Readings and homework

Here are the assigned readings and problems (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).

1. Vectors:
   • Date assigned: January 5 Thursday.
   • Date due: January 9 Monday.
   • Reading from the textbook: Review §§11.1–11.5 (as needed).
   • Handout: Vectors (optional).
   • Online notes: Vector operations (required).
   • 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: January 9 Monday.
   • Date due: January 10 Tuesday.
   • Reading from the textbook:
     • Pages 642–648 (§12.1);
     • From page 650 through Example 3 on page 652 (§12.2: introduction, Integrals of Vector Functions).
   • Handout: Point- and vector-valued functions through the end of the paragraph spanning pages 1 and 2.
   • Problems 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. 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. Applications of curves:
   • Date assigned: January 10 Tuesday.
   • Date due: January 11 Wednesday.
   • Reading from the textbook:
     • Pages 652–654 (§12.2: The Vector and Parametric Equations for Ideal Projectile Motion);
     • Page 656–659 (§12.3).
   • Handout: Point- and vector-valued functions, pages 2 and 3.
   • Problems 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 = ___.
4. Functions of several variables:
   • Date assigned: January 12 Thursday.
   • Date due: January 18 Wednesday.
   • Reading from the textbook: Pages 676–681 (§13.1).
   • Handout: Functions of several variables, through the top of page 3.
   • 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?
5. Limits and continuity:
   • Date assigned: January 18 Wednesday.
   • Date due: January 19 Thursday.
   • Reading from the textbook: Pages 684–690 (§13.2).
   • Handout: Functions of several variables, the rest of page 3 through page 5.
   • 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 (if anything) 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 (if anything) is the limit of f approaching (0, 0)?
6. Vector fields and differential forms:
   • Date assigned: January 18 Wednesday.
   • Date due: January 19 Thursday.
   • Handout: Differentials and differential forms, pages 1&2.
   • Reading from the textbook: Page 828 and most of page 829 (§15.2, Vector Fields).
   • Online notes: Examples of vector fields.
   • Problems due:
     1. Given u = ⟨M, N, O⟩, express u ⋅ dr as a differential form (using r = (x, y, z)).
     2. Given v = ⟨M, N⟩, express v ⋅ dr and v × dr as differential forms (using r = (x, y)).
7. Partial derivatives:
   • Date assigned: January 23 Monday.
   • Date due: January 24 Tuesday.
   • Handouts:
     • Functions of several variables, page 6;
     • Differentials and differential forms, pages 3 and 4 and most of page 5.
   • Reading from the textbook:
     • Pages 693–702 (§13.3), especially the Examples;
     • Skim pages 704–710 (§13.4).
   • Problems 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!)
8. Directional derivatives:
   • Date assigned: January 24 Tuesday.
   • Date due: January 25 Wednesday.
   • Reading from the textbook:
     • From page 713 through the end of Example 3 on page 717 (all of §13.5 except 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);
     • The bottom of page 829 and Example 1 on page 830 (§15.2, Gradient Fields).
   • Handout: Differentials and differential forms, the rest of page 5, page 6, and the top of page 7.
   • Problems 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)?
9. Tangents:
   • Date assigned: January 25 Wednesday.
   • Date due: January 26 Thursday.
   • Reading 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).
   • Handout: Differentials and differential forms, the rest of page 7 and page 8.
   • 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).
10. Linearization:
    • Date assigned: January 30 Monday.
    • Date due: January 31 Tuesday.
    • Reading from the textbook: The rest of page 723 and through page 727 (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)_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?
11. Optimization:
    • Date assigned: January 31 Tuesday.
    • Date due: February 1 Wednesday.
    • Handout: Optimization.
    • Reading from the textbook: Pages 730–736 (§13.7).
    • 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.
12. Constraints:
    • Date assigned: February 1 Wednesday.
    • Date due: February 2 Thursday.
    • Reading from the textbook: Pages 739–746 (§13.8).
    • 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.)
13. Integration along curves:
    • Date assigned: February 6 Monday.
    • Date due: February 7 Tuesday.
    • Handout: Integration on curves, pages 1 and 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).
    • 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 clockwise, then what should be the bounds on your integral in the variable t? (That is, is it ∫₀^2π or ∫_2π⁰?)
14. More integration on curves:
    • Date assigned: February 7 Tuesday.
    • Date due: February 8 Wednesday.
    • Handout: Integration on curves, the very bottom of page 2 and page 3.
