Readings and homework

Here are the assigned readings and exercises (Reading 1, Reading 2, Reading 3, Reading 4, Reading 5, Reading 6, Reading 7, Reading 8, Reading 9, Reading 10, Reading 11, Reading 12, Reading 13, Reading 14, Reading 15, Reading 16, Reading 17, Reading 18, Reading 19, Reading 20, Reading 21, Reading 22, Reading 23, Reading 24, Reading 25, Reading 26, Reading 27, Reading 28, Reading 29, Reading 30); but anything whose assigned date is in the future is subject to change!

1. Vectors:
   • Date assigned: April 1 Monday.
   • Date due: April 2 Tuesday.
   • Reading from the textbook: As needed: Review §§11.1–11.5.
   • Reading from my notes: Optional: Through the top of page 17 (through §1.12).
   • Online notes: Required: Vector operations.
   • Exercises due:
     1. Give a formula for the vector from the point (x₁, y₁) to the point (x₂, y₂).
     2. If u and v are vectors in 2 dimensions, then is u × v a scalar or a vector?
     3. If u and v are vectors in 3 dimensions, then is u × v a scalar or a vector?
2. Parametrized curves:
   • Date assigned: April 2 Tuesday.
   • Date due: April 3 Wednesday.
   • Reading from the textbook: Pages 642–648 (§12.1).
   • Reading from my notes: The rest of page 17 and the first two lines of page 18 (the first half of §1.13).
   • Exercises due:
     1. If C is a point-valued function, so that P = C(t) is a point (for each scalar value of t), then what type of value does its derivative C′ take?; that is, is dP/dt = C′(t) a point, a scalar, a vector, or what?
     2. If c is a vector-valued function, so that r = c(t) is a vector (for each scalar value of t), then what type of value does its derivative c′ take?; that is, is dr/dt = c′(t) a point, a scalar, a vector, or what?
3. Integrating parametrized curves:
   • Date assigned: April 3 Wednesday.
   • Date due: April 4 Thursday.
   • Reading from the textbook: Pages 650–654 (§12.2).
   • Reading from my notes: The rest of page 18 (the second half of §1.13).
   • Exercises due: If f is a vector-valued function, so that v = f(t) is a vector (for each scalar value of t), then:
     1. What type of value can its definite integrals take?; that is, can ∫^b_t=a f(t) dt = ∫^b_t=a v dt (where a and b are scalars) be a point, a scalar, a vector, or what?
     2. What type of value can its indefinite integrals take?; that is, can ∫ f(t) dt = ∫ v dt be a point, a scalar, a vector, or what?
4. Arclength:
   • Date assigned: April 4 Thursday.
   • Date due: April 8 Monday.
   • Reading from the textbook: Pages 656–659 (§12.3).
   • Reading from my notes: The bottom of page 21 and page 22 (§1.16).
   • Exercises due: Section 12.3 of the textbook uses several variables, including r, s, t, T, and v, to describe various quantities on the path of a parametrized curve. Fill in the right-hand side of each of these equations with the appropriate one of these variables:
     1. dr/dt = ___.
     2. v/|v| = ___.
     3. dr/ds = ___.
5. Functions of several variables:
   • Date assigned: April 8 Monday.
   • Date due: April 9 Tuesday.
   • Reading from the textbook: Pages 676–681 (§13.1).
   • Reading from my notes: Page 23 and through the top of page 25 (§2 through §2.1).
   • Exercises due:
     1. If f(2, 3) = 5, then what number or point must belong to the domain of f and what number or point must belong to the range of f?
     2. If f(2, 3) = 5, then what point must be on the graph of f?
6. Limits and continuity:
   • Date assigned: April 10 Wednesday.
   • Date due: April 11 Thursday.
   • Reading from the textbook: Pages 684–690 (§13.2).
   • Reading from my notes: The rest of page 25 through page 27 (§§2.2–2.4).
   • Exercises due:
     1. Suppose that the limit of f approaching (2, 3) is 5 (in symbols, lim_{(x,y)→(2,3)} f(x, y) = 5), and the limit of g approaching (2, 3) is 7 (so lim_{(x,y)→(2,3)} g(x, y) = 7). What (if anything) is the limit of f + g approaching (2, 3)? (so lim_{(x,y)→(2,3)} (f(x, y) + g(x, y)) = ___).
