MATH-2080-ES31&WBP01

Course administration:

• Canvas pages (where you must log in):
  • face-to-face (Section ES31),
  • online (Section WBP01).
• Help with DjVu (if you have trouble reading the files on this page).
• Course policies (TBA).
• Face-to-face class hours: Mondays through Thursdays from 9:00 to 9:50 in ESQ 203.
• Face-to-face class Zoom meeting: 999-4907-1395.
• Face-to-face final exam time: December 16 Wednesday from 9:00 to 10:40 in ESQ 203.

• Name: Toby Bartels, PhD.
• Canvas messages.
• Email: TBartels@Southeast.edu.
• Voice mail: 1-402-323-3452.
• Text messages: 1-402-805-3021.
• Virtual office hours:
  • Mondays and Wednesdays from 1:00 PM to 2:00,
  • Tuesdays and Thursdays from 10:30 to 12:00, and
  • by appointment.

Reading homework

1. General review:
   • Reading from the textbook: As needed: Review §§11.1–11.5.
   • Reading from my notes: Optional: Chapter 1 (through page 17).
   • Online notes: Required: Vector operations.
   • Exercises due on August 25 Tuesday:
     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?
   • Exercises from the textbook due on August 26 Wednesday: 11.2.5, 11.3.1, 11.4.1, 11.4.15, 11.5.23, 11.5.39.
2. Parametrized curves:
   • Reading from the textbook: Chapter 12 through Section 12.1 (pages 662–668).
   • Reading from my notes: Chapter 2 through the first half of Section 2.1 (all of page 19 and the first two lines of page 20).
   • Exercises due on August 26 Wednesday:
     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?
   • Exercises from the textbook due on August 27 Thursday: 12.1.5, 12.1.7, 12.1.9, 12.1.11, 12.1.15, 12.1.17, 12.1.19, 12.1.21, 12.1.23, 12.1.24, 12.1.37.
3. Standard parametrizations:
   • Reading: Online notes.
   • Exercises due on August 27 Thursday:
     1. For the oriented line segment from (x₁, y₁) to (x₂, y₂), write down the usual parametrization.
     2. More generally, for the oriented line segment from P₁ to P₂, write down the usual parametrization.
     3. For the circle in the 2-dimensional plane whose centre is (h, k) and whose radius is r, write down the usual parametrization.
     4. If f is continuous function whose domain is [a, b], write down write down the usual parametrization for the graph of f.
   • Exercises from the textbook due on August 31 Monday: 15.1.1, 15.1,3, 15.1.5, 15.1.7.
4. Integrating parametrized curves:
   • Reading from the textbook: Section 12.2 (pages 671–675).
   • Reading from my notes: The second half of Section 2.1 (the rest of page 20).
   • Exercises due on August 31 Monday: 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?
   • Exercises from the textbook due on September 1 Tuesday: 12.2.1, 12.2.3, 12.2.11, 12.2.17, 12.2.21, 12.2.25, 12.2.26.
5. Arclength:
   • Reading from the textbook: Section 12.3 (pages 678–680).
   • Reading from my notes: Section 2.4 (pages 23&24).
   • Exercises due on September 1 Tuesday: 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 = ___.
   • Exercises from the textbook due on September 2 Wednesday: 12.3.1, 12.3.5, 12.3.8, 12.3.9, 12.3.11, 12.3.14, 12.3.18.
6. Functions of several variables:
   • Reading from my notes: Chapter 3 through Section 3.1 (pages 27–29).
   • Reading from the textbook:
     • Chapter 13 through Section 13.1 "Domains and Ranges" (pages 697&698);
     • Section 13.1 "Graphs, Level Curves, and Contours of Functions of Two Variables" through the end of Section 13.1 (pages 700–702).
   • Exercises due September 2 Wednesday:
     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?
