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: Tuesdays and Thursdays from 11:00 to 12:40 in room U109.
• Final exam time: May 7 Thursday from 11:00 to 12:40 in room U109 (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: Tuesdays and Thursdays from 1:10 PM to 3:15 (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, same as above).
• Problem set from the textbook due on January 15 Thursday or ASAP thereafter (submit this 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.

The numbering below is wrong; the lessons should be grouped in pairs, but I don't want to bother here.

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.
   • Reading homework due on January 15 Thursday (submit this 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?
   • Problem set from the textbook due on January 20 Tuesday (submit this through MyLab): 11.2.5, 11.3.1, 11.4.1, 11.4.15, 11.5.23, 11.5.39.
2. Parametrized curves:
   • Objectives:
     • Evaluate a point- or vector-valued function;
     • Graph a parametrized curve in 2 dimensions;
     • Interpret a graph of a parametrized curve in 2 or 3 dimensions;
     • Find equations for a parametrized curve;
     • Find the velocity and acceleration of a parametrized curve.
   • 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).
   • Reading homework due on January 15 Thursday (submit this 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?
   • Problem set from the textbook due on January 20 Tuesday (submit this 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, 12.1.37.
3. Standard parametrizations:
   • Objectives:
     • Write the standard parametrization of a line segment;
     • Write linear parametrizations of a ray or line;
     • Write the standard parametrization of a circle;
     • Parametrize the graph of a function.
   • Reading from my notes: Section 2.4 (pages 21&22).
   • Reading homework due on January 20 Tuesday (submit this 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].
   • Problem set from the textbook due on January 22 Thursday (submit this through MyLab): 12.1.23, 12.1.24, 15.1.1, 15.1.3, 15.1.5, 15.1.7.
4. Integrating parametrized curves:
   • Objectives:
     • Evaluate definite integrals of a vector-valued function;
     • Find semidefinite integrals of a vector-valued function;
     • Recover position and velocity from acceleration and initial data.
   • Reading from my notes: Section 2.3 (page 20).
   • Reading from the textbook: Section 12.2 (pages 671–675).
   • Reading homework due on January 20 Tuesday (submit this 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?
   • Problem set from the textbook due on January 22 Thursday (submit this 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:
   • Objectives:
     • Find speed and direction of a parametrized curve;
     • Find the arclength element of a parametrized curve;
     • Parametrize a curve by arclength;
     • Evaluate the length of a parametrized curve.
   • Reading from the textbook: Section 12.3 (pages 678–680).
   • Reading from my notes: Section 2.7 (pages 25&26).
   • Reading homework due on January 22 Thursday (submit this 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 = ___.
   • Problem set from the textbook due on January 27 Tuesday (submit this 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:
   • Objectives:
     • Determine the size of a matrix;
     • Form identity and zero matrices;
     • Determine whether matrices can be added, subtracted, and multiplied;
     • Add, subtract, and multiply matrices.
   • Reading from my notes: Section 1.13 (pages 17&18).
   • Reading homework due on January 22 Thursday (submit this 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.
   • Problem set from an external website due on January 27 Tuesday: Take the Mathopolis quiz on multiplying matrices, and tell me how it went in a message in class or on Canvas.
7. Polar coordinates:
   • Objectives:
     • Convert between rectangular and polar coordinates in 2 dimensions;
     • Convert between rectangular, cylindrical, and spherical coordinates in 3 dimensions.
   • Readings from my notes:
     • Section 2.8 (pages 26&27);
     • Section 2.10 (page 28).
   • Reading homework due on January 27 Tuesday (submit this 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 cylindrical 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.
   • Problem set from the textbook due on January 29 Thursday (submit this through MyLab): 14.4.1, 14.4.2, 14.4.5, 14.4.7, 14.7.1, 14.7.7, 14.7.13.