     2. Suppose that the limit of f approaching (0, 0) horizontally is 4 (in symbols, lim_{(x,y)→(0,0),y=0} f(x, y) = 4), and the limit of f approaching (0, 0) vertically is 6 (so lim_{(x,y)→(0,0),x=0} f(x, y) = 6). What (if anything) is the limit of f approaching (0, 0)? (so lim_{(x,y)→(0,0)} f(x, y) = ___).
7. Vector fields and differential forms:
   • Date assigned: April 11 Thursday.
   • Date due: April 15 Monday.
   • Reading from my notes: Pages 29&30 (§3 through §3.3).
   • Reading from the textbook: Page 828 and most of page 829, except for Figure 15.10, and Figure 15.15 on page 830 (§15.2, Vector Fields, except for Figures 15.10 and 15.14).
   • Online notes: Examples of vector fields.
   • Exercises due:
     1. Given F(x, y, z) = ⟨u, v, w⟩, express F(x, y, z) ⋅ d(x, y, z) as a differential form.
     2. Given G(x, y) = ⟨M, N⟩, express G(x, y) ⋅ d(x, y) and G(x, y) × d(x, y) as differential forms.
8. Partial derivatives:
   • Date assigned: April 15 Monday.
   • Date due: April 16 Tuesday.
   • Readings from my notes:
     • Page 28 (§§2.5&2.6);
     • Page 31 and through most of page 33 (§§3.4&3.5).
   • Reading from the textbook: Pages 693–702 (§13.3), especially the Examples.
   • Exercises due:
     1. If f is a function of two variables and the partial derivatives of f are D₁f(x, y) = 2y and D₂f(x, y) = 2x, then what is the differential of f(x, y)? (If you're trying to figure out a formula for the function f, then you're doing too much work!)
     2. If u is a variable quantity and the differential of u is du = x² dx + y³ dy, then what are the partial derivatives of u with respect to x and y? (If you're trying to figure out a formula for the quantity u, then you're doing too much work!)
9. Directional derivatives:
   • Date assigned: April 16 Tuesday.
   • Date due: April 17 Wednesday.
   • Readings from the textbook:
     • From page 713 through the end of Example 3 on page 717 (all of §13.5 before Gradients and Tangents to Level Curves);
     • From the bottom of page 718 through page 720 (§13.5: Functions of Three Variables, The Chain Rule for Paths);
     • Figure 15.10 on page 829;
     • The bottom of page 829 and Example 1 on page 830 (§15.2, Gradient Fields).
   • Reading from my notes: The rest of page 33 and through the top of page 35 (§§3.6&3.7).
   • Exercises due: Suppose that ∇f(2, 3) = ⟨3/5, 4/5⟩.
     1. In which direction u is the directional derivative D_uf(2, 3) the greatest?
     2. In which directions u is the directional derivative D_uf(2, 3) equal to zero?
     3. In which direction u is the directional derivative D_uf(2, 3) the least (with a large absolute value but negative)?
10. Matrices and the Chain Rule:
    • Date assigned: April 18 Thursday.
    • Date due: April 22 Monday.
    • Reading from the textbook: Pages 704–710 (§13.4).
    • Reading from my notes: Most of page 35 and the top half of page 36 (§3.8).
    • Exercise due: If u = f(x, y, z) and v = g(x, y, z), then what is the matrix d(u, v)/d(x, y, z)? (Express the entries of this matrix using any notation for partial derivatives.)
11. Tangents:
    • Date assigned: April 22 Monday.
    • Date due: April 23 Tuesday.
    • Readings from the textbook:
      • The rest of pages 717 and 718 (§13.5, Gradients and tangents to level curves);
      • From page 721 through Example 3 on page 723 (§13.6, Tangent planes and normal lines).
    • Reading from my notes: The rest of page 36 and the top half of page 37 (§3.9).
    • Exercises due: Fill in each blank with ‘line’ or ‘plane’.
      1. If ∇f(a, b) exists but is not zero, then f has a tanget ___ and a normal ___ through (a, b).
      2. If ∇f(a, b, c) exists but is not zero, then f has a tanget ___ and a normal ___ through (a, b, c).
12. Local optimization:
    • Date assigned: April 23 Tuesday.
