MATH-2080-ES31

Here is material about the administration of the course:

• Canvas page (where you must log in for full access, available while the course is in session).
• Help with DjVu (if you have trouble reading the files below).
• Course policies (DjVu).
• Class hours: Mondays through Thursdays from 1:00 PM to 1:50 in ESQ 100C.

• Name: Toby Bartels, PhD;
• Canvas messages.
• Email: TBartels@Southeast.edu.
• Voice mail: 1-402-323-3452.
• Text messages: 1-402-805-3021.
• Office hours:
  • Mondays and Wednesdays from 9:30 to 10:30,
  • Tuesdays and Thursdays from 2:30 PM to 4:00, and
  • by appointment,
  in ESQ 112 and online. (I am often available outside of those times; feel free to send a message any time and to check for me in the office whenever it's open.)

Reading homework

• Chapter 1 (on vectors);
• Chapter 2 (on parametrized curves);
• Chapter 3 (on functions of several variables);
• The full set.

1. General review:
   • Date due: January 14 Tuesday.
   • Reading from the textbook: As needed: Review §§11.1–11.5.
   • Reading from my notes: Optional: Chapter 1 (through page 17).
   • Online notes: Required: Vector operations.
   • Exercises due:
     1. Give a formula for the vector from the point (x₁, y₁) to the point (x₂, y₂).
     2. If u and v are vectors in 2 dimensions, then is u × v a scalar or a vector?
     3. If u and v are vectors in 3 dimensions, then is u × v a scalar or a vector?
2. Parametrized curves:
   • Date due: January 15 Wednesday.
   • Reading from the textbook: Chapter 12 through Section 12.1 (pages 662–668).
   • Reading from my notes: Chapter 2 through the first half of Section 2.1 (all of page 19 and the first two lines of page 20).
   • Exercises due:
     1. If C is a point-valued function, so that P = C(t) is a point (for each scalar value of t), then what type of value does its derivative C′ take?; that is, is dP/dt = C′(t) a point, a scalar, a vector, or what?
     2. If c is a vector-valued function, so that r = c(t) is a vector (for each scalar value of t), then what type of value does its derivative c′ take?; that is, is dr/dt = c′(t) a point, a scalar, a vector, or what?
3. Integrating parametrized curves:
   • Date due: January 16 Thursday.
   • Reading from the textbook: Section 12.2 (pages 671–675).
   • Reading from my notes: The second half of Section 2.1 (the rest of page 20).
   • Exercises due: 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?
4. Arclength:
   • Date due: January 21 Tuesday.
   • Reading from the textbook: Section 12.3 (pages 678–680).
   • Reading from my notes: Section 2.4 (pages 23&24).
   • Exercises due: Section 12.3 of the textbook uses several variables, including r, s, t, T, and v, to describe various quantities on the path of a parametrized curve. Fill in the right-hand side of each of these equations with the appropriate one of these variables:
     1. dr/dt = ___.
     2. v/|v| = ___.
     3. dr/ds = ___.
5. Functions of several variables:
   • Date due: January 22 Wednesday.
   • Reading from my notes: Chapter 3 through Section 3.1 (pages 27–29).
   • Reading from the textbook:
     • Chapter 13 through Section 13.1 "Domains and Ranges" (pages 697&698);
     • Section 13.1 "Graphs, Level Curves, and Contours of Functions of Two Variables" through the end of Section 13.1 (pages 700–702).
   • Exercises due:
     1. If f(2, 3) = 5, then what number or point must belong to the domain of f and what number or point must belong to the range of f?
     2. If f(2, 3) = 5, then what point must be on the graph of f?
6. Topology in several variables:
   • Date due: January 23 Thursday.
   • Reading from the textbook: Section 13.1 "Functions of Two Variables" (pages 698&699).
   • Reading from my notes: Sections 3.2&3.3 (pages 29&30).
   • Exercises due: Let R be a (binary) 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.
7. Limits in several variables:
   • Date due: January 27 Monday.
