# MATH-2080-WBP01

Welcome to the permanent home page for Section WBP01 of MATH-2080 (Calculus 3) at Southeast Community College in the Fall term of 2021. I am Toby Bartels, the instructor.

## Contact information

I am often available outside of those times; feel free to send a message any time.

The official textbook for the course is the 4th Edition of University Calculus: Early Transcendentals by Hass et al published by Addison Wesley (Pearson). You will automatically get an online version of this textbook through Canvas, although you can use a print version instead if you like. This comes with access to Pearson MyLabs, integrated into Canvas, on which many of the assignments appear. There is also a packet of course notes (DjVu).

### Curves and functions

1. General review:
• Reading from the textbook: As needed: Review §§11.1–11.5.
• Reading from my notes: Optional: Through Chapter 1 (pages 1–17).
• Online notes: Required: Vector operations.
• Exercises due on August 25 Wednesday (submit these 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 26 Thursday (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, 11.2.5, 11.3.1, 11.4.1, 11.4.15, 11.5.23, 11.5.39.
2. Parametrized curves:
• Reading from the textbook: Chapter 12 through Section 12.1 (pages 662–668).
• Reading from my notes: Chapter 2 through the first half of Section 2.1 (all of page 19 and the first two lines of page 20).
• Exercises due on August 26 Thursday (submit these 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 27 Friday (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:
• Exercises due on August 30 Monday (submit these on Canvas):
1. For the oriented line segment from (x1, y1) to (x2, y2), write down the usual parametrization.
2. More generally, for the oriented line segment from P1 to P2, write down the usual parametrization.
3. For the circle in the 2-dimensional plane whose centre is (h, k) and whose radius is r, write down the usual parametrization.
4. If f is continuous function whose domain is [a, b], write down the usual parametrization for the graph of f.
• Exercises from the textbook due on August 31 Tuesday (submit these through MyLab): 15.1.1, 15.1,3, 15.1.5, 15.1.7.
4. Integrating parametrized curves:
• Reading from the textbook: Section 12.2 (pages 671–675).
• Reading from my notes: The second half of Section 2.1 (the rest of page 20).
• Exercises due on August 31 Tuesday (submit these 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 ∫bt=af(t) dt = ∫bt=av dt (where a and b are scalars) be a point, a scalar, a vector, or what?
2. What type of value can its indefinite integrals take?; that is, can ∫ f(t) dt = ∫ v dt be a point, a scalar, a vector, or what?
• Exercises from the textbook due on September 1 Wednesday (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 2.4 (pages 23&24).
• Exercises due on September 1 Wednesday (submit these 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 September 2 Thursday (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. Functions of several variables:
• Reading from my notes: Chapter 3 through Section 3.1 (pages 27–29).
• 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 on September 2 Thursday (submit these on Canvas):
1. If f(2, 3) = 5, then what number or point must belong to the domain of f and what number or point must belong to the range of f?
2. If f(2, 3) = 5, then what point must be on the graph of f?
• Exercises from the textbook due on September 3 Friday (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.
7. Topology in several variables:
• Reading from the textbook: Section 13.1 "Functions of Two Variables" (pages 698&699).
• Reading from my notes: Sections 3.2&3.3 (pages 29&30).
• Exercises due on September 7 Tuesday (submit these 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 8 Wednesday (submit these through MyLab): 13.1.17, 13.1.19, 13.1.23, 13.1.25, 13.1.27, 13.2.31, 13.2.33, 13.2.35, 13.2.39.
8. Limits in several variables:
• Section 13.2 (pages 705–711).
• Reading from my notes: Section 3.4 (pages 30&31).
• Exercises due on September 8 Wednesday (submit these 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)? (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=0f(x, y) = 4), and the limit of f approaching (0, 0) vertically is 6 (so lim(x,y)→(0,0),x=0f(x, y) = 6). What (if anything) is the limit of f approaching (0, 0)? (so lim(x,y)→(0,0)f(x, y) = ___).
• Exercises from the textbook due on September 9 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.
9. Vector fields:
• Reading from the textbook: Section 15.2 through "Vector Fields" (pages 854&855 and Figures 15.7–15.16), except Figure 15.11.
• Online notes: Examples of vector fields.
• Exercises due on September 9 Thursday (submit these 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 10 Friday (submit these through MyLab): 15.2.5, 15.2.47, 15.2.49, 15.2.51.
10. Linear differential forms:
• Reading from my notes: Chapter 4 through Section 4.3 (pages 33&34).
• Exercises due on September 13 Monday (submit these 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 14 Tuesday (submit these on Canvas too):
1. Evaluate 3x dx + 4x2y dy at (x, y) = (2, 6) along ⟨dx, dy⟩ = ⟨0.003, 0.005⟩. (Answer.)
2. Evaluate 2xy dx + 2yz dy + 2xz dz at (x, y, z) = (−1, 3, 2) along ⟨dx, dy, dz⟩ = ⟨0.01, 0.02, −0.01⟩.
3. Evaluate x2 dx + xy dy + xz dz at (x, y, z) = (4, 3, −2). (Answer.)
4. Evaluate 5x2 dx − 3xy dy at (x, y) = (1, 2).
11. Differentials:
• Readings from my notes: Section 4.4 (pages 35&36).
• Exercises due on September 14 Tuesday (submit these 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 eu using e, u, and du.
4. Write the differential of ln u = logeu using u and du.
5. Write the differentials of sin u and cos u using u, du, and trigonometric operations.
• Exercises not from the textbook due on September 15 Wednesday (submit these on Canvas too):
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).
Quiz 1, covering the material in Problem Sets 1–11, is available on September 17 Friday.