8. Functions of several variables:
   • Objectives:
     • Evaluate a function of 2 or 3 variables;
     • Find the domain of a function of 2 or 3 variables;
     • Graph the domain of a function of 2 variables;
     • Interpret a graph of a function of 2 variables;
     • Find an equation of a contour of a function of 2 or 3 variables;
     • Sketch a contour graph of a function of 2 variables;
     • Interpret a contour graph of a function of 2 or 3 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).
   • Reading homework due on January 27 Tuesday (submit this 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?
   • Problem set from the textbook due on January 29 Thursday (submit this 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.
9. Topology in several variables:
   • Objectives:
     • Find the boundary of the domain of a function of 2 or 3 variables;
     • Determine whether the domain of a function of 2 or 3 variables is open;
     • Determine whether the domain of a function of 2 or 3 variables is closed;
     • Determine whether the domain of a function of 2 or 3 variables is bounded;
     • Find where an analytically-defined function of 2 or 3 variables is continuous.
   • 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).
   • Reading homework due on January 29 Thursday (submit this 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.
   • Problem set from the textbook due on February 3 Tuesday (submit this 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.
10. Limits in several variables:
    • Objectives:
      • Evaluate the limit of a continuous function of several variables;
      • Evaluate the limit of a function of several variables by factoring;
      • Evaluate the limit of a function of 2 variables using polar coordinates;
      • Find witness curves for a limit of a function of several variables that does not exist.
    • Reading from the textbook: Section 13.2 (pages 705–711).
    • Reading from my notes: Section 3.4 (pages 34&35).
    • Reading homework due on January 29 Thursday (submit this 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) = ___.)
    • Problem set from the textbook due on February 3 Tuesday (submit this 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.
11. Vector fields:
    • Objectives:
      • Evaluate a vector field;
      • Graph a vector field in 2 dimensions;
      • Interpret the graph of a vector field in 2 or 3 dimensions;
      • Express a vector field as the product of a scalar field and a direction field.
    • 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.
    • Reading homework due on February 3 Tuesday (submit this 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.
    • Problem set from the textbook due on February 5 Thursday (submit this through MyLab): 15.2.5, 15.2.47, 15.2.49, 15.2.51.
12. Linear differential forms:
    • Objectives:
      • Evaluate a differential form at a point along a vector;
      • Evaluate a differential form at a point;
      • Express a vector field as a linear differential form;
      • Express a vector field in 2 dimensions as a linear differential pseudoform.
    • Reading from my notes: Chapter 4 through Section 4.3 (pages 39–41).
    • Reading homework due on February 3 Tuesday (submit this in class or on Canvas):
      1. Given 𝐅(x, y, z) = ⟨u, v, w⟩, express 𝐅(x, y, z) ⋅ ⟨dx, dy, dz⟩ as a differential form.
      2. Given 𝐆(x, y) = ⟨M, N⟩, express 𝐆(x, y) ⋅ ⟨dx, dy⟩ and 𝐆(x, y) × ⟨dx, dy⟩ as differential forms.
    • Problem set not from the textbook due on February 5 Thursday (submit this 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

13. Differentials:
    • Objective: Find the differential of an expression in several variables.
    • Reading from my notes: Section 4.4 (pages 41&42).
    • Reading homework due on February 5 Thursday (submit this 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.
    • Problem set not from the textbook due on February 12 Thursday (submit this 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).
14. Partial derivatives:
    • Objectives:
      • Find partial derivatives of an expression with several variables;
      • Find partial derivatives of a function of several variables;
      • Evaluate 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).
    • Reading homework due on February 5 Thursday (submit this 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!)
    • Problem set from the textbook due on February 12 Thursday (submit this 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.
15. Levels of differentiability:
    • Objectives:
      • Find a second partial derivative;
      • Compare mixed second partial derivatives;
      • Determine whether a function of several variables is differentiable.
    • 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).
    • Reading homework due on February 12 Thursday (submit this 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.
    • Problem set from the textbook due on February 17 Tuesday (submit this through MyLab): 13.3.43, 13.3.45, 13.3.61, 13.3.85, 13.3.93, 13.3.101.