   • Section 13.2 (pages 705–711).
   • Reading from my notes: Section 3.4 (pages 30&31).
   • Exercises due:
     1. Suppose that the limit of f approaching (2, 3) is 5 (in symbols, lim_{(x,y)→(2,3)} f(x, y) = 5), and the limit of g approaching (2, 3) is 7 (so lim_{(x,y)→(2,3)} g(x, y) = 7). What (if anything) is the limit of f + g approaching (2, 3)? (so lim_{(x,y)→(2,3)} (f(x, y) + g(x, y)) = ___).
     2. Suppose that the limit of f approaching (0, 0) horizontally is 4 (in symbols, lim_{(x,y)→(0,0),y=0} f(x, y) = 4), and the limit of f approaching (0, 0) vertically is 6 (so lim_{(x,y)→(0,0),x=0} f(x, y) = 6). What (if anything) is the limit of f approaching (0, 0)? (so lim_{(x,y)→(0,0)} f(x, y) = ___).
8. Vector fields:
   • Date due: January 28 Tuesday.
   • Reading from the textbook: Section 15.2 through "Vector Fields" (pages 854&855 and Figures 15.7–15.16), except Figure 15.11.
   • Online notes: Examples of vector fields.
   • Exercises due: Sketch a graph of the following vector fields:
     1. F(x, y) = ⟨x, y⟩ = xi + yj;
     2. G(x, y) = ⟨−y, x⟩ = −yi + xj.
9. Linear differential forms:
   • Date due: January 29 Wednesday.
   • Reading from my notes: Chapter 4 through Section 4.3 (pages 33&34).
   • Exercises due:
     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.
10. Differentials:
    • Date due: January 30 Thursday.
    • Readings from my notes: Section 4.4 (pages 35&36).
    • Exercises due:
      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.
11. Partial derivatives:
    • Date due: February 3 Monday.
    • Readings from my notes: Section 4.5 (pages 36&37).
    • Reading from the textbook: Section 13.3 through "Functions of More than Two Variables" (pages 714–719).
    • Exercises due:
      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!)
12. Levels of differentiability:
    • Date due: February 4 Tuesday.
    • Readings from my notes: Sections 3.5&3.6 (page 32).
    • Reading from the textbook: The rest of Section 13.3 (pages 719–723).
    • Exercises due: 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.
13. Directional derivatives:
    • Date due: February 5 Wednesday.
    • Readings from the textbook:
      • Section 13.5 through "Calculation and Gradients" (pages 736–738);
      • Section 13.5 from "Functions of Three Variables" (pages 742&743).
    • Reading from my notes: Sections 4.6&4.7 (pages 37–39).
    • Exercises due: Suppose that ∇f(2, 3) = ⟨3/5, 4/5⟩.
      1. In which direction u is the directional derivative D_uf(2, 3) the greatest?
      2. In which directions u is the directional derivative D_uf(2, 3) equal to zero?
      3. In which direction u is the directional derivative D_uf(2, 3) the least (with a large absolute value but negative)?
14. Gradient vector fields:
    • Date due: February 11 Tuesday.
    • Readings from the textbook:
      • Section 15.2 Figure 15.11 (page 855);
      • Section 15.2 "Gradient Fields" (pages 855&856).
    • Exercises due:
      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 = ___.
15. Matrices:
    • Date due: February 12 Wednesday.
    • Reading from my notes: Section 1.13 (page 17).
    • Exercises due: 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.
16. The Chain Rule:
    • Date due: February 13 Thursday.
    • Reading from the textbook: Section 13.4 (pages 726–733).
    • Reading from my notes: Section 4.8 (pages 39&40).
    • Exercise due: 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.)
17. Tangent flats and normal lines:
    • Date due: February 18 Tuesday.
    • Readings from the textbook:
      • Section 13.5 "Gradients and Tangents to Level Curves" (pages 740&741);
      • Section 13.6 "Tangent Planes and Normal Lines" (pages 744–746).
    • Reading from my notes: Section 4.9 (pages 40&41).