    • Reading from the textbook:
      • Pages 821–826 (§15.1);
      • the rest of page 836 and page 837 (§15.2, Flux Across a Simple Closed Plane Curve).
    • 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 should you integrate along the curve?
15. Conservative vector fields and exact differential forms:
    • Date assigned: February 8 Wednesday.
    • Date due: February 9 Thursday.
    • Handout: Integration on curves, pages 4 and 5.
    • Reading from the textbook: Pages 840–849 (§15.3).
    • 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.
16. Double integrals:
    • Date assigned: February 13 Monday.
    • Date due: February 14 Tuesday.
    • Reading from the textbook:
      • Pages 755–759 (§14.1);
      • Page 760 and the top half of page 761 (§14.2: introduction);
      • The theorem on page 762 (Theorem 14.2);
      • From the paragraph before Example 2 on page 763 through page 766 (§14.2: Example 2; Finding Limits of Integration; Properties of Double Integrals).
    • Problems due: Exercises 81&82 from Section 14.2 on page 769.
17. Triple integrals:
    • Date assigned: February 14 Tuesday.
    • Date due: February 15 Wednesday.
    • Reading 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).
    • Problem due: In how many ways can you order 3 variables of integration? List them.
18. Areas, volumes, and averages:
    • Date assigned: February 15 Wednesday.
    • Date due: February 16 Thursday.
    • Handout: Fubini theorems.
    • Reading from the textbook:
      • Pages 761–763 (§14.2: the rest of Volumes);
      • Pages 769–772 (§14.3);
      • The middle of page 780 (§14.5: Volume of a Region in Space);
      • The bottom of page 784 and page 785 (§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 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.)
19. Moments:
    • Date assigned: February 20 Monday.
    • Date due: February 21 Tuesday.
    • Reading from the textbook: Pages 788–793 (§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 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.
20. Coordinate transformations:
    • Date assigned: February 21 Tuesday.
    • Date due: February 22 Wednesday.
    • Handout: Change of variables in multiple integrals, pages 1–4.
    • Reading from the textbook: Pages 806–814 (§14.8).
    • 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 are formulas in the handout and the textbook that you can use, or you can work it out using the more proper expression involving dx and dy given above.)
21. Polar coordinates:
    • Date assigned: February 22 Wednesday.
    • Date due: February 23 Thursday.
    • Handout: Change of variables in multiple integrals, page 5.
    • Reading from the textbook:
      • Pages 773–777 (§14.4), especially the Examples;
      • Pages 795–802 (§14.7), especially the Examples.
    • 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.
22. Surfaces:
    • Date assigned: February 27 Monday.
    • Date due: February 28 Tuesday.
    • Reading from the textbook:
      • Pages 632–635 (§11.6);
      • Page 863 and through Example 3 page 865 (§15.5: introduction; Parametrizations of 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).
23. Integrals across surfaces:
    • Date assigned: February 28 Tuesday.
    • Date due: March 1 Wednesday.
    • Reading from the textbook: Most of page 878 and through Example 6 on page 881 (§15.6: Orientation of a Surface; Surface Integrals of Vector Fields).
    • Handout: Integration on surfaces, pages 1–4.
    • Problems 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 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.)
24. Integrals on surfaces:
    • Date assigned: March 1 Wednesday.
    • Date due: March 2 Thursday.
    • Handout: Integration on surfaces, the very bottom of page 4 and all of page 5.
    • Reading from the textbook:
      • The rest of page 865 and through page 871 (§15.5: Surface Area; Implicit Surfaces);
      • From page 874 through the end of Example 4 on page 878 (§15.6: introduction; Surface Integrals);
      • The rest of page 881 through page 883 (§15.6: Moments and Masses of Thin Shells).
    • 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.)
25. Green's Theorem:
    • Date assigned: March 6 Monday.
    • Date due: March 7 Tuesday.
    • Handout: The Stokes theorems, from page 1 through the top of page 6.
    • Reading from the textbook: Pages 851–861 (§15.4).
    • 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.)
26. Stokes's Theorem:
    • Date assigned: March 7 Tuesday.
    • Date due: March 8 Wednesday.
    • Handout: The Stokes theorems, the rest of page 6 and the top of page 7.
    • Reading from the textbook: Pages 885–895 (§15.7).
    • 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.)
27. Gauss's Theorem:
    • Date assigned: March 8 Wednesday.
    • Date due: March 9 Thursday.
    • Handout: The Stokes theorems, the rest of page 7.
    • Reading from the textbook: Pages 897–906 (§15.8).
    • 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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