   • Exercises from the textbook due on September 3 Thursday: 13.1.3, 13.1.5, 13.1.6, 13.1.8, 13.1.11, 13.1.14, 13.1.16, 13.1.31, 13.1.33, 13.1.34, 13.1.39, 13.1.41, 13.1.43, 13.1.51, 13.1.53, 13.1.59, 13.1.61.
7. Topology in several variables:
   • Reading from the textbook: Section 13.1 "Functions of Two Variables" (pages 698&699).
   • Reading from my notes: Sections 3.2&3.3 (pages 29&30).
   • Exercises due September 3 Thursday: Let R be a relation, thought of as a set of points in Rⁿ. Recall that a point P is in the boundary (or frontier) of R if, among the points arbitrarily close to P (including P itself), there are both at least one point in R and one point not in R. For each of the following examples, state whether R is open (Yes or No) and whether R is closed (Yes or No):
     1. There is at least one point in the boundary of R, and all of them are in R.
     2. There is at least one point in the boundary of R, and none of them are in R.
     3. There are points in the boundary of R, and at least one of them is in R and at least one of them is not.
     4. There are no points in the boundary of R.
   • Exercises from the textbook due on September 8 Tuesday: 13.1.17, 13.1.19, 13.1.23, 13.1.25, 13.1.27, 13.2.31, 13.2.33, 13.2.35, 13.2.39.
8. Limits in several variables:
   • Section 13.2 (pages 705–711).
   • Reading from my notes: Section 3.4 (pages 30&31).
   • Exercises due on September 8 Tuesday:
     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) = ___).
   • Exercises from the textbook due on September 9 Wednesday: 13.2.1, 13.2.5, 13.2.13, 13.2.17, 13.2.25, 13.2.27, 13.2.43, 13.2.47, 13.2.59.
9. Vector fields:
   • Reading from the textbook: Section 15.2 through "Vector Fields" (pages 854&855 and Figures 15.7–15.16), except Figure 15.11.
   • Online notes: Examples of vector fields.
   • Exercises due on September 9 Wednesday: Sketch a graph of the following vector fields:
     1. F(x, y) = ⟨x, y⟩ = xi + yj;
     2. G(x, y) = ⟨−y, x⟩ = −yi + xj.
   • Exercises from the textbook due on September 10 Thursday: 15.2.5, 15.2.47, 15.2.49, 15.2.51.
10. Linear differential forms:
    • Reading from my notes: Chapter 4 through Section 4.3 (pages 33&34).
    • Exercises due on September 10 Thursday:
      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.
    • Exercises not from the textbook due on September 14 Monday:
      1. Evaluate 3x dx + 4x²y dy at (x, y) = (2, 6) along ⟨dx, dy⟩ = ⟨0.003, 0.005⟩. (Answer.)
      2. Evaluate 2xy dx + 2yz dy + 2xz dz at (x, y, z) = (−1, 3, 2) along ⟨dx, dy, dz⟩ = ⟨0.01, 0.02, −0.01⟩.
      3. Evaluate x² dx + xy dy + xz dz at (x, y, z) = (4, 3, −2). (Answer.)
      4. Evaluate 5x² dx − 3xy dy at (x, y) = (1, 2).
11. Differentials:
    • Readings from my notes: Section 4.4 (pages 35&36).
    • Exercises due on September 14 Monday:
      1. If n is a constant, write a formula for the differential of uⁿ using n, u, and du.
      2. Write the differentials of u + v and uv using u, v, du, and dv.
      3. If e ≈ 2.71828 is the natural base, then write the differential of e^u using e, u, and du.
      4. Write the differential of ln u = log_e u using u and du.
      5. Write the differentials of sin u and cos u using u, du, and trigonometric operations.
    • Exercises not from the textbook due on September 15 Tuesday:
      1. Find the differential of 3x + 5y. (Answer.)
      2. Find the differential of −2x + 6y.
      3. Find d(3p² − 4q − 18). (Answer.)
      4. Find d(2s³ + 5t − 2).
      5. Evaluate d(2xy + 3x²) at (x, y) = (2, 3). (Answer.)
      6. Evaluate d(3xy − 2y²) at (x, y) = (−1, 2).