### Differentiation

1. Partial derivatives:
• Readings from my notes: Section 4.5 (pages 36&37).
• Reading from the textbook: Section 13.3 through "Functions of More than Two Variables" (pages 714–719).
• Exercises due on February 2 Tuesday (submit these 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 February 3 Wednesday (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.
2. Levels of differentiability:
• Readings from my notes: Sections 3.5&3.6 (page 32).
• Reading from the textbook: The rest of Section 13.3 (pages 719–723).
• Exercises due on February 3 Wednesday (submit these 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 February 4 Thursday (submit these through MyLab): 13.3.43, 13.3.45, 13.3.61, 13.3.85, 13.3.93, 13.3.101.
3. Directional derivatives:
• Section 13.5 through "Calculation and Gradients" (pages 736–738);
• Section 13.5 from "Functions of Three Variables" (pages 742&743).
• Reading from my notes: Sections 4.6&4.7 (pages 37–39).
• Exercises due on February 8 Monday (submit these on Canvas): Suppose that ∇f(2, 3) = ⟨3/5, 4/5⟩.
1. In which direction u is the directional derivative Duf(2, 3) the greatest?
2. In which directions u is the directional derivative Duf(2, 3) equal to zero?
3. In which direction u is the directional derivative Duf(2, 3) the least (with a large absolute value but negative)?
• Exercises from the textbook due on February 9 Tuesday (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.
• Section 15.2 Figure 15.11 (page 855);
• Section 15.2 "Gradient Fields" (pages 855&856).
• Exercises due on February 9 Tuesday (submit these 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 February 10 Wednesday (submit these through MyLab): 15.2.1, 15.2.2, 15.2.3, 15.2.4.
5. Matrices:
• Reading from my notes: Section 1.13 (page 17).
• Exercises due on February 10 Wednesday (submit these 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 February 11 Thursday: Take the Mathopolis quiz on multiplying matrices, and Submit a message on Canvas telling me how it went.
6. The Chain Rule:
• Reading from the textbook: Section 13.4 (pages 726–733).
• Reading from my notes: Section 4.8 (pages 39&40).
• Exercise due on February 11 Thursday (submit this 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 February 15 Monday (submit these through MyLab): 13.4.1, 13.4.3, 13.4.7, 13.4.9, 13.4.41.
7. Tangent flats and normal lines:
• Section 13.5 "Gradients and Tangents to Level Curves" (pages 740&741);
• Section 13.6 "Tangent Planes and Normal Lines" (pages 744–746).
• Reading from my notes: Section 4.9 (pages 40&41).
• Exercises due on February 15 Monday (submit these on Canvas): Fill in each blank with ‘line’ or ‘plane’.
1. If ∇f(a, b) exists but is not zero, then f has a tanget ___ and a normal ___ through (a, b).
2. If ∇f(a, b, c) exists but is not zero, then f has a tanget ___ and a normal ___ through (a, b, c).
• Exercises from the textbook due on February 16 Tuesday (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.
8. Linearization:
• Section 13.6 "How to Linearize a Function of Two Variables" (pages 747–749).
• Section 13.6 "Functions of More Than Two Variables" (pages 750&751).
• Reading from my notes: Section 4.10 (pages 41–44).
• Exercises due on February 16 Tuesday (submit these 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 February 17 Wednesday (submit these through MyLab): 13.6.31, 13.6.33, 13.6.35, 13.6.39, 13.6.41.
9. Estimation:
• Section 13.6 "Estimating Change in a Specific Direction" (page 747).
• Section 13.6 "Differentials" (pages 749&750).
• Exercises due on February 17 Wednesday (submit these on Canvas):
1. If f is a function of two variables and f is differentiable at a point P0 = (x0, y0), then about how much does the value of f change at that point if you move a distance of Δs in the direction of the vector v? (Your answer should involve f or its gradient or partial derivatives, the distance Δs or ds, and v or its length or direction. If you have any other quantity in your answer, then explain how to get it from these.)
2. If (∂u/∂x)y = −3 and (∂u/∂y)x = 2, then is the quantity u more or less sensitive to small changes in x compared to changes in y?
• Exercises from the textbook due on February 18 Thursday (submit these through MyLab): 13.6.21, 13.6.23, 13.6.51, 13.6.55.
10. Local optimization:
• Reading from my notes: Section 4.11 (pages 44&45).
• Section 13.7 through "Derivative Tests for Local Extreme Values" (pages 754–758).
• Exercises due on February 18 Thursday (submit these 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 February 22 Monday (submit these through MyLab): 13.7.2, 13.7.7, 13.7.9, 13.7.15, 13.7.27, 13.7.43.
11. Constrained optimization:
• Reading from the textbook: The rest of Section 13.7 (pages 758–760).
• Exercise due on February 23 Tuesday (submit this 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 will be given partially by strict inequalities and partially by equations. Be sure to give the final total including the 9 that I've mentioned in this hint. A picture may help.)
• Exercises from the textbook due on February 24 Wednesday (submit these through MyLab): 13.7.31, 13.7.33, 13.7.37, 13.7.51, 13.7.59.
12. Lagrange multipliers:
• Reading from the textbook: Section 13.8 (pages 763–770).
• Exercises due on February 24 Wednesday (submit these 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 February 25 Thursday (submit these through MyLab): 13.8.1, 13.8.3, 13.8.9, 13.8.11, 13.8.15, 13.8.23.
Quiz 2, covering the material in Problem Sets 12–23, is available on March 4 Thursday.

