MATH-2080-LN01

Course administration

• Canvas page (where you must log in).
• Help with DjVu (if you have trouble reading the DjVu files on this page).
• SCC syllabus (DjVu).
• SCC Course Statements.
• SCC Student Technical Support Resources (DjVu).
• My course policies (DjVu).
• Class hours: Mondays through Thursdays from 1:00 PM to 1:50 in room U103.
• Final exam time: December 12 Friday from 1:00 PM to 2:40 in room U103 (or by appointment).

Contact information

• Name: Toby Bartels, PhD.
• Canvas messages.
• Email: TBartels@Southeast.edu.
• Voice mail: 1-402-323-3452.
• Text messages: 1-402-805-3021.
• Office hours: Weekdays from 10:00 to 10:50 (or by appointment) in room U9C (or over Zoom meeting 610-024-2876).

Readings

Try to read this introduction before the first day of class:

• Objectives:
  • Understand what to expect from this course;
  • Know how to submit assignments.
• Reading: My course policies (DjVu TBA, same as above).
• Exercises from the textbook due on August 19 Tuesday or ASAP thereafter (submit these through MyLab): O.1.1, O.1.2, O.1.3, O.1.4, O.1.5, O.1.6, O.1.7, O.1.8, O.1.10, O.1.11, O.1.12.

Curves and functions

1. Review of vectors:
   • Objectives:
     • Review vectors in 2 and 3 dimensions;
     • Subtract points to get a vector;
     • Add a vector to a point to get a point;
     • Take the cross product of two vectors in 2 dimensions.
   • Reading from the textbook: As needed: Review Chapter 11 through Section 11.5 (pages 614–649).
   • Reading from my notes: Optional: Through Section 1.12 (through page 17).
   • My additional online notes: Required: Vector operations.
   • Exercises due on August 19 Tuesday (submit these in class or on Canvas):
     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 20 Wednesday (submit these through MyLab): 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 Section 2.2 (page 19).
   • Exercises due on August 20 Wednesday (submit these in class or on Canvas):
     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 21 Thursday (submit these through MyLab): 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 from my notes: Section 2.4 (pages 21&22).
   • Exercises due on August 21 Thursday (submit these in class or on Canvas): Write down the usual parametrizations (including the bounds on the parameter) for:
     1. The oriented line segment from (x₁, y₁) to (x₂, y₂).
     2. More generally, the oriented line segment from P₁ to P₂.
     3. The circle in the 2-dimensional plane whose centre is (h, k) and whose radius is r.
     4. The graph of f, where f is a continuous function whose domain is [a, b].
   • Exercises from the textbook due on August 25 Monday (submit these through MyLab): 15.1.1, 15.1.3, 15.1.5, 15.1.7.
4. Integrating parametrized curves:
   • Reading from my notes: Section 2.3 (page 20).
   • Reading from the textbook: Section 12.2 (pages 671–675).
   • Exercises due on August 25 Monday (submit these in class or on Canvas): 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 August 26 Tuesday (submit these through MyLab): 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 2.7 (pages 25&26).
   • Exercises due on August 26 Tuesday (submit these in class or on Canvas): 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 August 27 Wednesday (submit these through MyLab): 12.3.1, 12.3.5, 12.3.8, 12.3.9, 12.3.11, 12.3.14, 12.3.18.
6. Matrices:
   • Reading from my notes: Section 1.13 (pages 17&18).
   • Exercises due on August 27 Wednesday (submit these in class or on Canvas): 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 August 28 Thursday: Take the Mathopolis quiz on multiplying matrices, and tell me how it went in a message in class or on Canvas.
7. Functions of several variables:
   • Reading from my notes: Chapter 3 through Section 3.1 (pages 29–31).
   • Reading from the textbook:
     • Chapter 13 through Section 13.1 "Domains and Ranges" (pages 697&698);
     • Section 13.1 from "Graphs, Level Curves, and Contours of Functions of Two Variables" onwards (pages 700–702).
   • Exercises due on August 28 Thursday (submit these in class or on Canvas): Suppose that f is a function of two variables, and f(2, 3) = 5.
     1. What number or point must belong to the domain of f?
     2. What number or point must belong to the range of f?
     3. What point must be on the graph of f?
   • Exercises from the textbook due on September 2 Tuesday (submit these through MyLab): 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.
8. 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 31&34).
   • Exercises due on September 2 Tuesday (submit these in class or on Canvas): 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 3 Wednesday (submit these through MyLab): 13.1.17, 13.1.19, 13.1.23, 13.1.25, 13.1.27, 13.2.33, 13.2.35, 13.2.39.
9. Limits in several variables:
   • Reading from the textbook: Section 13.2 (pages 705–711).
   • Reading from my notes: Section 3.4 (pages 34&35).
   • Exercises due on September 3 Wednesday (submit these in class or on Canvas):
     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)? (That is, 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)? (That is, lim_{(x,y)→(0,0)} f(x, y) = ___.)
   • Exercises from the textbook due on September 4 Thursday (submit these through MyLab): 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.
10. Vector fields:
    • Reading from the textbook: Section 15.2 through "Vector Fields" (pages 854&855), including Figures 15.7–15.16 (pages 854–856), except for Figure 15.11 (page 855).
    • Online notes: Examples of vector fields.
    • Exercises due on September 4 Thursday (submit these in class or on Canvas): 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 8 Monday (submit these through MyLab): 15.2.5, 15.2.47, 15.2.49, 15.2.51.
11. Linear differential forms:
    • Reading from my notes: Chapter 4 through Section 4.3 (pages 39–41).
    • Exercises due on September 8 Monday (submit these in class or on Canvas):
      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 9 Tuesday (submit these in class or on Canvas):
      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).