Differentiation

12. Partial derivatives:
    • Readings from my notes: Section 4.5 (pages 36&37).
    • Reading from the textbook: Section 13.3 through "Functions of More than Two Variables" (pages 714–719).
    • Exercises due on September 15 Tuesday:
      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!)
    • Exercises from the textbook due on September 16 Wednesday: 13.3.1, 13.3.2, 13.3.3, 13.3.9, 13.3.11, 13.3.23, 13.3.25, 13.3.29, 13.3.39, 13.3.63.
13. Levels of differentiability:
    • Readings from my notes: Sections 3.5&3.6 (page 32).
    • Reading from the textbook: The rest of Section 13.3 (pages 719–723).
    • Exercises due on September 16 Wednesday: For each of the following statements about functions on R², state whether it is always true or sometimes false:
      1. If a function is continuous, then it is differentiable.
      2. If a function is differentiable, then it is continuous.
      3. If a function's partial derivatives (defined as limits) all exist, then the function is differentiable.
      4. If a function's partial derivatives (defined as limits) all exist and are continuous, then the function is differentiable.
      5. If a differentiable function's second partial derivatives (defined as limits of the first partial derivatives) all exist, then the mixed partial derivatives are equal.
      6. If a differentiable function's second partial derivatives (defined as limits of the first partial derivatives) all exist and are continuous, then the mixed partial derivatives are equal.
    • Exercises from the textbook due on September 17 Thursday: 13.3.43, 13.3.45, 13.3.61, 13.3.85, 13.3.93, 13.3.101.
14. Directional derivatives:
    • Readings from the textbook:
      • Section 13.5 through "Calculation and Gradients" (pages 736–738);
      • Section 13.5 from "Functions of Three Variables" (pages 742&743).
    • Reading from my notes: Sections 4.6&4.7 (pages 37–39).
    • Exercises due on September 21 Monday: 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)?
    • Exercises from the textbook due on September 22 Tuesday: 13.5.1, 13.5.3, 13.5.5, 13.5.7, 13.5.11, 13.5.13, 13.5.15, 13.5.19, 13.5.23.
15. Gradient vector fields:
    • Readings from the textbook:
      • Section 15.2 Figure 15.11 (page 855);
      • Section 15.2 "Gradient Fields" (pages 855&856).
    • Exercises due on September 22 Tuesday:
      1. If u = f(x, y), where f is a differentiable function of two variables, and du = 2y dx + 2x dy, then what vector field is the gradient of f? That is, ∇f(x, y) = du/d(x, y) = _____.
      2. If v = g(x, y), where g is a differentiable function of two variables, and ∇g(x, y) = ⟨x², y³⟩ = x²i + y³j, then what are the partial derivatives of g? That is, D₁g(x, y) = ∂v/∂x = ___, and D₂g(x, y) = ∂v/∂y = ___.
    • Exercises from the textbook due on September 23 Wednesday: 15.2.1, 15.2.2, 15.2.3, 15.2.4.
16. Matrices:
    • Reading from my notes: Section 1.13 (page 17).
    • Exercises due on September 23 Wednesday: Fill in the blanks with words or short phrases:
      1. Suppose that A and B are matrices. The matrix product AB exists if and only if the number of _____ of A is equal to the the number of _____ of B.
      2. Suppose that v and w are vectors in Rⁿ. Let A be a 1-by-n row matrix whose entries are the components of v, and let B be an n-by-1 column matrix whose entries are the components of w. Then AB is a 1-by-1 matrix whose entry is the _____ of v and w.
    • Exercises from an external website due on September 24 Thursday: Take the Mathopolis quiz on multiplying matrices, and send me a message telling me how it went.
17. The Chain Rule:
    • Reading from the textbook: Section 13.4 (pages 726–733).
    • Reading from my notes: Section 4.8 (pages 39&40).
    • Exercise due on September 24 Thursday: 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.)
    • Exercises from the textbook due on September 28 Monday: 13.4.1, 13.4.3, 13.4.7, 13.4.9, 13.4.41.
18. Tangent flats and normal lines:
    • Readings from the textbook:
      • Section 13.5 "Gradients and Tangents to Level Curves" (pages 740&741);
      • Section 13.6 "Tangent Planes and Normal Lines" (pages 744–746).
    • Reading from my notes: Section 4.9 (pages 40&41).
    • Exercises due on September 28 Monday: 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).
    • Exercises from the textbook due on September 29 Tuesday: 13.5.25, 13.5.27, 13.6.1, 13.6.5, 13.6.11, 13.6.15, 13.6.17.