### Integration

1. Integration on curves:
• Reading from my notes: Chapter 5 through Section 5.2 (pages 47&48).
• Reading from the textbook: Section 15.2 "Line Integrals with Respect to dx, dy, or dz" (pages 857&858).
• Exercises due on February 25 Thursday (submit these 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 cos4t dt or ∫0 cos4t dt?
• Exercises from the textbook due on March 1 Monday (submit these through MyLab): 15.2.13, 15.2.15, 15.2.17, 15.2.23.
2. Integrating vector fields:
• Section 5.3 (page 48);
• Section 5.5 (page 49).
• Reading from the textbook: The rest of Section 15.2 (pages 856&857, 859–863).
• Exercises due on March 1 Monday (submit these 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 F(x, y) = ⟨x2, 3⟩ = x2i +3j across a curve in the (x, y)-plane, what differential form do you integrate along the curve?
3. To integrate inwards across a circle, should the circle be oriented clockwise or counterclockwise (using a counterclockwise coordinate system as usual)?
• Exercises from the textbook due on March 2 Tuesday (submit these through MyLab): 15.2.9, 15.2.11, 15.2.19, 15.2.21, 15.2.29, 15.2.33.
3. Integrating scalar fields:
• Reading from my notes: Section 5.4 (page 49).
• Reading from the textbook: Section 15.1 except for "Mass and Moment Calculations" (pages 847–850, pages 851&852).
• Exercises due on March 2 Tuesday (submit these 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 ∫0f(sin t, cos t) dt or ∫0f(sin t, cos t) dt?)
• Exercises from the textbook due on March 3 Wednesday (submit these through MyLab): 15.1.9, 15.1.13, 15.1.15, 15.1.21, 15.1.30.
4. Double integrals on rectangles:
• Reading from the textbook: Chapter 14 through Section 14.1 (pages 779–783).
• Exercises due on March 3 Wednesday (submit these on Canvas):
1. Rewrite ∫ba ∫dcf(x, y) dy dx as an iterated integral ending with dx dy.
2. Assuming that f is continuous everywhere, is it possible that these two iterated integrals could evaluate to different results?
• Exercises from the textbook due on March 4 Thursday (submit these through MyLab): 14.1.3, 14.1.6, 14.1.10, 14.1.19, 14.1.23.
5. Double integrals:
• Reading from the textbook: Section 14.2 (pages 784–790).
• Reading from my notes: Chapter 6 through Section 6.2 (pages 53&54).
• Exercises due on March 8 Monday (submit these on Canvas):
1. Suppose that a and b are real numbers with a ≤ b and g and h are functions, both continuous on [a, b], with g ≤ h on [a, b]. Let R be the region {x, y | a ≤ x ≤ b, g(x) ≤ y ≤ h(x)}, and suppose that f is a function of two variables, continuous on R. Write an iterated integral equal to the double integral of f on R.
2. Suppose that c and d are real numbers with c ≤ d and g and h are functions, both continuous on [c, d], with g ≤ h on [c, d]. Let R be the region {x, y | g(y) ≤ x ≤ h(y), c ≤ y ≤ d}, and suppose that f is a function of two variables, continuous on R. Write an iterated integral equal to the double integral of f on R.
• Exercises from the textbook due on March 9 Tuesday (submit these through MyLab): 14.2.1, 14.2.2, 14.2.7, 14.2.19, 14.2.23, 14.2.79.
6. Systems of inequalities:
• Reading from my notes: Section 6.3 (pages 55–57).
• Exercises due on March 9 Tuesday (submit these 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 March 10 Wednesday (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.
7. Triple integrals:
• Reading from the textbook: Section 14.5 except "Volume of a Region in Space" and "Average value of a function in space" (pages 803, 804–810, 810).
• Exercise due on March 10 Wednesday (submit this on Canvas): In how many ways can you order 3 variables of integration? List them.
• Exercises from the textbook due on March 11 Thursday (submit these through MyLab): 14.5.10, 14.5.15, 14.5.3, 14.5.5, 14.5.21.