    • Exercises due: Fill in each blank with ‘line’ or ‘plane’.
      1. If ∇f(a, b) exists but is not zero, then f has a tanget ___ and a normal ___ through (a, b).
      2. If ∇f(a, b, c) exists but is not zero, then f has a tanget ___ and a normal ___ through (a, b, c).
18. Linearization:
    • Date due: February 19 Wednesday.
    • The rest of Section 13.6 (pages 747–751).
    • Reading from my notes: Section 4.10 (pages 41–44).
    • Exercises due:
      1. If a function f is to have a good linear approximation in a region, then it's best if its partial derivatives of what order are close to zero in that region? (Its first partial derivatives, its second partial derivatives, its third partial derivatives, or what?)
      2. If (∂u/∂x)_y = −3 and (∂u/∂y)_x = 2, then is the quantity u more or less sensitive to changes in x compared to changes in y?
19. Local optimization:
    • Date due: February 20 Thursday.
    • Reading from my notes: Section 4.11 (pages 44&45).
    • Section 13.7 through "Derivative Tests for Local Extreme Values" (pages 754–758).
    • Exercises due: Consider a function f of two variables that is defined everywhere. Identify whether f has a local maximum, a local minimum, a saddle, or none of these at a point (a, b) with these characteristics:
      1. If the partial derivatives of f at (a, b) are both negative.
      2. If one of the partial derivatives of f at (a, b) is zero and the other is negative.
      3. If both partial derivatives of f at (a, b) are zero and the Hessian determinant of f at (a, b) is negative.
      4. If both partial derivatives of f at (a, b) are zero, the Hessian determinant of f at (a, b) is positive, and the unmixed second partial derivatives of f at (a, b) are negative.
      5. If both partial derivatives of f at (a, b) are zero, the Hessian determinant of f at (a, b) is positive, the unmixed second partial derivatives of f at (a, b) are positive, and the mixed second partial derivatives of f at (a, b) are negative.
20. Constrained optimization:
    • Date due: February 24 Monday.
    • Reading from the textbook: The rest of Section 13.7 (pages 758–760).
    • Exercise due: Suppose that you wish to maximize a continuous function on the region in 3 dimensions defined in rectangular coordinates by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and 0 ≤ z ≤ 1. How many different constrained regions will you have to check? (Hint: One constrained region to check is the 3-dimensional interior, given by the strict inequalities 0 < x < 1, 0 < y < 1, and 0 < z < 1. There are eight constrained regions given entirely by equations, each of which is a 0-dimensional point: x = 0, y = 0, and z = 0; x = 0, y = 0, and z = 1; x = 0, y = 1, and z = 0; x = 0, y = 1, and z = 1; x = 1, y = 0, and z = 0; x = 1, y = 0, and z = 1; x = 1, y = 1, and z = 0; and x = 1, y = 1, and z = 1. You still need to count the constrained regions of intermediate dimension, each of which will be given partially by strict inequalities and partially by equations. Be sure to give the final total including the 9 that I've already mentioned in this hint. A picture may help.)
21. Lagrange multipliers:
    • Date due: February 25 Tuesday.
    • Reading from the textbook: Section 13.8 (pages 763–770).
    • Exercises due: 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?
22. Integration on curves:
    • Date due: February 26 Wednesday.
    • Reading from my notes: Chapter 5 through Section 5.1 (page 47).
    • Reading from the textbook: Section 15.2 "Line Integrals with Respect to dx, dy, or dz" (pages 857&858).
    • Exercises due:
      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 2x dx counterclockwise along the circle parametrized by (x, y) = (sin t, cos 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 ∫^2π_t=0 2 sin t cos t dt or ∫⁰_t=2π 2 sin t cos t dt?
23. Integrating vector fields:
    • Date due: February 27 Thursday.
    • Reading from my notes: Sections 5.2&5.3 (pages 48&49).
    • Reading from the textbook: The rest of Section 15.2 (pages 856&857, 859–863).