16. Directional derivatives:
    • Objectives:
      • Evaluate a gradient of a scalar field;
      • Evaluate a directional derivative of a scalar field;
      • Evaluate and interpret the magnitude of a gradient.
    • 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).
    • Reading homework due on February 12 Thursday (submit this 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)?
    • Problem set from the textbook due on February 17 Tuesday (submit this 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.
17. Gradient vector fields:
    • Objectives:
      • Find the gradient of a scalar field;
      • Graph the gradient of a scalar field in 2 dimensions.
    • 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).
    • Reading homework due on February 17 Tuesday (submit this 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 = ___.
    • Problem set from the textbook due on February 19 Thursday (submit this through MyLab): 15.2.1, 15.2.2, 15.2.3, 15.2.4.
18. The Chain Rule:
    • Objectives:
      • Find a Jacobian matrix of several functions of several variables;
      • Substitute differentials in a differential form;
      • Evaluate a partial derivative of a composite function.
    • Reading from the textbook: Section 13.4 (pages 726–733).
    • Reading from my notes: Section 4.9 (pages 46&47).
    • Reading homework due on February 17 Tuesday (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.)
    • Problem set from the textbook due on February 19 Thursday (submit this through MyLab): 13.4.1, 13.4.3, 13.4.7, 13.4.9, 13.4.41.
19. Tangent flats and normal lines:
    • Objectives:
      • Find tangent lines to level curves of a function of 2 variables;
      • Find tangent planes to level surfaces of a function of 3 variables;
      • Parametrize normal lines to level curves of a function of 2 variables;
      • Parametrize normal lines to level surfaces of a function of 3 variables.
    • 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).
    • Reading homework due on February 19 Thursday (submit this 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).
    • Problem set from the textbook due on February 24 Tuesday (submit this through MyLab): 13.5.25, 13.5.27, 13.6.1, 13.6.5, 13.6.11, 13.6.15, 13.6.17.
20. Linearization:
    • Objectives:
      • Linearize a function of several variables;
      • Estimate the error in the linearization of a function of 2 variables.
    • 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).
    • Reading homework due on February 19 Thursday (submit this 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?)
    • Problem set from the textbook due on February 24 Tuesday (submit this through MyLab): 13.6.31, 13.6.33, 13.6.35, 13.6.39, 13.6.41.
21. Estimation:
    • Objectives:
      • Approximate a change in a function of several variables;
      • Find the sensitivity of a quantity on a variable.
    • Reading from the textbook:
      • Section 13.6 "Estimating Change in a Specific Direction" (page 747);
      • Section 13.6 "Differentials" (pages 749&750).
    • Reading homework due on February 24 Tuesday (submit this 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?
    • Problem set from the textbook due on February 26 Thursday (submit this through MyLab): 13.6.21, 13.6.23, 13.6.51, 13.6.55.
22. Local optimization:
    • Objectives:
      • Find the critical points of a function of several variables;
      • Find the Hessian determinant of a function of several variables;
      • Classify the critical points of a function of 2 variables.
    • 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).
    • Reading homework due on February 24 Tuesday (submit this 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.
    • Problem set from the textbook due on February 26 Thursday (submit this through MyLab): 13.7.2, 13.7.7, 13.7.9, 13.7.15, 13.7.27, 13.7.43.
23. Constrained optimization:
    • Objectives:
      • Identify the boundary and corners of a compact domain.
      • Find the extreme values of a continuous function on a compact domain.
    • Reading from the textbook: The rest of Section 13.7 (pages 758–760).
    • Reading homework due on February 26 Thursday (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.)
    • Problem set from the textbook due on March 3 Tuesday (submit this through MyLab): 13.7.31, 13.7.33, 13.7.37, 13.7.51, 13.7.59.
24. Lagrange multipliers:
    • Objectives:
      • Calculate the Lagrange multipliers on the boundary of the domain of a function of several variables;
      • Find extreme values of a function using Lagrange multipliers.