8. Areas, volumes, and averages:
• 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 March 11 Thursday (submit these 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 §14.2 of the textbook, as an iterated double integral in the variables x and y;
2. Using ideas from §14.5 of the textbook, as an iterated triple integral in the variables x, y, and z.
(If you want to check your answers somewhat: You shouldn't be able to evaluate your answer to #1, because I haven't told you which function f is; however, you should be able to begin evaluating your answer to #2 if you write the variables in an appropriate order, and this should turn it into your answer from #1, after which you shouldn't be able to go any further.)
• Exercises from the textbook due on March 22 Monday (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.
9. The area element:
• Reading from my notes: Sections 6.4&6.5 (pages 57–60).
• Exercises due on March 22 Monday (submit these 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. Let P, Q, and R be three points in R2; write ⟨a, b⟩ for the vector Q − P, and write ⟨c, d⟩ for the vector R − P. Express the area of the triangle with vertices P, Q, and R using only 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 March 23 Tuesday (submit these on Canvas too):
1. Evaluate dx ∧ dy along d(x, y) = ⟨−2, 3⟩, ⟨4, 6⟩. (Answer.)
2. Evaluate dx ∧ dy along d(x, y) = ⟨4, −9⟩, ⟨6, 3⟩.
3. Find the area of a triangle if two of the vectors along its sides are ⟨−2, 3⟩ and ⟨4, 6⟩. (Answer.)
4. Find the area of a triangle if two of the vectors along its sides are ⟨4, −9⟩ and ⟨6, 3⟩.
10. Coordinate transformations:
• Reading from my notes: Section 6.6 (pages 60&61).
• Reading from the textbook: Section 14.8 (pages 832–839).
• Exercise due on March 23 Tuesday (submit this 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, rather than some more sophisticated notation instead.)
• Exercises from the textbook due on March 24 Wednesday (submit these through MyLab): 14.8.1, 14.8.3, 14.8.7, 14.8.9, 14.8.17, 14.8.22.
11. Polar coordinates:
• Reading from my notes: Sections 2.5&2.6 (pages 24–26).
• Exercises due on March 24 Wednesday (submit these 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 these, express the rectangular coordinates x, y, and z in terms of the spherical coordinates ρ, φ, and θ.
• Exercises from the textbook due on March 25 Thursday (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.
12. Area integrals in polar coordinates:
• Reading from my notes: Section 6.7 (pages 62&63).
• Reading from the textbook: Section 14.4 (pages 796–801).
• Exercises due on March 25 Thursday (submit these on Canvas):
1. Give a formula for the area element in the plane in rectangular coordinates x and y.
2. Give a formula for the area element in the plane in polar coordinates r and θ.
• Exercises from the textbook due on March 29 Monday (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.
13. Volume integrals in polar coordinates:
• Reading from the textbook: Section 14.7 (pages 820–828).
• Exercises due on March 29 Monday (submit these on Canvas):
1. Give a formula for the volume element in space in rectangular coordinates x, y, and z.
2. Give a formula for the volume element in space in cylindrical coordinates r, θ, and z.
3. Give a formula for the volume element in space in spherical coordinates ρ, φ, and θ (using the American mathematicians' convention for which of these is which).
• Exercises from the textbook due on March 30 Tuesday (submit these through MyLab): 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.
Quiz 3, covering the material in Problem Sets 24–36, is available on April 1 Thursday.