    • Exercises due:
      1. To integrate the vector field F(x, y, z) = ⟨2x, −3x, 4xy⟩ = 2xi − 3xj + 4xyk along a curve in (x, y, z)-space, what differential form do you integrate along the curve?
      2. To integrate the vector field F(x, y) = ⟨x², 3⟩ = x²i +3j across a curve in the (x, y)-plane, what differential form do you integrate along the curve?
      3. To integrate inwards across a circle, should the circle be oriented clockwise or counterclockwise (using a counterclockwise coordinate system as usual)?
24. Integrating scalar fields:
    • Date due: March 2 Monday.
    • Reading from the textbook: Section 15.1 except for "Mass and Moment Calculations" (pages 847–850, pages 851&852).
    • Exercises due:
      1. To integrate the scalar field f(x, y, z) = 2x − 4xy on a curve in (x, y, z)-space, what differential form do you integrate along the curve?
      2. To integrate a scalar field on the circle parametrized 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 ∫^2π₀ f(sin t, cos t) dt or ∫⁰_2π f(sin t, cos t) dt?)
25. Double integrals on rectangles:
    • Date due: March 3 Tuesday.
    • Reading from the textbook: Chapter 14 through Section 14.1 (pages 779–783).
    • Exercises due:
      1. Rewrite ∫^b_a ∫^d_c f(x, y) dy dx as an iterated integral ending with dx dy.
      2. Assuming that f is continuous everywhere, is it possible that these two iterated integrals could evaluate to different results?
26. Double integrals:
    • Date due: March 4 Wednesday.
    • Reading from the textbook: Section 14.2 (pages 784–790).
    • Reading from my notes: Chapter 6 through Section 6.2 (pages 53&54).
    • Exercises due:
      1. Suppose that a and b are real numbers with a ≤ b and g and h are functions, both continuous on [a, b], with g ≤ h on [a, b]. Let R be the region {x, y | a ≤ x ≤ b, g(x) ≤ y ≤ h(x)}, and suppose that f is function of two variables continuous on R. Write an iterated integral equal to the double integral of f on R.
      2. Suppose that c and d are real numbers with c ≤ d and g and h are functions, both continuous on [c, d], with g ≤ h on [c, d]. Let R be the region {x, y | g(y) ≤ x ≤ h(y), c ≤ y ≤ d}, and suppose that f is function of two variables continuous on R. Write an iterated integral equal to the double integral of f on R.
27. Systems of inequalities:
    • Date due: March 11 Wednesday.
    • Reading from my notes: Section 6.3 (pages 55–57).
    • Exercises due: 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?
28. Triple integrals:
    • Date due: March 12 Thursday.
    • Reading from the textbook: Section 14.5 except "Volume of a Region in Space" and "Average value of a function in space" (pages 803, 804–810, 810).
    • Exercise due: In how many ways can you order 3 variables of integration? List them.
29. Areas, volumes, and averages:
    • Date due: March 31 Tuesday.
    • 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: Suppose that a < b and c < d are four real numbers, that f is a continuous function of two variables whose domain is the rectangle {x, y | a ≤ x ≤ b, c ≤ y ≤ d}, and that f takes only positive values. Write down expressions (in terms of a, b, c, d, and f) for the volume under the graph of f:
      1. Using ideas from §14.2 of the textbook, as an iterated double integral in the variables x and y;
      2. Using ideas from §14.5 of the textbook, as an iterated triple integral in the variables x, y, and z.
      (To check: You shouldn't be able to evaluate your answer to #1, because I haven't told you which function f is; however, you should be able to begin evaluating your answer to #2 if you write the variables in an appropriate order, and this should turn it into your answer from #1, after which you shouldn't be able to go any further.)
30. The area element:
    • Date due: April 1 Wednesday.
    • Reading from my notes: Sections 6.4&6.5 (pages 57–60).
    • Exercises due:
      1. Let P, Q, and R be three points in R²; write ⟨x₁, y₁⟩ for the vector Q − P, and write ⟨x₂, y₂⟩ for the vector R − P. Express the area of the triangle with vertices P, Q, and R using only x₁, x₂, y₁, and y₂.