    • Reading from the textbook: Section 13.8 (pages 763–770).
    • Reading homework due on February 26 Thursday (submit this 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.)
    • Problem set from the textbook due on March 3 Tuesday (submit this through MyLab): 13.8.1, 13.8.3, 13.8.9, 13.8.11, 13.8.15, 13.8.23, 13.8.29.

Integration

25. Geometry of curves:
    • Objectives: TBA.
    • Reading from the textbook:
      • Section 2.4 (pages TBA);
      • Section 2.5 (pages TBA).
    • Reading homework due on March 3 Tuesday (submit this in class or on Canvas): TBA.
    • Problem set from the textbook due on March 17 Tuesday (submit this through MyLab): 12.4.1, 12.4.2, 12.4.5, 12.4.20, 12.5.3, 12.5.6, 12.5.7, 12.5.9.
26. Integration on curves:
    • Objective: Integrate a linear differential form on a parametrized curve.
    • 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).
    • Reading homework due on March 3 Tuesday (submit this 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?
    • Problem set from the textbook due on March 17 Tuesday (submit this through MyLab): 15.2.13, 15.2.15, 15.2.17, 15.2.23.
27. Integrating vector fields:
    • Objectives:
      • Integrate a vector field along a parametrized curve;
      • Calculate the work done by a force field;
      • Integrate a vector field across a parametrized curve in 2 dimensions.
    • 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).
    • Reading homework due on March 17 Tuesday (submit this 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)?
    • Problem set from the textbook due on March 19 Thursday (submit this through MyLab): 15.2.9, 15.2.11, 15.2.19, 15.2.21, 15.2.29, 15.2.33.
28. Integrating scalar fields:
    • Objective: Integrate a scalar field on a parametrized curve.
    • 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).
    • Reading homework due on March 17 Tuesday (submit this 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?)
    • Problem set from the textbook due on March 19 Thursday (submit this through MyLab): 15.1.9, 15.1.13, 15.1.15, 15.1.21, 15.1.30.
29. Conservative vector fields and exact differential forms:
    • Objectives:
      • Determine whether a linear differential form is closed;
      • Determine whether a vector field is irrotational;
      • Antidifferentiate an exact differential form;
      • Find the potential functions of a conservative vector field;
      • Use an antidifferential to evaluate a line integral.
    • Reading from my notes: Section 5.6 (pages 58&59).
    • Reading from the textbook: Section 15.3 (pages 867–876).
    • Reading homework due on March 19 Thursday (submit this 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) ⋅ ⟨dx, dy⟩ must be exact.
      4. If 𝐅 is a vector field in two dimensions and the differential form 𝐅(x, y) ⋅ ⟨dx, dy⟩ is exact, then 𝐅 must be conservative.
    • Problem set from the textbook due on March 24 Tuesday (submit this 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.
30. Double integrals:
    • Objectives:
      • Evaluate an iterated integral in 2 variables;
      • Evaluate a double integral on a rectangle;
      • Set up a double integral as an iterated integral;
      • Evaluate a double integral on a region of Type 1;
      • Evaluate a double integral on a region of Type 2.
    • 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).
    • Reading homework due on March 19 Thursday (submit this 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.
    • Problem set from the textbook due on March 24 Tuesday (submit this 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.
31. Systems of inequalities:
    • Objectives:
      • Solve a system of inequalities in 2 variables;
      • Describe a region bounded by curves in 2 dimensions;
      • Reverse the order of integration in an interated integral in 2 variables.
    • Reading from my notes: Section 6.3 (pages 63–65).
    • Reading homework due on March 24 Tuesday (submit this 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?
    • Problem set from the textbook due on March 26 Thursday (submit this 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.
32. Triple integrals:
    • Objectives:
      • Evaluate an iterated integral in 3 variables;
      • Set up a triple integral as an iterated integral;
      • Solve a system of inequalities in 3 variables;
      • Evaluate a triple integral.