Differentiation

12. Differentials:
    • Reading from my notes: Section 4.4 (pages 41&42).
    • Exercises due on September 9 Tuesday (submit these in class or on Canvas):
      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 10 Wednesday (submit these in class or on Canvas):
      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).
13. Partial derivatives:
    • Reading from my notes: Section 4.5 (pages 42–44).
    • Reading from the textbook: Section 13.3 through "Functions of More than Two Variables" (pages 714–719).
    • Exercises due on September 10 Wednesday (submit these in class or on Canvas):
      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 11 Thursday (submit these through MyLab): 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.
14. Levels of differentiability:
    • Reading from my notes:
      • Sections 3.5&3.6 (page 35–37);
      • Section 4.6 (page 44).
    • Reading from the textbook: The rest of Section 13.3 (pages 719–723).
    • Exercises due on September 16 Tuesday (submit these in class or on Canvas): 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 Wednesday (submit these through MyLab): 13.3.43, 13.3.45, 13.3.61, 13.3.85, 13.3.93, 13.3.101.
15. Directional derivatives:
    • Reading from my notes: Sections 4.7&4.8 (pages 44–46).
    • Readings from the textbook:
      • Section 13.5 through "Calculation and Gradients" (pages 736–740);
      • Section 13.5 from "Functions of Three Variables" onwards (pages 742&743).
    • Exercises due on September 17 Wednesday (submit these in class or on Canvas): Suppose that ∇f(2, 3) = ⟨⅗, ⅘⟩.
      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 18 Thursday (submit these through MyLab): 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.
16. Gradient vector fields:
    • Readings from the textbook:
      • Section 15.2 Figure 15.11 (page 855);
      • Section 15.2 "Gradient Fields" (pages 855&856), including Figure 15.17 (page 856).
    • Exercises due on September 18 Thursday (submit these in class or on Canvas):
      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 22 Monday (submit these through MyLab): 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, 15.2.1, 15.2.2, 15.2.3, 15.2.4.
17. The Chain Rule:
    • Reading from the textbook: Section 13.4 (pages 726–733).
    • Reading from my notes: Section 4.9 (pages 46&47).
    • Exercise due on September 22 Monday (submit this in class or on Canvas): 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 23 Tuesday (submit these through MyLab): 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 through "Tangent Planes and Normal Lines" (pages 744–746).
    • Reading from my notes: Section 4.10 (page 48).
    • Exercises due on September 23 Tuesday (submit these in class or on Canvas): Fill in each blank with ‘line’ or ‘plane’.
      1. If ∇f(a, b) exists but is not zero, then f has a tangent ___ and a normal ___ through (a, b).
      2. If ∇f(a, b, c) exists but is not zero, then f has a tangent ___ and a normal ___ through (a, b, c).
    • Exercises from the textbook due on September 24 Wednesday (submit these through MyLab): 13.5.25, 13.5.27, 13.6.1, 13.6.5, 13.6.11, 13.6.15, 13.6.17.
19. Linearization:
    • Reading from the textbook:
      • 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 4.11 (pages 49–52).
    • Exercises due on September 24 Wednesday (submit these in class or on Canvas): 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 25 Thursday (submit these through MyLab): 13.6.31, 13.6.33, 13.6.35, 13.6.39, 13.6.41.
20. Estimation:
    • Reading from the textbook:
      • Section 13.6 "Estimating Change in a Specific Direction" (page 747);
      • Section 13.6 "Differentials" (pages 749&750).
    • Exercises due on September 25 Thursday (submit these in class or on Canvas):
