Homework assignments for Jeff Mermin's Math 2163 section (problems on Achieve; go here to enroll.)

Due dateReadingWebAssign ProblemsWritten assignment
This week's assignments
Monday, December 4 Chapter 16.1: Know what vector fields are, and be able to connect them to arrow plots.
Be able to recognize conservative vector fields, based on the cross-partials condition. Given a conservative vector field, be able to find a potential function using integration.
Assignment 36
Wednesday, December 6 Chapter 16.2: Be able to set up and evaluate vector line integrals. (Don't worry about scalar line integrals; we don't have time this semester.) Be able to recognize the situations where vector line integrals are appropriate.
Chapter 16.3: Be able to test vector fields for conservativeness based on the cross-partials condition, and to take advantage of conservative vector fields in vector integrals.
Assignment 37
Friday, December 8 Skim Chapter 17.1, and be aware of Green's theorem, which helps us address non-path-independent integrals.
Review, and bring your questions to class.
Assignment 38
Monday, December 11 Study for the final exam, which is at 10:00 in our usual classroom.Various review and extra credit Achieve assignmentsExtra credit written assignments:
(1):Change of variable (and solutions to the examples)
(2):Area


Previous assignments
Wednesday, August 23Chapters 12.1 and 12.2: Understand what vectors are and why it is useful to represent them using rectangular coordinates. Be able to add, subtract, and scale vectors, both conceptually and with coordinates. Understand the point-direction form for the equation of a line (called the "vector parametrization" on page 685), and be able to work with it. Assignment 1
Friday, August 25Chapter 12.3: Be able to compute the dot product of two vectors, and internalize its algebraic properties (Theorem 1). Understand the geometric interpretation of the dot product in terms of angles and projections (in particular, know what it means for the dot product of two vectors to be zero). Assignment 2
Monday, August 28Chapter 12.4: Memorize the formula for the cross product. (Use the definition in terms of a "symbolic determinant" (Definition, p. 701), or the i, j, k relations (Formulas 5 and 6, p. 703), or the formula (a,b,c) × (d,e,f) = (bf-ce, cd-af, ae-bd).) Practice on dozens of random vectors until it becomes automatic. Make sure that you're getting the correct sign in the second entry. Also internalize the key difference between the two products: The dot product of two vectors is a scalar, and the cross product is a vector. Assignment 3
Wednesday, August 30Chapter 12.4: Be comfortable with the algebraic (Theorem 2) and geometric (Theorem 1) properties of the cross product. Be able to interpret the cross product of two vectors in terms of parallelism, perpendicularity, and area. In particular, know what it means for the cross product of two vectors to be zero, and be able to save yourself some effort by recognizing this situation (i.e., avoid computing the cross product when the geometry tells you it will be zero).
Continue to practice computing both dot and cross products by hand, until you never make mistakes.
Assignment 4
Friday, September 1Chapter 12.5: Understand the equation of a plane, and be able to work with it. Understand that the normal vector is perpendicular to all vectors inside the plane. Be able to compute the equation of a line given two points, and of a plane given three points. Be able to find three points on a plane, or two points on a line, given its equation. Understand the ideas involved in the solutions to Examples 2 and 4. Assignment 5
Monday, September 4 Monday is Labor Day, so there's no class. But if you have the chance over the weekend, get started on this handout. Understand the nature of the questions about points, lines, and planes, and recognize the utility of the cross product in answering them: Given two vectors, we can generate a common perpendicular by taking the cross product.
Monday, September 4 Monday is Labor Day, so there's no class. But if you have the chance over the weekend, get started on this handout. Understand the nature of the questions about points, lines, and planes, and recognize the utility of the cross product in answering them: Given two vectors, we can generate a common perpendicular by taking the cross product.
Wednesday, September 6This handout: Skim the whole thing to understand what the questions are. Understand the key geometric value of the cross product: x × y is perpendicular to both x and y. Choose one of the seven sections, and work through it with at least one partner. Assignment 6
Friday, September 8This handout: Make sure you understand the suggested plans throughout. Work through at least one new section with a partner. Assignment 7
Monday, September 11This handout (and model solutions to section 6): Work through the remaining sections, and make sure you feel confident about this sort of problem. Assignment 8
Wednesday, September 13 Chapter 13.1: Know what vector-valued functions are, and be able to match graphs with parametrizations. Understand the distinction between a path (which has only one parametrization) and a curve (which is just a collection of points, and can have many parametrizations). Assignment 9Written Assignment 1:
Model solutions to section 6 are here.
