Readings and homework

Here are the assigned readings and exercises (undated assignments are tentative):

1. Vectors:
   • Date assigned: March 30 Wednesday.
   • Date due: March 31 Thursday.
   • Textbook: Review §§11.1–5.
   • Handout: Vectors (optional).
   • Online notes: Vector operations.
   • Problems 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: March 31 Thursday.
   • Date due: April 4 Monday.
   • Textbook: Pages 649–655 (§12.1)
   • Handout: Point- and vector-valued functions, first three paragraphs.
   • Problems due:
     • 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?
     • 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. Integration of vector-valued functions:
   • Date assigned: April 4 Monday.
   • Date due: April 5 Tuesday.
   • Textbook: Pages 657–661 (§12.2).
   • Handout: Point- and vector-valued functions, the remainder.
   • Problem due: 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?
4. Arclength:
   • Date assigned: April 5 Tuesday.
   • Date due: April 6 Wednesday.
   • Textbook: Pages 664–667 (§12.3).
   • Online notes: Arclength.
   • 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 = ___.
5. Functions of several variables:
   • Date assigned: April 7 Thursday.
   • Date due: April 11 Monday.
   • 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?
6. Limits and continuity:
   • Date assigned: April 11 Monday.
   • Date due: April 12 Tuesday.
   • Textbook: Pages 694–700 (§13.2).
   • Handout: Definitions for functions of several variables, through the top of page 3.
   • 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)?
7. Vector fields and differential forms:
   • Date assigned: April 12 Tuesday.
   • Date due: April 13 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)).
8. Partial derivatives:
   • Date assigned: April 14 Thursday.
   • Date due: April 18 Monday.
   • Textbook:
     • Pages 703–711 (§13.3), especially the Examples;
     • Skim pages 714–720 (§13.4).
   • Handouts:
     • Definitions for functions of several variables, the rest of 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 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 18 Monday.
   • Date due: April 19 Tuesday.
   • Textbook:
     • From page 723 through the top of page 727, then from the bottom of page 728 through page 729 (most of §13.5);
     • The two line in the middle of page 836, then the top half of page 837 (§15.2, Gradients).
   • Online notes: Differentials, gradients, and partial derivatives.
   • 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)?
10. Tangents:
    • Date assigned: April 19 Tuesday.
    • Date due: April 20 Wednesday.
    • Textbook:
      • The rest of pages 727&728 (§13.5, Gradients and tangents to level curves);
      • From page 730 through 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).
11. Linearization:
    • Date assigned: April 21 Thursday.
    • Date due: April 25 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?
12. Optimization:
    • Date assigned: April 25 Monday.
    • Date due: April 26 Tuesday.
    • 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.
13. Constraints:
    • Date assigned: April 26 Tuesday.
    • Date due: April 27 Wednesday.
    • Textbook: Pages 748–755 (§13.8).
    • Online notes: Lagrange multipliers.
    • Problem 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, and what are they? (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 find the constrained regions of intermediate dimension, each of which will be given partially by strict inequalities and partially by equations.)
14. Integration along curves:
    • Date assigned: April 28 Thursday.
    • Date due: May 2 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?)
15. More integration on curves:
    • Date assigned: May 2 Monday.
    • Date due: May 3 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?
16. Conservative vector fields and exact differential forms:
    • Date assigned: May 3 Tuesday.
    • Date due: May 4 Wednesday.
    • Textbook: Pages 847–856.
    • Handout: The Fundamental Theorem of Calculus.
    • 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.
17. Double integrals:
    • Date assigned: May 5 Thursday.
    • Date due: May 9 Monday.
    • 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.
18. Triple integrals:
    • Date assigned: May 9 Monday.
    • Date due: May 10 Tuesday.
    • 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.
19. Areas, volumes, and averages:
    • Date assigned: May 10 Tuesday.
    • Date due: May 11 Wednesday.
    • 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.
      (To check: You should not be able to evaluate the integral in #1 without knowing which function f is; you should be able to evaluate the integral in #2 partly, and this will turn it into the integral from #1.)
20. Moments:
    • Date assigned: May 12 Thursday.
    • Date due: May 16 Monday.
    • Textbook: Pages 795–800 (§14.6).
    • Problems 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 moment of inertia I₀ of a two-dimensional plate in terms of the moments of inertia I_x and I_y.
21. Coordinate transformations:
    • Date assigned: May 16 Monday.
    • Date due: May 17 Tuesday.
    • Textbook: Pages 814–821 (§14.8).
    • Handout: Change of variables in multiple integrals.
    • Problem 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. (There is a formula in the textbook that you can use, or you can work it out systematically from the more proper expression involving dx and dy given above.)
22. Polar coordinates:
    • Date assigned: May 17 Tuesday.
    • Date due: May 18 Wednesday.
    • Textbook:
      • Pages 780–784 (§14.4);
      • Pages 802–810 (§14.7);
    • 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.
23. Surfaces:
    • Date assigned: May 19 Thursday.
    • Date due: May 23 Monday.
    • Textbook:
      • Pages 638–641 (§11.6);
      • 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).
    • 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 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 23 Monday.
    • Date due: May 25 Wednesday.
    • Textbook: From pages 883 to the middle of page 885 (§15.6, Orientation and Surface integral for flux).
    • Handout: The wedge product, pages 1–4.
    • Problems due:
      1. If you parametrize a closed surface containing the origin using the spherical coordinates φ and θ and orient (by which I technically mean pseudoorient) this surface outwards, then (using the right-hand rule to interpret this as an honest orientation) does this orientation correspond to dφ ∧ dθ (that is increasing φ followed by increasing θ) or to dθ ∧ dφ (that is increasing θ followed by increasing φ)?
      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 23 Monday.
    • Date due: May 25 Wednesday.
    • 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, page 5.
    • 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 and/or partial derivatives. (There are multiple correct answers to this throughout the readings from the textbook and the handout.)
26. Green's Theorem:
    • Date assigned: May 26 Thursday.
    • Date due: May 31 Tuesday.
    • Textbook: Pages 858–867 (§15.4).
    • Handout: The Stokes theorems, pages 1–5.
    • Problem 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.)
27. Stokes's Theorem:
    • Date assigned: May 31 Tuesday.
    • Date due: June 1 Wednesday.
    • Textbook: Pages 889–898 (§15.7).
    • Handout: The Stokes theorems, the bottom of page 5 and most of page 6.
    • 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.)
28. Gauss's Theorem:
    • Date assigned: June 1 Wednesday.
    • Date due: June 2 Thursday.
    • Textbook: Pages 900–909 (§15.8).
    • Handout: The Stokes theorems, the bottom of page 6 and page 7.
    • 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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