      1. If f is a function of two variables and f is differentiable at P₀ = (x₀, y₀), then approximately how much does the value of f change if you move from the point P₀ in the direction of the vector v for a distance of Δs? (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 is −3 and (∂u‍/‍∂y)_x is 2, then is the quantity u more sensitive or less sensitive to small changes in x compared to changes in y?
    • Exercises from the textbook due on September 29 Monday (submit these through MyLab): 13.6.21, 13.6.23, 13.6.51, 13.6.55.
21. Local optimization:
    • Reading from my notes: Section 4.12 (pages 52–54).
    • Reading from the textbook: Section 13.7 through "Derivative Tests for Local Extreme Values" (pages 754–758).
    • Exercises due on September 29 Monday (submit these in class or on Canvas): 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 September 30 Tuesday (submit these through MyLab): 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 September 30 Tuesday (submit this in class or on Canvas): 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 is 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 1 Wednesday (submit these through MyLab): 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 1 Wednesday (submit these in class or on Canvas): 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 2 Thursday (submit these through MyLab): 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 Section 5.2 (pages 55&56).
    • Reading from the textbook: Section 15.2 "Line Integrals with Respect to dx, dy, or dz" (pages 857&858).
    • Exercises due on October 7 Tuesday (submit these in class or on Canvas):
      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 x³ 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 cos⁴ t dt or ∫_2π⁰ cos⁴ t dt?
    • Exercises from the textbook due on October 8 Wednesday (submit these through MyLab): 15.2.13, 15.2.15, 15.2.17, 15.2.23.
25. Integrating vector fields:
    • Readings from my notes:
      • Section 5.3 (page 56);
      • Section 5.5 (page 57).
    • Readings from the textbook:
      • Section 15.2 "Line Integrals of Vector Fields" (pages 856&857);
      • The rest of Section 15.2 (pages 859–863).
    • Exercises due on October 8 Wednesday (submit these in class or on Canvas):
      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 G(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 9 Thursday (submit these through MyLab): 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: Section 5.4 (page 57).
    • Readings from the textbook:
      • Chapter 15 through Section 15.1 "Additivity" (pages 847–850);
      • Section 15.1 "Line Integrals in the Plane" (pages 851&852).
    • Exercises due on October 9 Thursday (submit these in class or on Canvas):
      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 15 Wednesday (submit these through MyLab): 15.1.9, 15.1.13, 15.1.15, 15.1.21, 15.1.30.
27. Double integrals:
    • Reading from the textbook: Chapter 14 through Section 14.2 (pages 779–790).
    • Reading from my notes: Chapter 6 through Section 6.2 (pages 61–63).
    • Exercises due on October 15 Wednesday (submit these in class or on Canvas):
      1. Rewrite ∫^b_a ∫^d_c f(x, y) dy dx as an iterated integral ending with dx dy.
      2. 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.
      3. 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 16 Thursday (submit these through MyLab): 14.1.3, 14.1.6, 14.1.10, 14.1.19, 14.1.23, 14.2.1, 14.2.2, 14.2.7, 14.2.19, 14.2.23, 14.2.79.
28. Systems of inequalities:
    • Reading from my notes: Section 6.3 (pages 63–65).
    • Exercises due on October 16 Thursday (submit these in class or on Canvas): 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 20 Monday (submit these through MyLab): 14.2.9, 14.2.11, 14.2.13, 14.2.17, 14.2.35, 14.2.41, 14.2.49, 14.2.51.
29. Triple integrals:
    • Readings from the textbook:
      • Section 14.5 "Triple Integrals" (page 803);
      • Section 14.5 "Finding Limits of Integration in the Order dz dy dx" (pages 804–810);
      • Section 14.5 "Properties of Triple Integrals" (page 810).
