Readings and homework

Here are the assigned readings and exercises:

1. Vectors:
   • Date assigned: January 6 Wednesday.
   • Date due: January 7 Thursday.
   • Textbook: Review §§11.1–5.
   • Online notes: Vector operations.
   • Problems due: none (this time only).
2. Parametrized curves:
   • Date assigned: January 7 Thursday.
   • Date due: January 11 Monday.
   • Textbook:
     • Pages 649–655 (§12.1);
     • Pages 657–661 (§12.2).
   • Online notes: Point- and vector-valued functions.
   • Problems due:
     1. If r is a point-valued function, so that P = r(t) is a point (for each scalar value of t), then what type of value does its derivative r′ take?; that is, is dP/dt = r′(t) a point, a scalar, a vector, or what?
     2. If r is a vector-valued function, so that v = r(t) is a vector (for each scalar value of t), then what type of values do its definite integrals have?; that is, is ∫^b_t=a r(t) dt a point, a scalar, a vector, or what?
3. Functions of several variables:
   • Date assigned: January 11 Monday.
   • Date due: January 12 Tuesday.
   • Textbook: Pages 686–691 (§13.1).
   • Online notes: The hierarchy of functions and relations.
   • Problems 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?
4. Limits and continuity:
   • Date assigned: January 14 Thursday.
   • Date due: January 19 Tuesday.
   • Textbook: Pages 694–700 (§13.2).
   • Handout: Definitions for functions of several variables, pages 1&2.
   • Problems 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. What is the limit of f + g approaching (2, 3)? (Hint: Look at Theorem 1.)
     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. What is the limit of f approaching (0, 0)?
5. Vector fields and differential forms:
   • Date assigned: January 19 Tuesday.
   • Date due: January 20 Wednesday.
   • Textbook: Pages 835&836 (§15.2, Vector Fields).
   • Handout: Differential 1-forms.
   • Problems due:
     1. Given u = ⟨2x, −3x, 4xy⟩ = 2xi − 3xj + 4xyk, express u ⋅ dr as a differential form (using r = (x, y, z)).
     2. Given v = ⟨x², 3⟩ = x²i +3j, express v ⋅ dr and v × dr as differential forms (using r = (x, y)).
6. Partial derivatives:
   • Date assigned: January 21 Thursday.
   • Date due: January 25 Monday.
   • Textbook: Pages 703–711 (§13.3), especially the Examples.
   • Handouts:
     • Definitions for functions of several variables, page 3;
     • Differentials.
   • Problems due:
     1. If f is a function of two variables and the partial derivatives of f are D₁f(x, y) = ∂f(x, y)/∂x = 2y and D₂f(x, y) = ∂f(x, y)/∂y = 2x, then what is the differential of u = f(x, y)?
     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?
7. Directional derivatives:
   • Date assigned: January 25 Monday.
   • Date due: January 26 Tuesday.
   • Textbook:
     • Skim pages 714–720 (§13.4);
     • Read from page 723 to the top of page 727, then from the bottom of page 728 to page 729 (most of §13.5);
     • Read the middle two lines of page 836 and the top half of page 837 (§15.2, Gradients).
   • Problems due: Suppose that ∇f(2, 3) = ⟨3, 4⟩ = 3i + 4j.
     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)?
8. Tangents:
   • Date assigned: January 26 Tuesday.
   • Date due: January 27 Wednesday.
   • Textbook:
     • The rest of pages 727&728 (§13.5, Gradients and tangents to level curves),
     • From page 730 to the top of page 733 (§13.6, Tangent planes and normal lines).
   • Online notes: Tangents and normal lines.
   • Problems 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).
9. Linearization:
   • Date assigned: January 28 Thursday.
   • Date due: February 1 Monday.
   • Textbook: From the bottom of page 733 to page 737 (most of the rest of §13.6).
   • Handout: Taylor's Theorem in several variables.
   • Problems 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 = −3 and ∂u/∂y = 2, then is the quantity u sensitive more to changes in x or to changes in y?
10. Optimization:
    • Date assigned: February 1 Monday.
    • Date due: February 4 Thursday.
    • Textbook: Pages 740–745 (§13.7).
    • Online notes: Optimization.
    • Problems 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.
11. Lagrange multipliers:
    • Date assigned: February 4 Thursday.
    • Date due: February 8 Monday.
    • Textbook: Pages 748–755 (§13.8).
    • Online notes: Review Optimization.
    • Problem due: Suppose that you wish to maximize a continuous function on the region in 3 dimensions defined in rectangular coordinates by x, y, z ≥ 0 and x + y + z ≤ 1. How many different constrained regions will you have to check, and what are they? (Hint: One constrained region to check is the 3-dimensional interior, given by x, y, z > 0 and x + y + z < 1, where the constraints are given entirely by strict inequalities. Each of four other constrained regions to check is a 0-dimensional point, one given by x, y, z = 0, one given by x, y = 0 and z = 1, one given by x, z = 0 and y = 1, and one given by x = 1 and y, z = 0; in each of these, the constraints are given entirely by equations. You still need to find the constrained regions of intermediate dimension, each of which will be given partially by strict inequalities and partially by equations.)
12. Double integrals:
    • Date assigned: February 8 Monday.
    • Date due: February 9 Tuesday.
    • Textbook:
      • Pages 763–767 (§14.1);
      • From page 768 to the top of page 769, Theorem 14.2 on page 770, pages 771–774 (most of §14.2).
    • Problems due: Problems 14.2.81&82 from page 776.