    • 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).
    • Reading homework due on March 24 Tuesday (submit this in class or on Canvas): In how many ways can you order 3 variables of integration? List them.
    • Problem set from the textbook due on March 26 Thursday (submit this through MyLab): 14.5.10, 14.5.15, 14.5.3, 14.5.5, 14.5.21.
33. Areas, volumes, and averages:
    • Objectives:
      • Find the area of a compact region in 2 dimensions;
      • Find the volume of a compact region in 3 dimensions;
      • Find the average value of a function of 2 variables;
      • Find the average value of a function of 3 variables.
    • 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).
    • Reading homework due on March 26 Thursday (submit this 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.)
    • Problem set from the textbook due on March 31 Tuesday (submit this 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.
34. The area element:
    • Objectives:
      • Express the area element as a generalized differential form;
      • Express the volume element as a generalized differential form;
      • Evaluate a differential 2-form at a point along 2 vectors.
    • Reading from my notes: Sections 6.4&6.5 (pages 65–69).
    • Reading homework due on March 26 Thursday (submit this 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⟩.
    • Problem set not from the textbook due on March 31 Tuesday (submit this 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 ⟨dx, dy⟩ = ⟨−2, 3⟩, ⟨4, 6⟩. (Answer.)
      6. Evaluate dx ∧ dy along ⟨dx, dy⟩ = ⟨4, −9⟩, ⟨6, 3⟩.
35. Coordinate transformations:
    • Objectives:
      • Find a Jacobian determinant of several functions of several variables;
      • Convert an area or volume integral to different variables.
    • Reading from my notes: Section 6.6 (pages 69&70).
    • Reading from the textbook: Section 14.8 (pages 832–839).
    • Reading homework due on March 31 Tuesday (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 precisely 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 precise 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.)
    • Problem set from the textbook due on April 2 Thursday (submit this through MyLab): 14.8.1, 14.8.3, 14.8.7, 14.8.9, 14.8.17, 14.8.22.
36. Integrals in polar coordinates:
    • Objectives:
      • Convert an area integral to polar coordinates;
      • Evaluate a double integral using polar coordinates;
      • Convert a volume integral to cylindrical or spherical coordinates;
      • Evaluate a triple integral using 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).
    • Reading homework due on March 31 Tuesday (submit this 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.
    • Problem set from the textbook due on April 2 Thursday (submit this 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

37. Parametrized surfaces:
    • Objectives:
      • Parametrize the graph of a function of 2 variables;
      • Parametrize a surface in cylindrical or spherical coordinates;
      • Parametrize a quadratic surface.
    • 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).
    • Reading homework due on April 2 Thursday (submit this 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).
    • Problem set from the textbook due on April 9 Thursday (submit this 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.
38. Integrals along surfaces:
    • Objective: Integrate a differential 2-form on an oriented surface in 3 dimensions.
    • Reading from my notes: Sections 7.2–7.4 (pages 68–70).
    • Reading homework due on April 2 Thursday (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.
    • Problem set not from the textbook due on April 9 Thursday (submit this 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).
39. Flux across surfaces:
    • Objectives:
      • Find the pseudo-oriented surface elements on a parametrized surface in 3 dimensions;
      • Find the orientation of a pseudo-oriented surface in a right-handed 3-dimensional coordinate system;
      • Integrate a vector field in 3 dimensions across a surface.
    • 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).
    • Reading homework due on April 9 Thursday (submit this 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.)
    • Problem set from the textbook due on April 14 Tuesday (submit this through MyLab): 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:
    • Objectives:
      • Find the surface-area element on a parametrized surface in 3 dimensions;
      • Find the area of a surface in 3 dimensions;
      • Integrate a scalar field on a surface in 3 dimensions.
    • 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).
    • Reading homework due on April 9 Thursday (submit this 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.
    • Problem set from the textbook due on April 14 Tuesday (submit this through MyLab): 15.5.19, 15.5.21, 15.6.1, 15.6.5, 15.6.7, 15.6.11, 15.6.15.