    • Exercises due on October 20 Monday (submit these in class or on Canvas): In how many ways can you order 3 variables of integration? List them.
    • Exercises from the textbook due on October 21 Tuesday (submit these through MyLab): 14.5.10, 14.5.15, 14.5.3, 14.5.5, 14.5.21.
30. 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 21 Tuesday (submit these in class or on Canvas): 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 Section 14.2 of the textbook, as an iterated double integral in the variables x and y;
      2. Using ideas from Section 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 22 Wednesday (submit these through MyLab): 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.
31. The area element:
    • Reading from my notes: Sections 6.4&6.5 (pages 65–69).
    • Exercises due on October 22 Wednesday (submit these in class or on Canvas): Write all answers explicitly in terms of scalars and operations on scalars; don't leave the final answer as a dot product, cross product, or wedge product.
      1. Suppose that ⟨a, b⟩ and ⟨c, d⟩ are two vectors along two non-parallel sides of a parallelogram. Express the area of the parallelogram using a, b, c, and d.
      2. In the (x, y)-plane, evaluate the differential form |‍dx ∧ dy| along the vectors ⟨a, b⟩ and ⟨c, d⟩.
    • Exercises not from the textbook due on October 23 Thursday (submit these in class or on Canvas):
      1. Evaluate the 2-dimensional scalar cross product ⟨−2, 3⟩ × ⟨4, 6⟩. (Answer.)
      2. Evaluate the 2-dimensional scalar cross product ⟨4, −9⟩ × ⟨6, 3⟩.
      3. Find the area of a parallelogram if two of the vectors along its sides are ⟨−2, 3⟩ and ⟨4, 6⟩. (Answer.)
      4. Find the area of a parallelogram if two of the vectors along its sides are ⟨4, −9⟩ and ⟨6, 3⟩.
      5. Evaluate dx ∧ dy along d(x, y) = ⟨−2, 3⟩, ⟨4, 6⟩. (Answer.)
      6. Evaluate dx ∧ dy along d(x, y) = ⟨4, −9⟩, ⟨6, 3⟩.
32. Coordinate transformations:
    • Reading from my notes: Section 6.6 (pages 69&70).
    • Reading from the textbook: Section 14.8 (pages 832–839).
    • Exercise due on October 23 Thursday (submit this in class or on Canvas): 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, not a shorthand notation such as a Jacobian.)
    • Exercises from the textbook due on October 27 Monday (submit these through MyLab): 14.8.1, 14.8.3, 14.8.7, 14.8.9, 14.8.17, 14.8.22.
33. Polar coordinates:
    • Readings from my notes:
      • Section 2.8 (pages 26&27);
      • Section 2.10 (page 28).
    • Exercises due on October 27 Monday (submit these in class or on Canvas): 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 #1 and #2, express the rectangular coordinates x, y, and z in terms of the spherical coordinates ρ, φ, and θ.
    • Exercises from the textbook due on October 28 Tuesday (submit these through MyLab): 14.4.1, 14.4.2, 14.4.5, 14.4.7, 14.7.1, 14.7.3, 14.7.13.
34. Integrals in polar coordinates:
    • Readings from my notes: Section 6.7 (pages 70–72).
    • Readings from the textbook:
      • Section 14.4 (pages 796–801);
      • Section 14.7 (pages 820–828).
    • Exercises due on October 28 Tuesday (submit these in class or on Canvas): Use the U.S. mathematicians' conventions for polar coordinates.
      1. Recall that the area element in the (x, y)-plane is đA = |‍dx ∧ dy|‍, or dx dy for short. Give a formula for the area element using polar coordinates r and θ instead.
      2. Recall that the volume element in (x, y, z)-space is đV = |‍dx ∧ dy ∧ dz|‍, or dx dy dz for short.
         1. Give a formula for the volume element using cylindrical coordinates r, θ, and z;
         2. Give a formula for the volume element using spherical coordinates ρ, φ, and θ.
    • Exercises from the textbook due on October 29 Wednesday (submit these through MyLab): 14.4.9, 14.4.17, 14.4.20, 14.4.23, 14.4.25, 14.4.27, 14.4.29, 14.4.33, 14.4.37,14.7.23, 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