19. Linearization:
    • Section 13.6 "How to Linearize a Function of Two Variables" (pages 747–749).
    • Section 13.6 "Functions of More Than Two Variables" (pages 750&751).
    • Reading from my notes: Section 4.10 (pages 41–44).
    • Exercises due on September 29 Tuesday: Let f be a function of two variables, and let P₀ = (x₀, y₀) be a point at which f is differentiable.
      1. Write down a formula for the linear approximation of f near P₀; use the gradient ∇f or its components D₁f and D₂f (in addition to f and either P₀ or its coordinates x₀ and y₀).
      2. Suppose that f is infinitely differentiable on a region containing P₀. If the linearization of f near P₀ is to be a good approximation in this region, then what order of partial derivatives of f should be close to zero in that region? (That is, should its first partial derivatives be close to zero, its second partial derivatives, its third partial derivatives, or what?)
    • Exercises from the textbook due on September 30 Wednesday: 13.6.31, 13.6.33, 13.6.35, 13.6.39, 13.6.41.
20. Estimation:
    • Section 13.6 "Estimating Change in a Specific Direction" (page 747).
    • Section 13.6 "Differentials" (pages 749&750).
    • Exercises due on September 30 Wednesday:
      1. If f is a function of two variables and f is differentiable at a point P₀ = (x₀, y₀), then about how much does the value of f change at that point if you move a distance of Δs in the direction of the vector v? (Your answer should involve f or its gradient or partial derivatives, the distance Δs or ds, and v or its length or direction. If you have any other quantity in your answer, then explain how to get it from these.)
      2. If (∂u/∂x)_y = −3 and (∂u/∂y)_x = 2, then is the quantity u more or less sensitive to small changes in x compared to changes in y?
    • Exercises from the textbook due on October 1 Thursday: 13.6.21, 13.6.23, 13.6.51, 13.6.55.
21. Local optimization:
    • Reading from my notes: Section 4.11 (pages 44&45).
    • Section 13.7 through "Derivative Tests for Local Extreme Values" (pages 754–758).
    • Exercises due on October 1 Thursday: Consider a function f of two variables that is infinitely differentiable 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 nonzero.
      2. If one of the partial derivatives of f at (a, b) is zero and the other is nonzero.
      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, and the unmixed second partial derivatives of f at (a, b) are positive.
    • Exercises from the textbook due on October 5 Monday: 13.7.2, 13.7.7, 13.7.9, 13.7.15, 13.7.27, 13.7.43.
22. Constrained optimization:
    • Reading from the textbook: The rest of Section 13.7 (pages 758–760).
    • Exercise due on October 5 Monday: 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 this triple of strict inequalities: (0 < x < 1, 0 < y < 1, 0 < z < 1). There are eight constrained regions given entirely by equations, each of which is a 0-dimensional point: (x = 0, y = 0, z = 0); (x = 0, y = 0, z = 1); (x = 0, y = 1, z = 0); (x = 0, y = 1, z = 1); (x = 1, y = 0, z = 0); (x = 1, y = 0, z = 1); (x = 1, y = 1, z = 0); (x = 1, y = 1, 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 mentioned in this hint. A picture may help.)
    • Exercises from the textbook due on October 6 Tuesday: 13.7.31, 13.7.33, 13.7.37, 13.7.51, 13.7.59.
23. Lagrange multipliers:
    • Reading from the textbook: Section 13.8 (pages 763–770).
    • Exercises due on October 6 Tuesday: For simplicity, assume that all of the functions that appear in the following exercises are differentiable everywhere and never have a zero gradient.
      1. If you wish to use Lagrange multipliers to maximize f(x, y) subject to the constraint that g(x, y) = 0, then what system of equations do you need to solve?
      2. If you wish to use Lagrange multipliers to maximize f(x, y, z) subject to the constraint that g(x, y, z) = 0, then what system of equations do you need to solve?
      3. If you wish to use Lagrange multipliers to maximize f(x, y, z) subject to the constraint that g(x, y, z) = 0 and h(x, y, z) = 0, then what system of equations do you need to solve? (For simplicity, assume that the gradients of g and h are never parallel or antiparallel.)
    • Exercises from the textbook due on October 7 Wednesday: 13.8.1, 13.8.3, 13.8.9, 13.8.11, 13.8.15, 13.8.23.