### More integration

1. Parametrized surfaces:
• Section 11.6 (pages 651–655);
• Section 15.5 through "Parametrizations of Surfaces (pages 890&891).
• Reading from my notes: Chapter 7 through Section 7.1 (page 65).
• Exercises due on March 30 Tuesday (submit these 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 March 31 Wednesday (submit these through MyLab): 15.5.1, 15.5.3, 15.5.5, 15.5.9, 15.5.13.
2. Integrals along surfaces:
• Reading from my notes: Sections 7.2–7.4 (pages 66–68).
• Exercise due on March 31 Wednesday (submit this 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 and du ∧ dv.
• Exercises not from the textbook due on April 1 Thursday: TBA.
3. Flux across surfaces:
• Section 15.6 introduction (page 900);
• Section 15.6 from "Orientation of a Surface" to "Computing a Surface Integral for a Level Surface" (pages 904–906).
• Reading from my notes: Section 7.5 (pages 68&69).
• Exercises due on April 5 Monday (submit these 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 dS = n dσ on a parametrized surface in terms of the coordinates (x, y, z) and/or the parameters (u, v) and their differentials and/or partial derivatives. (There are multiple correct answers to this throughout the readings from the textbook and my notes.)
• Exercises from the textbook due on April 6 Tuesday (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.
4. Integrals on surfaces:
• Reading from my notes: Section 7.6 (pages 69&70).
• The rest of Section 15.5 (pages 891–898);
• Section 15.6 "Surface Integrals" (pages 900–903).
• Exercises due on April 6 Tuesday (submit these on Canvas):
1. Write down a formula for the surface area element dσ = |dS| on a parametrized surface in terms of the coordinates (x, y, z) and/or the parameters (u, v) and their differentials and/or partial derivatives. (There are multiple correct answers to this throughout the readings from the textbook and my notes.)
2. If f is a continuous function of two variables with a compact domain R, write down a double integral for the surface area of the graph of f, using f and its partial derivatives.
• Exercises from the textbook due on April 7 Wednesday (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.
5. Moments:
• 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 April 7 Wednesday (submit these on Canvas):
1. Give the formulas for the centre of mass (, ȳ, ) of a three-dimensional solid in terms of the total mass M and the moments Mx,y, Mx,z, and My,z.
2. Give a formula for the polar moment of inertia I0 of a two-dimensional plate in terms of the moments of inertia Ix and Iy about the coordinate axes.
• Exercises from the textbook due on April 8 Thursday (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.
6. Conservative vector fields and exact differential forms:
• Reading from my notes: Section 5.6 (pages 50&51).
• Reading from the textbook: Section 15.3 (pages 867–876).
• Exercises due on April 8 Thursday (submit these 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 April 12 Monday (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.
7. Exterior differentials:
• Reading from my notes: Chapter 8 through Section 8.1 (pages 71–73).
• Exercises due on April 12 Monday (submit these 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 April 13 Tuesday (submit these on Canvas too): 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.
8. Green's Theorem:
• Reading from my notes: Section 8.3 (pages 74&75).
• Reading from the textbook: Section 15.4 (pages 878–887).