    • Date due: April 24 Wednesday.
    • Reading from my notes: Most of page 40 and page 41 (§3.11).
    • Reading from the textbook: Pages 730–734 (§13.7: introduction; Derivative Tests for Local Extreme Values).
    • Exercises due: Consider a function f of two variables that is defined everywhere. Identify whether f has a local maximum, a local minimum, a saddle, or none of these at a point (a, b) with these characteristics:
      1. If the partial derivatives of f at (a, b) are both negative.
      2. If one of the partial derivatives of f at (a, b) is zero and the other is negative.
      3. If both partial derivatives of f at (a, b) are zero and the Hessian determinant of f at (a, b) is negative.
      4. If both partial derivatives of f at (a, b) are zero, the Hessian determinant of f at (a, b) is positive, and the unmixed second partial derivatives of f at (a, b) are negative.
      5. If both partial derivatives of f at (a, b) are zero, the Hessian determinant of f at (a, b) is positive, the unmixed second partial derivatives of f at (a, b) are positive, and the mixed second partial derivatives of f at (a, b) are negative.
13. Constrained optimization:
    • Date assigned: April 24 Wednesday.
    • Date due: April 25 Thursday.
    • Readings from the textbook:
      • The rest of page 734 and through page 736 (§13.7: Absolute Maxima and Minima on Closed Bounded Regions).
      • Pages 739–746 (§13.8).
    • Exercise due: Suppose that you wish to maximize a continuous function on the region in 3 dimensions defined in rectangular coordinates by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and 0 ≤ z ≤ 1. How many different constrained regions will you have to check? (Hint: One constrained region to check is the 3-dimensional interior, given by the strict inequalities 0 < x < 1, 0 < y < 1, and 0 < z < 1. There are eight constrained regions given entirely by equations, each of which is a 0-dimensional point: x = 0, y = 0, and z = 0; x = 0, y = 0, and z = 1; x = 0, y = 1, and z = 0; x = 0, y = 1, and z = 1; x = 1, y = 0, and z = 0; x = 1, y = 0, and z = 1; x = 1, y = 1, and z = 0; and x = 1, y = 1, and z = 1. You still need to count the constrained regions of intermediate dimension, each of which will be given partially by strict inequalities and partially by equations. Be sure to give the final total including the 9 that I've already mentioned in this hint. A picture may help.)
14. Linearization:
    • Date assigned: April 29 Monday.
    • Date due: April 30 Tuesday.
    • Reading from the textbook: The rest of page 723 and through page 727 (the rest of §13.6).
    • Reading from my notes: The rest of page 37 and through the rest of page 40 (§3.10).
    • Exercises due:
      1. If a function f is to have a good linear approximation in a region, then it's best if its partial derivatives of what order are close to zero in that region? (Its first partial derivatives, its second partial derivatives, its third partial derivatives, or what?)
      2. If (∂u/∂x)_y = −3 and (∂u/∂y)_x = 2, then is the quantity u more or less sensitive to changes in x compared to changes in y?
15. Integration along curves:
    • Date assigned: April 30 Tuesday.
    • Date due: May 1 Wednesday.
    • Reading from my notes: Page 43 and all but the bottom two lines of page 44 (§4 through most of §4.2).
    • Reading from the textbook: From page 830 through Example 7 on page 836 (§15.2: Line Integrals of Vector Fields; Line Integrals with Respect to dx, dy, or dz; Work Done by a Force over a Curve in Space; Flow Integrals and Circulation for Velocity Fields).
    • Exercises due:
      1. If you wish to integrate the vector field F(x, y, z) = ⟨2x, −3x, 4xy⟩ = 2xi − 3xj + 4xyk along a curve in (x, y, z)-space, then what differential form are you integrating?
      2. If you wish to integrate a vector field F along the circle parametrized by (x, y) = (sin t, cos t) for 0 ≤ t ≤ 2π, and if you orient the circle counterclockwise (assuming that the coordinate system is also oriented counterclockwise), then what should be the bounds on your integral in the variable t? (That is, is it ∫^2π₀ F(sin t, cos t) · ⟨cos t, −sin t⟩ dt or ∫⁰_2π F(sin t, cos t) · ⟨cos t, −sin t⟩ dt?)