41. Moments:
    • Objectives:
      • Find the mass element on a curve, plane region, surface, or solid;
      • Find the mass of a curve, plane region, surface, or solid;
      • Find the moments of a curve, plane region, surface, or solid;
      • Find the centre of mass of a curve, plane region, surface, or solid;
      • Find the moments of inertia of a curve, plane region, surface, or solid.
    • 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).
    • Reading homework due on April 14 Tuesday (submit this 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.
    • Problem set from the textbook due on April 16 Thursday (submit this through MyLab): 14.6.3, 14.6.13, 14.6.19, 14.6.25, 14.7.99, 15.1.35, 15.6.45.
42. Exterior differentials:
    • Objective: Find the exterior differential of an exterior differential form.
    • Reading from my notes: Chapter 8 through Section 8.1 (pages 79–81).
    • Reading homework due on April 14 Tuesday (submit this 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.
    • Problem set not from the textbook due on April 16 Thursday (submit this 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.
43. Green's Theorem:
    • Objectives:
      • Use the differential-form form of Green's Theorem;
      • Use the partial-derivative form of Green's Theorem;
      • Use the circulation-curl form of Green's Theorem;
      • Use the flux-divergence form of Green's Theorem.
    • Reading from my notes: Section 8.3 (pages 82–84).
    • Reading from the textbook: Section 15.4 (pages 878–887).
    • Reading homework due on April 16 Thursday (submit this 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.)
    • Problem set from the textbook due on April 21 Tuesday (submit this 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.
44. The Kelvin–Stokes Curl Theorem:
    • Objectives:
      • Find the curl of a vector field in 3 dimensions;
      • Use Stokes's Theorem to integrate a curl across a surface;
      • Use Stokes's Theorem to integrate a vector field along a closed curve.
    • Reading from my notes: Section 8.4 (page 84&85).
    • Reading from the textbook: Section 15.7 (pages 910–921).
    • Reading homework due on April 16 Thursday (submit this 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 appropriately for #1, then is the cylindrical coordinate θ increasing or decreasing overall along C?
    • Problem set from the textbook due on April 21 Tuesday (submit this 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.
45. The Ostrogradsky–Gauss Divergence Theorem:
    • Objectives:
      • Find the divergence of a vector field in 3 dimensions;
      • Use Gauss's Theorem to integrate a divergence on a solid region;
      • Use Gauss's Theorem to integrate a vector field across a closed surface.
    • Reading from my notes: Section 8.5 (page 85).
    • Reading from the textbook: Section 15.8 (pages 923–931).
    • Reading homework due on April 21 Tuesday (submit this 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 appropriately for #1, then is the spherical coordinate ρ increasing or decreasing overall through S?
    • Problem set from the textbook due on April 23 Thursday (submit this through MyLab): 15.8.5, 15.8.6, 15.8.9, 15.8.11, 15.8.13, 15.8.17, 15.8.22.
46. Cohomology:
    • Objectives:
      • Find the curl of a gradient in 2 or 3 dimensions;
      • Find the divergence of a curl in 3 dimensions.
    • Reading from my notes: Section 8.2 (pages 81&82).
    • Reading homework due on April 21 Tuesday (submit this 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⟩ = 2x³y² 𝐢 + 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.)
    • Problem set from the textbook due on April 23 Thursday (submit this through MyLab): 15.3.25, 15.4.45, 15.7.27, 15.8.23.

Quizzes

1. Curves and functions:
   • Date: February 10 Tuesday.
   • Corresponding problem sets: 1–6.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
2. Differentiation:
   • Date: March 5 Thursday.
   • Corresponding problem sets: 7–12.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
3. Integration:
   • Date: April 7 Tuesday.
   • Corresponding problem sets: 13–18.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
4. More integration:
   • Date: April 28 Tuesday.
   • Corresponding problem sets: 19–23.
   • 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 (DjVu TBA).

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