35. 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 67).
    • Exercises due on October 29 Wednesday (submit these in class or on Canvas):
      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 October 30 Thursday (submit these through MyLab): 11.6.1, 11.6.3, 11.6.7, 11.6.9, 11.6.11, 15.5.1, 15.5.3, 15.5.5, 15.5.9, 15.5.13.
36. Integrals along surfaces:
    • Reading from my notes: Sections 7.2–7.4 (pages 68–70).
    • Exercise due on November 4 Tuesday (submit this in class or on Canvas): 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 (or Jacobians) and du ∧ dv.
    • Exercises not from the textbook due on November 5 Wednesday (submit these in class or on Canvas):
      1. Find the integral of x dx ∧ dy + y dy ∧ dz on the triangle in (x, y, z)-space with vertices (0, 0, 1), (0, 1, 0), and (1, 0, 0), oriented clockwise when viewed from the origin (in a right-handed coordinate system). (Answer.)
      2. Find the integral of x dx ∧ dy − y dy ∧ dz on the triangle in (x, y, z)-space with vertices (0, 0, 1), (0, 1, 0), and (1, 0, 0), oriented clockwise when viewed from the origin (in a right-handed coordinate system).
      3. Find the integral of dy ∧ dz on the portion of the unit sphere in the first octant, oriented clockwise when viewed from the origin (in a right-handed coordinate system). (Answer.)
      4. Find the integral of dx ∧ dy on the portion of the unit sphere in the first octant, oriented clockwise when viewed from the origin (in a right-handed coordinate system).
37. Flux across surfaces:
    • Reading from the textbook: 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 (pages 76&77).
    • Exercises due on November 5 Wednesday (submit these in class or on Canvas):
      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 đS = n đσ 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 6 Thursday (submit these through MyLab): 15.6.19, 15.6.23, 15.6.25, 15.6.33, 15.6.35, 15.6.37, 15.6.41.
38. Integrals on surfaces:
    • Reading from my notes: Section 7.6 (pages 77&78).
    • Readings from the textbook:
      • The rest of Section 15.5 (pages 891–898);
      • Section 15.6 through "Surface Integrals" (pages 900–903).
    • Exercises due on November 6 Thursday (submit these in class or on Canvas):
      1. Write down a formula for the surface-area element đσ = ‖‍đS‖ 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 10 Monday (submit these through MyLab): 15.5.19, 15.5.21, 15.6.1, 15.6.5, 15.6.7, 15.6.11, 15.6.15.
39. Moments:
    • Readings 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 10 Monday (submit these in class or on Canvas):
      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 due on November 11 Tuesday (submit these through MyLab): 14.6.3, 14.6.13, 14.6.19, 14.6.25, 14.7.99, 15.1.35, 15.6.45.
40. Conservative vector fields and exact differential forms:
    • Reading from my notes: Section 5.6 (pages 58&59).
    • Reading from the textbook: Section 15.3 (pages 867–876).
    • Exercises due on November 11 Tuesday (submit these in class or on Canvas): 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 12 Wednesday (submit these through MyLab): 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.
41. Exterior differentials:
    • Reading from my notes: Chapter 8 through Section 8.1 (pages 79–81).