16. More integration on curves:
    • Date assigned: May 1 Wednesday.
    • Date due: May 6 Monday.
    • Reading from my notes: the rest of page 44 and page 45 (the rest of §4.2 and §4.3).
    • Readings from the textbook:
      • Page 821 and through Example 3 on page 824 (§15.1 through Additivity);
      • The second half of page 825 and page 826 (§15.1: Line Integrals in the Plane);
      • The rest of page 836 and page 837 (§15.2, Flux Across a Simple Closed Plane Curve).
    • Exercises due:
      1. If you wish to integrate a scalar field (that is a function of several variables) on the circle parametrized by (x, y) = (sin t, cos t) for 0 ≤ t ≤ 2π, then what should be the bounds on your integral in the variable t? (That is, is it ∫^2π₀ f(sin t, cos t) dt or ∫⁰_2π f(sin t, cos t) dt?)
      2. If you wish to integrate the vector field F(x, y) = ⟨x², 3⟩ = x²i +3j across a curve in the (x, y)-plane, and if you orient the plane as usual, then what differential form should you integrate along the curve?
17. Double integrals:
    • Date assigned: May 6 Monday.
    • Date due: May 7 Tuesday.
    • Readings from the textbook:
      • Page 755 and most of page 756 (§14.1: introduction; Double Integrals);
      • Page 757 after Figure 14.3 and through page 759 (§14.1: Fubini's Theorem for Calculating Double Integrals);
      • Page 760 and the top half of page 761 (§14.2: introduction; Double Integrals over Bounded, Nonrectangular Regions);
      • The theorem on page 762 (Theorem 14.2);
      • The paragraph before Example 2 on page 763 and through page 766 (§14.2: Example 2; Finding Limits of Integration; Properties of Double Integrals).
    • Reading from my notes: Pages 49 and 50 (§5 through §5.2).
    • Exercises due:
      1. Rewrite ∫^b_a ∫^d_c f(x, y) dy dx as an iterated integral ending with dx dy.
      2. Assuming that f is continuous everywhere, is it possible that these two iterated integrals could evaluate to different results?
18. Systems of inequalities:
    • Date assigned: May 8 Wednesday.
    • Date due: May 9 Thursday.
    • Reading from my notes: page 51 and the top of page 52 (§5.3).
    • Exercises due:
      1. If you wish to integate a function f of two variables on the region R = {x, y | 0 ≤ x ≤ 2, x² ≤ y ≤ 2x}, then what iterated integral do you use?
      2. Given only x² ≤ y ≤ 2x, what equation (or inequality) would you solve to find 0 ≤ x ≤ 2?
19. Triple integrals:
    • Date assigned: Also May 8 Wednesday.
    • Date due: Also May 9 Thursday.
    • Readings from the textbook:
      • Page 779 and the top of page 780 (§14.5: introduction; Triple Integrals);
      • From the bottom of page 780 to the end of Example 3 on page 784 (§14.5: Finding Limits of Integration in the Order dz dy dx);
      • The middle of page 785 (§14.5: Properties of Triple Integrals).
    • Exercise due: In how many ways can you order 3 variables of integration? List them.
20. Areas, volumes, and averages:
    • Date assigned: May 9 Thursday.
    • Date due: May 13 Monday.
    • Readings from the textbook:
      • The bottom of page 756 and the top of page 757 (§14.1: Double Integrals as Volumes);
      • The rest of pages 761–763 (§14.2: the rest of Volumes);
      • Pages 769–772 (§14.3);
      • The rest of page 780 (§14.5: Volume of a Region in Space);
      • The rest of pages 784&785 (§14.5: Average value of a function in space).
    • Exercises due: Suppose that a < b and c < d are four real numbers, that f is a continuous function of two variables whose domain is the rectangle {x, y | a ≤ x ≤ b, c ≤ y ≤ d}, and that f takes only positive values. Write down expressions (in terms of a, b, c, d, and f) for the volume under the graph of f:
      1. Using ideas from §14.2, as an iterated double integral in the variables x and y;
      2. Using ideas from §14.5, as an iterated triple integral in the variables x, y, and z.
      (To check: You shouldn't be able to evaluate your answer to #1, because I haven't told you which function f is; however, you should be able to begin evaluating your answer to #2 if you write the variables in an appropriate order, and this should turn it into your answer from #1, after which you shouldn't be able to go any further.)