Integration

24. Integration on curves:
    • Reading from my notes: Chapter 5 through the first three paragraphs of Section 5.2 (pages 47&48).
    • Reading from the textbook: Section 15.2 "Line Integrals with Respect to dx, dy, or dz" (pages 857&858).
    • Exercises due on October 7 Wednesday:
      1. To integrate a differential form M(x, y) dx + N(x, y) dy along a parametrized curve (x, y) = (f(t), g(t)) for a ≤ t ≤ b, oriented in the direction of increasing t, what integral in the variable t do you evaluate?
      2. To integrate the differential form x² dy clockwise around the unit circle circle, parametrized (as usual) by (x, y) = (cos t, sin t) for 0 ≤ t ≤ 2π (using a counterclockwise coordinate system as usual), what are the bounds on the integral in the parameter t? That is, is it ∫₀^2π cos³ t dt or ∫_2π⁰ cos³ t dt?
    • Exercises from the textbook due on October 8 Thursday: 15.2.13, 15.2.15, 15.2.17, 15.2.23.
25. Integrating vector fields:
    • Reading from my notes:
      • The next two paragraphs of Section 5.2 (page 48);
      • Section 5.3 (page 49).
    • Reading from the textbook: The rest of Section 15.2 (pages 856&857, 859–863).
    • Exercises due on October 8 Thursday:
      1. To integrate the vector field F(x, y, z) = ⟨2x, −3x, 4xy⟩ = 2xi − 3xj + 4xyk along a curve in (x, y, z)-space, what differential form do you integrate along the curve?
      2. To integrate the vector field F(x, y) = ⟨x², 3⟩ = x²i +3j across a curve in the (x, y)-plane, what differential form do you integrate along the curve?
      3. To integrate inwards across a circle, should the circle be oriented clockwise or counterclockwise (using a counterclockwise coordinate system as usual)?
    • Exercises from the textbook due on October 12 Monday: 15.2.9, 15.2.11, 15.2.19, 15.2.21, 15.2.29, 15.2.33.
26. Integrating scalar fields:
    • Reading from my notes: The rest of Section 5.2 (pages 48&49).
    • Reading from the textbook: Section 15.1 except for "Mass and Moment Calculations" (pages 847–850, pages 851&852).
    • Exercises due on October 12 Monday:
      1. To integrate the scalar field f(x, y, z) = 2x − 4xy on a curve in (x, y, z)-space, what (nonlinear) differential form do you integrate along the curve?
      2. To integrate a scalar field f on the unit circle, parametrized (clockwise) by (x, y) = (sin t, cos t) for 0 ≤ t ≤ 2π, what should be the bounds on your integral in the variable t? (That is, is it ∫₀^2π f(sin t, cos t) dt or ∫_2π⁰ f(sin t, cos t) dt?)
    • Exercises from the textbook due on October 13 Tuesday: 15.1.9, 15.1.13, 15.1.15, 15.1.21, 15.1.30.
27. Double integrals on rectangles:
    • Reading from the textbook: Chapter 14 through Section 14.1 (pages 779–783).
    • Exercises due on October 13 Tuesday:
      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?
    • Exercises from the textbook due on October 14 Wednesday: 14.1.3, 14.1.6, 14.1.10, 14.1.19, 14.1.23.
28. Double integrals:
    • Reading from the textbook: Section 14.2 (pages 784–790).
    • Reading from my notes: Chapter 6 through Section 6.2 (pages 53&54).
    • Exercises due on October 14 Wednesday:
      1. Suppose that a and b are real numbers with a ≤ b and g and h are functions, both continuous on [a, b], with g ≤ h on [a, b]. Let R be the region {x, y | a ≤ x ≤ b, g(x) ≤ y ≤ h(x)}, and suppose that f is a function of two variables, continuous on R. Write an iterated integral equal to the double integral of f on R.
      2. Suppose that c and d are real numbers with c ≤ d and g and h are functions, both continuous on [c, d], with g ≤ h on [c, d]. Let R be the region {x, y | g(y) ≤ x ≤ h(y), c ≤ y ≤ d}, and suppose that f is a function of two variables, continuous on R. Write an iterated integral equal to the double integral of f on R.
    • Exercises from the textbook due on October 15 Thursday: 14.2.1, 14.2.2, 14.2.7, 14.2.19, 14.2.23, 14.2.79.
29. Triple integrals:
    • Reading from the textbook: Section 14.5 except "Volume of a Region in Space" and "Average value of a function in space" (pages 803, 804–810, 810).
    • Exercise due on October 21 Wednesday: In how many ways can you order 3 variables of integration? List them.
    • Exercises from the textbook due on October 22 Thursday: 14.5.10, 14.5.15, 14.5.3, 14.5.5, 14.5.21.