13. Triple integrals:
    • Date assigned: February 9 Tuesday.
    • Date due: February 10 Wednesday.
    • Textbook: From page 786 through all but the bottom of page 790, the very middle of page 791 (most of §14.5).
    • Online notes: Fubini theorems.
    • Problem due: In how many ways can you order 3 variables of integration? List them.
14. Areas and volumes:
    • Date assigned: February 11 Thursday.
    • Date due: February 15 Monday.
    • Textbook:
      • Pages 769&770 (§14.2, Volumes);
      • Pages 777–779 (§14.3);
      • The bottom of page 791 and the top of page 792 (§14.5, Average value of a function in space).
    • Problems 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.
      Then evaluate these integrals as far as you can without knowing what f is, and check that they give the same result.
15. Moments:
    • Date assigned: February 15 Monday.
    • Date due: February 16 Tuesday.
    • Textbook: Pages 795–800 (§14.6).
    • Problems due:
      1. Give the formulas for the centre of mass (x̄, #x0233;, 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 moment of inertia I₀ of a two-dimensional plate in terms of the moments of inertia I_x and I_y.
16. Polar coordinates:
    • Date assigned: February 16 Tuesday.
    • Date due: February 17 Wednesday.
    • Textbook:
      • Pages 780–784 (§14.4);
      • Pages 802–810 (§14.7);
      • Optional: Pages 814–821 (§14.8).
    • Handout: Change of variables in multiple integrals (optional).
    • Problems due:
      1. Give a formula for the area element in the plane in rectangular coordinates x and y.
      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.
      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, θ, and φ.
17. Integration along curves:
    • Date assigned: February 18 Thursday.
    • Date due: February 22 Monday.
    • Textbook: Pages 837–842 (most of §15.2).
    • Handout: Integration on curves, pages 1 and 2.
    • Problems 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 differential form or a vector field along the circle parametrized by (x, y) = (cos t, sin t) for 0 ≤ t ≤ 2π, and if you orient the circle counterclockwise, then what should be the bounds on your integral in the variable t? (That is, is it ∫₀^2π or ∫_2π⁰?)
18. More integration on curves:
    • Date assigned: February 22 Monday.
    • Date due: February 23 Tuesday.
    • Textbook:
      • Pages 828–832 (§15.1);
      • From the bottom of page 842 to page 844 (§15.2, Flux across a simple plane curve).
    • Handout: Integration on curves, the very bottom of page 2 and page 3.
    • Problems due:
      1. If you wish to integrate a scalar field (that is a function of several variables) on the circle parametrized by (x, y) = (cos t, sin t) for 0 ≤ t ≤ 2π, then what should be the bounds on your integral in the variable t? (That is, is it ∫₀^2π or ∫_2π⁰?)
      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 counterclockwise as usual, then what differential form are you integrating?
19. Arclength:
    • Date assigned: February 23 Tuesday.
    • Date due: February 24 Wednesday.
    • Textbook: Pages 664–667 (§12.3).
    • Problems due: This section 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 = ___.
20. Integrals across surfaces:
    • Date assigned: February 25 Thursday.
    • Date due: February 29 Monday.
    • Textbook:
      • Page 870 and the examples on page 871 (§15.5, Parametrizations of surfaces);
      • The middle of page 875 (§15.5, the beginning of Implicit surfaces);
      • From pages 883 to the middle of page 885 (§15.6, Orientation and Surface integral for flux).
    • Handout: The wedge product, from page 1 through the last full paragraph on page 3.
    • Problems due:
      1. Write down a parametrization of the sphere x² + y² + z² = 1 using the spherical coordinates φ and θ.
      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. (There are multiple correct answers to this throughout the readings from the textbook and the handout.)
21. Integrals on surfaces:
    • Date assigned: February 29 Monday.
    • Date due: March 1 Tuesday.
    • Textbook:
      • Pages 871–877 (most of §15.5);
      • From page 880 through the top of page 883 (§15.6, Surface integrals);
      • From the middle of page 885 through page 887 (§15.6, Moments and masses of thin shells).
    • Handout: The wedge product, the very bottom of page 3 and page 4.
    • Problem due: 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. (There are multiple correct answers to this throughout the readings from the textbook and the handout.)
22. Conservative vector fields and exact differential forms:
    • Date assigned: March 1 Tuesday.
    • Date due: March 2 Wednesday.
    • Textbook: Pages 847–856.
    • Handout: Exterior differential forms, page 4 (except for the top part).
    • Problems due (true or false):
      1. If f is a scalar field (a function of several variables), then its gradient, the vector field ∇f, must be conservative.
      2. If u is a 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.
23. Green's Theorem:
    • Date assigned: March 3 Thursday.
    • Date due: March 7 Monday.
    • Textbook: Pages 858–867 (§15.4).
    • Handout: Exterior differential forms, page 5 and the top of page 6.
    • Problem due: Write down as many different forms of the general statement of Green's Theorem as you can think of.
24. Stokes's Theorem:
    • Date assigned: March 7 Monday.
    • Date due: March 8 Tuesday.
    • Textbook: Pages 889–898 (§15.7).
    • Handout: Exterior differential forms, the bottom of page 5 (again).
    • Problems 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 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.)
25. Gauss's Theorem:
    • Date assigned: March 8 Tuesday.
    • Date due: March 9 Wednesday.
    • Textbook: Pages 900–909 (§15.8).
    • Handout: Exterior differential forms, page 6.
    • Problems 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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