21. Polar coordinates:
    • Date assigned: May 13 Monday.
    • Date due: May 14 Tuesday.
    • Readings from the textbook:
      • Pages 773–777 (§14.4);
      • Pages 795–802 (§14.7).
    • Exercises due:
      1. Give a formula for the area element in the plane in rectangular coordinates x and y. (Answer: dx dy, or more properly |dx ∧ dy|; either is acceptable, as are dy dx and |dy ∧ dx|.)
      2. Give a formula for the area element in the plane in polar coordinates r and θ.
      3. Give a formula for the volume element in space in rectangular coordinates x, y, and z. (Answer: dx dy dz, or more properly |dx ∧ dy ∧ dz|; either is acceptable, as are the the other orders.)
      4. Give a formula for the volume element in space in cylindrical coordinates r, θ, and z.
      5. Give a formula for the volume element in space in spherical coordinates ρ, φ, and θ (using the U.S. mathematicians' convention for which of these is which).
22. Coordinate transformations:
    • Date assigned: May 15 Wednesday.
    • Date due: May 16 Thursday.
    • Reading from my notes: The rest of page 52 and through page 57 (§§5.4–5.7).
    • Reading from the textbook: Pages 806–814 (§14.8).
    • Exercise due: If x = f(u, v) and y = g(u, v), where f and g are continuously differentiable everywhere, then write the area element dx dy (which is more properly written |dx ∧ dy|) in terms of u, v, their differentials, and the partial derivatives of f and g (which you can also think of as the partial derivatives of x and y with respect to u and v). (There are formulas in both the notes and the textbook that you can use, or you can work it out from first principles using the more proper expression involving dx and dy given above. You may use any correct formula, as long as it explicitly uses partial derivatives as directed, rather than some more sophisticated notation instead.)
23. Parametrized surfaces:
    • Date assigned: May 16 Thursday.
    • Date due: May 20 Monday.
    • Readings from the textbook:
      • Pages 632–635 (§11.6);
      • Page 863 and through Example 3 page 865 (§15.5: introduction; Parametrizations of Surfaces).
    • Reading from my notes: Page 59 and the top half of page 60 (§6 through §6.2).
    • Exercises due:
      1. Write down a parametrization of the sphere x² + y² + z² = 1 using the spherical coordinates φ and θ (using the U.S. mathematicians' convention for which of these is which).
      2. Write down a parametrization of the portion of the cone x² + y² = z² where 0 ≤ z ≤ 1 using cylindrical coordinates (either z and θ or r and θ).
24. Integrals across surfaces:
    • Date assigned: May 20 Monday.
    • Date due: May 21 Tuesday.
    • Reading from the textbook:
      • The middle of page 874 (§15.6: introduction);
      • Most of page 878 and through Example 6 on page 881 (§15.6: Orientation of a Surface; Surface Integrals of Vector Fields).
    • Reading from my notes: The rest of page 60 and through most of page 62 (§§6.3–6.5).
    • Exercises due:
      1. If you parametrize a closed surface containing the origin using the spherical coordinates φ and θ (using the U.S. mathematicians' convention for which of these is which) and orient (by which I technically mean pseudoorient) this surface outwards, then (using the right-hand rule in a right-handed coordinate system to interpret this as an honest orientation) does this orientation correspond to increasing φ followed by increasing θ (that is dφ ∧ dθ) or to increasing θ followed by increasing φ (that is dθ ∧ dφ)?
      2. Write down a formula for the pseudooriented surface element dS = n dσ on a parametrized surface in terms of the coordinates (x, y, z) and/or the parameters (u, v) and their differentials and/or partial derivatives. (There are multiple correct answers to this throughout the readings from the textbook and the handout.)
25. Integrals on surfaces:
    • Date assigned: May 21 Tuesday.
    • Date due: May 22 Wednesday.
    • Reading from my notes: The rest of page 62 and page 63 (§6.6).