      2. In the (x, y)-plane, evaluate the differential form |dx ∧ dy| along the vectors ⟨x₁, y₁⟩ and ⟨x₂, y₂⟩.
31. Coordinate transformations:
    • Date due: April 2 Thursday.
    • Reading from my notes: Section 6.6 (page 61).
    • Reading from the textbook: Section 14.8 (pages 832–839).
    • Exercise due: If x = f(u, v) and y = g(u, v), where f and g are continuously differentiable everywhere, then write the area element dx dy (which is more properly written |dx ∧ dy|) in terms of u, v, their differentials, and the partial derivatives of f and g (which you can also think of as the partial derivatives of x and y with respect to u and v). (There are formulas in both my notes and the textbook that you can use, or you can work it out from first principles using the more proper expression involving dx and dy given above. You may use any correct formula, as long as it explicitly uses partial derivatives as directed, rather than some more sophisticated notation instead.)
32. Polar coordinates:
    • Date due: April 6 Monday.
    • Reading from my notes: Sections 2.5&2.6 (pages 24–26).
    • Exercises due: Use the U.S. mathematicians' conventions for polar coordinates.
      1. Express the rectangular coordinates x and y in terms of the polar coordinates r and θ.
      2. Express the cyclindrical coordinates z and r in terms of the spherical coordinates ρ and φ.
      3. Combining these, express the rectangular coordinates x, y, and z in terms of the spherical coordinates ρ, φ, and θ.
33. Integration in polar coordinates:
    • Date due: April 7 Tuesday.
    • Reading from my notes: Section 6.7 (page 62).
    • Readings from the textbook:
      • Section 14.4 (pages 796–801), especially the Examples;
      • Section 14.7 (pages 820–828), especially the Examples.
    • Exercises due:
      1. Give a formula for the area element in the plane in rectangular coordinates x and y. (Answer: dx dy, or more properly |dx ∧ dy|; either is acceptable, as are dy dx and |dy ∧ dx|.)
      2. Give a formula for the area element in the plane in polar coordinates r and θ.
      3. Give a formula for the volume element in space in rectangular coordinates x, y, and z. (Answer: dx dy dz, or more properly |dx ∧ dy ∧ dz|; either is acceptable, as are the the other orders.)
      4. Give a formula for the volume element in space in cylindrical coordinates r, θ, and z.
      5. Give a formula for the volume element in space in spherical coordinates ρ, φ, and θ (using the U.S. mathematicians' convention for which of these is which).
34. Parametrized surfaces:
    • Date due: April 8 Wednesday.
    • Readings from the textbook:
      • Section 11.6 (pages 651–655);
      • Section 15.5 through "Parametrizations of Surfaces (pages 890&891);
      • Section 15.6 "Orientation of a Surface" (page 904).
    • Reading from my notes: Chapter 7 through Section 7.2 (pages 63&64).
    • Exercises due:
      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).
35. Integrals along surfaces:
    • Date due: April 14 Tuesday.
    • Reading from my notes: Sections 7.3&7.4 (pages 64&65).
    • Exercise due: If x = f(u, v), y = g(u, v), and z = h(u, v), where f, g, and h are differentiable functions, express each of dy ∧ dz, dz ∧ dx, and dx ∧ dy using partial derivatives and du ∧ dv.
36. Integrals across surfaces:
    • Date due: April 15 Wednesday.
    • Reading from the textbook:
      • Section 15.6 introduction (page 900);
      • Section 15.6 from "Surface Integrals of Vector Fields" to "Computing a Surface Integral for a Level Surface" (pages 904–906).
    • Reading from my notes: Section 7.5 (page 66).
    • Exercises due:
      1. If you parametrize a closed surface containing the origin using the spherical coordinates φ and θ (using the U.S. mathematicians' convention for which of these is which) and orient (by which I technically mean pseudoorient) this surface outwards, then (using the right-hand rule in a right-handed coordinate system to interpret this as an honest orientation) does this orientation correspond to increasing φ followed by increasing θ (that is dφ ∧ dθ) or to increasing θ followed by increasing φ (that is dθ ∧ dφ)?