30. Systems of inequalities:
    • Reading from my notes: Section 6.3 (pages 55–57).
    • Exercises due on October 22 Thursday: Suppose that you wish to integrate a function f of two variables on the region R = {x, y | x² ≤ y ≤ 2x}.
      1. Given only x² ≤ y ≤ 2x, what equation (or inequality) would you solve to find that you also have 0 ≤ x ≤ 2?
      2. Now that you have both x² ≤ y ≤ 2x and 0 ≤ x ≤ 2, what iterated integral do you evaluate?
    • Exercises from the textbook due on October 26 Monday: 14.2.9, 14.2.11, 14.2.13, 14.2.17, 14.2.35, 14.2.41, 14.2.49, 14.2.51.
31. Areas, volumes, and averages:
    • Readings from the textbook:
      • Section 14.3 (pages 793–795);
      • Section 14.5 "Volume of a Region in Space" (pages 803&804);
      • Section 14.5 "Average value of a function in space" (page 810).
    • Exercises due on October 26 Monday: Suppose that a, b, c, and d are four real numbers with a ≤ b and c ≤ d, 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(x, y) ≥ 0 whenever a ≤ x ≤ b and c ≤ y ≤ d. 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 of the textbook, as an iterated double integral in the variables x and y;
      2. Using ideas from §14.5 of the textbook, as an iterated triple integral in the variables x, y, and z.
      (If you want to check your answers somewhat: 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.)
    • Exercises from the textbook due on October 27 Tuesday: 14.3.1, 14.3.3, 14.3.5, 14.3.11, 14.3.20, 14.3.21, 14.2.57, 14.2.63, 14.5.25, 14.5.29, 14.5.33, 14.5.37.
32. The area element:
    • Reading from my notes: Sections 6.4&6.5 (pages 57–60).
    • Exercises due on October 27 Tuesday:
      1. Let P, Q, and R be three points in R²; write ⟨x₁, y₁⟩ for the vector Q − P, and write ⟨x₂, y₂⟩ for the vector R − P. Express the area of the triangle with vertices P, Q, and R using only x₁, x₂, y₁, and y₂.
      2. In the (x, y)-plane, evaluate the differential form |dx ∧ dy| along the vectors ⟨x₁, y₁⟩ and ⟨x₂, y₂⟩.
    • Exercises not from the textbook due on October 28 Wednesday: cancelled.
33. Coordinate transformations:
    • Reading from my notes: Section 6.6 (pages 60&61).
    • Reading from the textbook: Section 14.8 (pages 832–839).
    • Exercise due on October 28 Wednesday: 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 my 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.)
    • Exercises from the textbook due on October 29 Thursday: 14.8.1, 14.8.3, 14.8.7, 14.8.9, 14.8.17, 14.8.22.
34. Polar coordinates:
    • Reading from my notes: Sections 2.5&2.6 (pages 24–26).
    • Exercises due on October 29 Thursday: Use the U.S. mathematicians' conventions for polar coordinates.
      1. Express the rectangular coordinates x and y in terms of the polar coordinates r and θ.
      2. Express the cyclindrical coordinates z and r in terms of the spherical coordinates ρ and φ.
      3. Combining these, express the rectangular coordinates x, y, and z in terms of the spherical coordinates ρ, φ, and θ.
    • Exercises from the textbook due on November 2 Monday: 14.4.1, 14.4.2, 14.4.5, 14.4.7, 14.7.1, 14.7.3, 14.7.13.
35. Area integrals in polar coordinates:
    • Reading from my notes: Section 6.7 (pages 62&63).
    • Reading from the textbook: Section 14.4 (pages 796–801).
    • Exercises due on November 2 Monday:
      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.
    • Exercises from the textbook due on November 3 Tuesday: 14.4.9, 14.4.17, 14.4.20, 14.4.23, 14.4.25, 14.4.27, 14.4.29, 14.4.33, 14.4.37.
36. Volume integrals in polar coordinates:
    • Reading from the textbook: Section 14.7 (pages 820–828).
    • Exercises due on November 3 Tuesday:
      1. Give a formula for the volume element in space in rectangular coordinates x, y, and z.
      2. Give a formula for the volume element in space in cylindrical coordinates r, θ, and z.
      3. Give a formula for the volume element in space in spherical coordinates ρ, φ, and θ (using the American mathematicians' convention for which of these is which).
    • Exercises from the textbook due on November 4 Wednesday: 14.7.23, 14.7.25, 14.7.29, 14.7.33, 14.7.35, 14.7.45, 14.7.60, 14.7.61, 14.7.63, 14.7.71, 14.7.85, 14.7.87.