    • Exercises due on November 12 Wednesday (submit these in class or on Canvas): 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 13 Thursday (submit these in class or on Canvas): Find the exterior differential (aka exterior derivative) of each of the following exterior differential forms:
      1. 2x dx + 3y dx + 4x dy + 5y dy. (Answer.)
      2. 3x dx + 2y dx − 5x dy − 4y dy.
      3. 2xy dx + 3yz dy + 4xz dz. (Answer.)
      4. 4xz dx + 3xy dy + 2yz dz.
      5. 2x dx ∧ dy + 3y dx ∧ dz + 4z dy ∧ dz. (Answer.)
      6. 2z dx ∧ dy + 3y dx ∧ dz + 4x dy ∧ dz.
42. Green's Theorem:
    • Reading from my notes: Section 8.3 (pages 82–84).
    • Reading from the textbook: Section 15.4 (pages 878–887).
    • Exercises due on November 13 Thursday (submit these in class or on Canvas): 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 17 Monday (submit these through MyLab): 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.
43. Stokes's Theorem:
    • Reading from my notes: Section 8.4 (page 84&85).
    • Reading from the textbook: Section 15.7 (pages 910–921).
    • Exercises due on November 17 Monday (submit these in class or on Canvas): 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 18 Tuesday (submit these through MyLab): 15.7.7, 15.7.9, 15.7.11, 15.7.13, 15.7.15, 15.7.19, 15.7.23, 15.7.33.
44. Gauss's Theorem:
    • Reading from my notes: Section 8.5 (page 85).
    • Reading from the textbook: Section 15.8 (pages 923–931).
    • Exercises due on November 18 Tuesday (submit these in class or on Canvas): 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 19 Wednesday (submit these through MyLab): 15.8.5, 15.8.6, 15.8.9, 15.8.11, 15.8.13, 15.8.17, 15.8.22.
45. Cohomology:
    • Reading from my notes: Section 8.2 (pages 81&82).
    • Exercises due on November 19 Wednesday (submit these in class or on Canvas):
      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 20 Thursday (submit these through MyLab): 15.3.25, 15.4.45, 15.7.27, 15.8.23.

Quizzes

1. Curves and functions:
   • Review date: September 11 Thursday.
   • Date taken: September 15 Monday.
   • Corresponding problem sets: 1–11.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
2. Differentiation:
   • Review date: October 2 Thursday.
   • Date taken: October 6 Monday.
   • Corresponding problem sets: 12–23.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
3. Integration:
   • Review date: October 30 Thursday.
   • Date taken: November 3 Monday.
   • Corresponding problem sets: 24–34.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
4. More integration:
   • Review date: November 20 Thursday.
   • Date taken: November 24 Monday.
   • Corresponding problem sets: 35–45.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.

Final exam

For the exam, you may use one sheet of notes that you wrote yourself, but you may not use your book or anything else not written by you. You certainly should not talk to other people! Calculators are allowed (although you shouldn't really need one), but not communication devices (like cell phones).

The exam consists of questions similar in style and content to those in the practice exam on MyLab.

The permanent URI of this web page is https://tobybartels.name/MATH-2080/2025FA/.