      2. Write down a formula for the pseudooriented surface element dS = n dσ on a parametrized surface in terms of the coordinates (x, y, z) and/or the parameters (u, v) and their differentials and/or partial derivatives. (There are multiple correct answers to this throughout the readings from the textbook and my notes.)
37. Integrals on surfaces:
    • Date due: April 16 Thursday.
    • Reading from my notes: Section 6.6 (pages 66&67).
    • Readings from the textbook:
      • The rest of Section 15.5 (pages 891–898);
      • Section 15.6 "Surface Integrals" (pages 900–903).
    • Exercises due:
      1. Write down a formula for the surface area element dσ = |dS| on a parametrized surface in terms of the coordinates (x, y, z) and/or the parameters (u, v) and their differentials and/or partial derivatives. (There are multiple correct answers to this throughout the readings from the textbook and my notes.)
      2. If f is a continuous function of two variables with a compact domain R, write down a double integral for the surface area of the graph of f, using f and its partial derivatives.
38. Moments:
    • Date due: April 20 Monday.
    • Reading from the textbook:
      • Section 14.6 (page 813–818);
      • Section 15.1 "Mass and Moment Calculations" (pages 850&851);
      • Section 15.6 "Moments and Masses of Thin Shells" (pages 906–908).
    • Exercises due:
      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.
39. Conservative vector fields and exact differential forms:
    • Date due: April 21 Tuesday.
    • Reading from my notes: Section 5.4 (pages 50&51).
    • Reading from the textbook: Section 15.3 (pages 867–876).
    • Exercises due (true or false):
      1. If f is a differentiable scalar field (a function of several variables), 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.
40. Exterior differentials:
    • Date due: April 22 Wednesday.
    • Reading from my notes: Chapter 8 through Section 8.2 (pages 69–71).
    • Exercises due: 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.
41. Green's Theorem:
    • Date due: April 27 Monday.
    • Reading from my notes: Section 8.3 (pages 72&73).
    • Reading from the textbook: Section 15.4 (pages 878–887).
    • Exercise due: Write down as many different versions of the general statement of Green's Theorem as you can think of. (There are some in both the textbook and 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.)
42. Stokes's Theorem:
    • Date due: April 28 Tuesday.
    • Reading from my notes: Section 8.4 (page 74).
    • Reading from the textbook: Section 15.7 (pages 910–921).
    • Exercises due:
      1. Suppose that you have a compact surface in 3-dimensional space, the z-axis passes through this surface, the surface is oriented (by which I really mean pseudooriented) so that z is increasing along the z-axis through the surface, and you orient the boundary of this surface using the right-hand rule in a right-handed coordinate system as usual. Is the cylindrical coordinate θ increasing or decreasing overall along the boundary curve?
      2. Given f(x, y, z) = 2x³y² cos(e^sin(z)), what is ∇ × ∇f, the curl of the gradient of f? (Hint: If you're doing a bunch of calculations, then you're making this too hard.)
43. Gauss's Theorem:
    • Date due: April 29 Wednesday.
    • Reading from my notes: Section 8.5 (page 75).
    • Reading from the textbook: Section 15.8 (pages 923–931).
    • Exercises due:
      1. Suppose that you have a compact region in 3-dimensional space, bounded by a closed surface, the origin lies within this region, and you orient (by which I really mean pseudoorient) the boundary in the usual way for a closed surface bounding a compact region. Is the spherical coordinate ρ increasing or decreasing overall through the boundary surface?
      2. Given F(x, y, z) = ⟨2x³y², cos(e^sin(z)), sin(e^cos(z))⟩ = 2x³y²i + cos(e^sin(z))j + sin(e^cos(z))k, what is ∇ ⋅ ∇ × F, the divergence of the curl of F? (Hint: If you're doing a bunch of calculations, then you're making this too hard.)

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