• Exercise due on April 14 Wednesday (submit this 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 April 15 Thursday (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.
9. Stokes's Theorem:
• Reading from my notes: Section 8.4 (page 76).
• Reading from the textbook: Section 15.7 (pages 910–921).
• Exercises due on April 15 Thursday (submit these 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 April 19 Monday (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.
10. Gauss's Theorem:
• Reading from my notes: Section 8.5 (page 77).
• Reading from the textbook: Section 15.8 (pages 923–931).
• Exercises due on April 19 Monday (submit these 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 April 20 Tuesday (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.
11. Cohomology:
• Reading from my notes: Section 8.2 (page 73).
• Exercises due on April 20 Tuesday (submit these 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 esin 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 esin z, sin ecos z⟩ = 2x3y2i + cos esin zj + sin ecos zk, 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 April 21 Wednesday (submit these through MyLab): 15.3.25, 15.4.45, 15.7.27, 15.8.23.
Quiz 4, covering the material in Problem Sets 37–47, is available on April 22 Thursday.

## Quizzes

1. Curves and functions:
• Review date: February 4 Thursday (in class or through Zoom).
• Date due on MyLab: February 8 Monday (before 1:00 PM).
• Corresponding problems sets: 1–11.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
• Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result except #3 & #6.
2. Differentiation:
• Review date: March 4 Thursday (in class or through Zoom).
• Date due on MyLab: March 8 Monday (before 1:00 PM).
• Corresponding problems sets: 12–23.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
• Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result.
3. Integration:
• Review date: April 1 Thursday (in class or through Zoom).
• Date due on MyLab: April 5 Monday (before 1:00 PM).
• Corresponding problems sets: 24–36.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
• Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result except #5.
4. More integration:
• Review date: April 22 Thursday (in class or through Zoom).
• Date due on MyLab: April 26 Monday (before 1:00 PM).
• Corresponding problems sets: 37, 39–47.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
• Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result.

## Final exam

There is a comprehensive final exam at the end of the term. (You'll arrange to take it some time from April 30 to May 6.) To speed up grading at the end of the term, the exam will be multiple choice, with no partial credit.

For the exam, you may use one sheet of notes that you wrote yourself; please take a scan or a picture of this (both sides) and submit it on Canvas. However, 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 final exam on MyLab.

The final exam will be proctored. In the face-to-face class, you can take it on May 4 Tuesday, in our normal classroom at the normal time but lasting until 2:40 PM. If you have access to a computer with a webcam, then you can schedule a time with me to take the exam in a Zoom meeting. If you're near Lincoln, then we can schedule a time for you to take the exam in person. If none of these will work for you, then contact me as soon as possible to make alternate arrangements.

This web page and the files linked from it were written by Toby Bartels, last edited on 2021 August 31. Toby reserves no legal rights to them.

The permanent URI of this web page is `http://tobybartels.name/MATH-2080/2021FA/`.