More integration

37. Parametrized surfaces:
    • Readings from the textbook:
      • Section 11.6 (pages 651–655);
      • Section 15.5 through "Parametrizations of Surfaces (pages 890&891).
    • Reading from my notes: Chapter 7 through Section 7.1 (page 65).
    • Exercises due on November 4 Wednesday:
      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).
    • Exercises from the textbook due on November 5 Thursday: 15.5.1, 15.5.3, 15.5.5, 15.5.9, 15.5.13.
38. Integrals along surfaces:
    • Reading from my notes: Sections 7.2–7.4 (pages 66&67).
    • Exercise due on November 5 Thursday: If x = f(u, v), y = g(u, v), and z = h(u, v), where f, g, and h are differentiable functions, express each of dy ∧ dz, dz ∧ dx, and dx ∧ dy using partial derivatives and du ∧ dv.
    • Exercises not from the textbook due on November 9 Monday: none.
39. Flux across surfaces:
    • Reading from the textbook:
      • Section 15.6 introduction (page 900);
      • Section 15.6 from "Orientation of a Surface" to "Computing a Surface Integral for a Level Surface" (pages 904–906).
    • Reading from my notes: Section 7.5 (page 68).
    • Exercises due on November 9 Monday:
      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 my notes.)
    • Exercises from the textbook due on November 10 Tuesday: 15.6.19, 15.6.23, 15.6.25, 15.6.33, 15.6.35, 15.6.37, 15.6.41.
40. Integrals on surfaces:
    • Reading from my notes: Section 7.6 (pages 68&69).
    • Readings from the textbook:
      • The rest of Section 15.5 (pages 891–898);
      • Section 15.6 "Surface Integrals" (pages 900–903).
    • Exercises due on November 10 Tuesday:
      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 my notes.)
      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.
    • Exercises from the textbook due on November 11 Wednesday: 15.5.19, 15.5.21, 15.6.1, 15.6.5, 15.6.7, 15.6.11, 15.6.15.
41. Moments:
    • Reading from the textbook:
      • Section 14.6 (page 813–818);
      • Section 15.1 "Mass and Moment Calculations" (pages 850&851);
      • Section 15.6 "Moments and Masses of Thin Shells" (pages 906–908).
    • Exercises due on November 11 Wednesday:
      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.
    • Exercises from the textbook on November 12 Thursday: 14.6.3, 14.6.13, 14.6.19, 14.6.25, 14.7.99, 15.1.35, 15.6.45.
42. Conservative vector fields and exact differential forms:
    • Reading from my notes: Section 5.4 (pages 50&51).
    • Reading from the textbook: Section 15.3 (pages 867–876).
    • Exercises due on November 16 Monday: True or false:
      1. If f is a differentiable scalar field, 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.
    • Exercises from the textbook due on November 17 Tuesday: 15.3.1, 15.3.3, 15.3.5, 15.3.7, 15.3.8, 15.3.11, 15.3.13, 15.3.17, 15.3.21.
43. Exterior differentials:
    • Reading from my notes: Chapter 8 through Section 8.1 (pages 71–73).
    • Exercises due on November 17 Tuesday: Write down the exterior differentials of the following exterior differential forms:
      1. x,
      2. dx,
      3. x dy,
      4. x dy + y dz,
      5. x dy ∧ dz.
    • Exercises not from the textbook due on November 18 Wednesday: none.
44. Green's Theorem:
    • Reading from my notes: Section 8.3 (pages 74&75).
    • Reading from the textbook: Section 15.4 (pages 878–887).
    • Exercise due on November 18 Wednesday: 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 my notes. 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.)