    • Readings from the textbook:
      • The rest of page 865 and through page 871 (§15.5: Surface Area; Implicit Surfaces);
      • The rest of page 874 through the end of Example 4 on page 878 (§15.6: Surface Integrals).
    • Exercises due:
      1. Write down a formula for the surface area element dσ = |dS| on a parametrized surface in terms of the coordinates (x, y, z) and/or the parameters (u, v) and their differentials and/or partial derivatives. (There are multiple correct answers to this throughout the readings from the textbook and the handout.)
      2. If f is a continuous function of two variables with a compact domain R, write down a double integral for the surface area of the graph of f, using f and its partial derivatives.
26. Moments:
    • Date assigned: May 22 Wednesday.
    • Date due: May 23 Thursday.
    • Reading from the textbook:
      • Pages 788–793 (§14.6);
      • The rest of pages 824&825 (§15.1: Mass and Moment Calculations);
      • The rest of page 881 through page 883 (§15.6: Moments and Masses of Thin Shells).
    • Exercises due:
      1. Give the formulas for the centre of mass (x̄, ȳ, z̄) of a three-dimensional solid in terms of the total mass M and the moments M_x,y, M_x,z and M_y,z.
      2. Give a formula for the polar moment of inertia I₀ of a two-dimensional plate in terms of the moments of inertia I_x and I_y about the coordinate axes.
27. Conservative vector fields and exact differential forms:
    • Date assigned: May 28 Tuesday.
    • Date due: May 29 Wednesday.
    • Reading from my notes: Pages 46&47 (§4.4).
    • Reading from the textbook: Pages 840–849 (§15.3).
    • Exercises due (true or false):
      1. If f is a differentiable scalar field (a function of several variables), then its gradient, the vector field ∇f, must be conservative.
      2. If u is a differentiable scalar quantity, then its differential, the differential form du, must be exact.
      3. If F is a conservative vector field in two dimensions, then the differential form F(x, y) ⋅ d(x, y) must be exact.
      4. If F is a vector field in two dimensions and the differential form F(x, y) ⋅ d(x, y) is exact, then F must be conservative.
28. Green's Theorem:
    • Date assigned: May 29 Wednesday.
    • Date due: May 30 Thursday.
    • Reading from my notes: From page 65 through the top of page 70 (§7 through §7.3).
    • Reading from the textbook: Pages 851–861 (§15.4).
    • Exercise due: Write down as many different versions of the general statement of Green's Theorem as you can think of. (There are some in both the textbook and the handout. I'll give full credit for at least two that are different beyond a trivial change in notation, but there are really more than that.)
29. Stokes's Theorem:
    • Date assigned: May 30 Thursday.
    • Date due: June 3 Monday.
    • Reading from my notes: The rest of page 70 and the very top of page 71 (§7.4).
    • Reading from the textbook: Pages 885–895 (§15.7).
    • Exercises due:
      1. Suppose that you have a compact surface in 3-dimensional space, the z-axis passes through this surface, the surface is oriented (by which I really mean pseudooriented) so that z is increasing along the z-axis through the surface, and you orient the boundary of this surface using the right-hand rule in a right-handed coordinate system as usual. Is the cylindrical coordinate θ increasing or decreasing overall along the boundary curve?
      2. Given f(x, y, z) = 2x³y² cos(e^sin(z)), what is ∇ × ∇f, the curl of the gradient of f? (Hint: If you're doing a bunch of calculations, then you're making this too hard.)
30. Gauss's Theorem:
    • Date assigned: June 3 Monday.
    • Date due: June 4 Tuesday.
    • Reading from my notes: The rest of page 71 (§7.5).
    • Reading from the textbook: Pages 897–906 (§15.8).
    • Exercises due:
      1. Suppose that you have a compact region in 3-dimensional space, the origin lies within this region, and you orient (by which I really mean pseudoorient) the boundary as usual. Is the spherical coordinate ρ increasing or decreasing overall through the boundary surface?
      2. Given F(x, y, z) = ⟨2x³y², cos(e^sin(z)), sin(e^cos(z))⟩ = 2x³y²i + cos(e^sin(z))j + sin(e^cos(z))k, what is ∇ ⋅ ∇ × F, the divergence of the curl of F? (Hint: If you're doing a bunch of calculations, then you're making this too hard.)

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