is zero. This conservation is why conservative vector fields are called ‘conservative’.)

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