    • Exercises from the textbook due on November 19 Thursday: 15.4.7, 15.4.9, 15.4.13, 15.4.15, 15.4.17, 15.4.21, 15.4.27, 15.4.29, 15.4.32.
45. Stokes's Theorem:
    • Reading from my notes: Section 8.4 (page 76).
    • Reading from the textbook: Section 15.7 (pages 910–921).
    • Exercises due on November 19 Thursday: In 3-dimensional space, let S be a surface bounded by a closed curve C.
      1. If F is a differentiable vector field defined on (at least) S, then the integral of F along C equals the integral of the _____ of F across S, if the orientations are appropriately matched.
      2. If the z-axis passes through S, you orient (or really pseudo-orient) S so that z is increasing along the z-axis through the surface, and you orient C so that Stokes's Theorem holds, then is the cylindrical coordinate θ increasing or decreasing overall along C?
    • Exercises from the textbook due on November 23 Monday: 15.7.7, 15.7.9, 15.7.11, 15.7.13, 15.7.15, 15.7.19, 15.7.23, 15.7.33.
46. Gauss's Theorem:
    • Reading from my notes: Section 8.5 (page 77).
    • Reading from the textbook: Section 15.8 (pages 923–931).
    • Exercises due on November 23 Monday: In 3-dimensional space, let D be a region bounded by a closed surface S.
      1. If F is a differentiable vector field defined on (at least) D, then the integral of F across S equals the integral of the _____ of F on D, if the orientation is appropriate.
      2. If the origin lies within D and you orient (or really pseudo-orient) S so that Gauss's Theorem holds, then is the spherical coordinate ρ increasing or decreasing overall through S?
    • Exercises from the textbook due on November 24 Tuesday: 15.8.5, 15.8.6, 15.8.9, 15.8.11, 15.8.13, 15.8.17, 15.8.22.
47. Cohomology:
    • Reading from my notes: Section 8.2 (page 73).
    • Exercises due on November 24 Tuesday:
      1. Fill in the blank: If α is an exterior differential form, then d ∧ d ∧ α (the exterior differential of the exterior differential of α) is ___. (Assume that α is at least twice differentiable so that this second-order differential exists.)
      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.)
      3. 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.)
    • Exercises from the textbook due on November 30 Monday: 15.3.25, 15.4.45, 15.7.27, 15.8.23.

Quizzes

1. Curves and functions:
   • Review date: September 17 Thursday (in class or through Zoom).
   • Date due on MyLab: September 20 Sunday (and into the next morning).
   • Corresponding problems sets: 1–11.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result except #3 & #6.
2. Differentiation:
   • Review date: October 15 Thursday (in class or through Zoom).
   • Date due on MyLab: October 20 Tuesday (and into the next morning).
   • Corresponding problems sets: 12–23.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result.
3. Integration:
   • Review date: November 12 Thursday (in class or through Zoom).
   • Date due on MyLab: November 16 Sunday (and into the next morning).
   • Corresponding problems sets: 24–36.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result except #5.
4. More integration:
   • Review date: December 3 Thursday (in class or through Zoom).
   • Date due on MyLab: December 6 Sunday (and into the next morning).
   • Corresponding problems sets: 37–47.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result.

Final exam

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