Your group will turn in section 2 and one other section of your choice (that is, section 1, 3, 4, 5, or 7).
Friday, September 15Chapter 13.2: Understand that limits, derivatives, and integrals of vector-valued functions can all be computed "component-wise", so calculus works out as well as we might hope for these things. Then make sure you remember how to do them from Calculus I and II. Assignment 10
Monday, September 18 Chapter 13.3: Know the formula for the arc-length of a parametrized curve, and be able to set it up given any parametrization. Understand that the resulting integral is almost always impossible, but be able to recognize some of the cases, like those in the book, where it can be evaluated.
Know what the phrase "arc-length parametrization" means, and understand why I haven't assigned any problems on it.
Skim Chapters 13.4 and 13.5, enough that you'll recognize the big ideas in the future.
Assignment 11
Wednesday, September 20Review Chapters 12 and (especially) 13.1 through 13.3, and bring your questions to class. Assignment 12
Friday, September 22Study for the first exam.
Monday, September 25 Chapter 14.1: Understand what functions of two variables are, and be able to tell their graphs apart using traces, level curves, or contour maps. Assignment 13
Wednesday, September 27Chapter 14.1: Be able to sketch the contour graph of a function, and to read off important features from a contour map.
Chapter 14.2: Understand that the ideas of continuity and limits are exactly the same for multivariable functions as they were for single-variable functions. But also know that the definition of limit is considerably more complicated and that in the interesting cases limits very rarely exist. (In particular, understand that Example 7 is a thing that happens.)
Assignment 14
Friday, September 29 Chapter 14.3: Understand that a partial derivative is just a derivative with the other independant variables treated as constants, so that everything you learned in Calculus I still applies. Know the notation and definitions for higher-order partial derivatives, and internalize Clairaut's theorem (but read exercise 80). Assignment 15
Monday, October 2 Chapter 14.4: Know the chain rule: If z is a function of x and y, then dz=zxdx + zydy. (This isn't called the chain rule in the book, but it's the key to all of chapter 14.) Understand what differentials (dx, dy, dz) are, and be able to use them for estimation and finding tangent planes. Assignment 16
Wednesday, October 4 Chapter 14.4: Understand how to think of the differentials dx, dy, and dz as limits of Δx, Δy, and Δz, and continue to emphasize the chain rule dz=zxdx + zydy in your thinking about how they're related. Also observe how the chain rule shows up in the formula for a tangent plane. Assignment 17
Friday, October 6 Chapter 14.5: Know what the gradient vector is, and be able to use it to find "directional derivatives". Understand that directional derivatives are another way to do estimation, and contain essentially the same information as the tangent plane. Assignment 18
Monday, October 9 Chapters 14.4 and 14.5: Understand the relationship between directional derivatives and the chain rule (dz=zxdx + zydy): A directional derivative specifies a value for \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\), so the chain rule (if we know the partials) tells us \(\frac{dz}{dt}\). Practice computing directional derivatives both from this perspective and by taking dot products with the gradient, and understand the relationship between the direction of the gradient and the size of the directional derivative. Assignment 19
Wednesday, October 11 Chapter 14.6: Know what the book calls the chain rule, and understand its relationship to the real chain rule: If z is a function of x and y, then dz=zxdx + zydy. Be aware that this leads to a formula for implicit differentiation, but don't bother learning the formula. Assignment 20
Friday, October 13 Because it should be fall break and isn't, I expect most of you to be exhausted and overwhelmed. This Friday is a catch-up day. Nothing is due in our class, there is no quiz, and I will not lecture on any new material. I will happily answer any questions about any of the material we've discussed so far; if there are no questions, I will remind you about important techniques from Calculus I and/or Algebra III. If you need the time to catch up on sleep, or to stay afloat in your other classes, please use it for that purpose.
If you have time to work ahead in Calculus, begin reading Chapter 14.7 about optimization. Focus on the big ideas, and try to understand how the contour map of a saddle point is qualitatively different than that of a local maximum or minimum.
Monday, October 16 Chapter 14.7: Understand what critical points are. Be able to recognize critical points, and master the algebraic procedure for finding them. Know the three different types of critical point, and know what their contour maps look like. Assignment 20
Wednesday, October 18 Chapter 14.7: Memorize the second derivative test (which is much more complicated than the one from Calculus I) for classifying critical points as local maxima, local minima, or saddle points. Understand how dramatically more complicated the situation has become. Assignment 21
Friday, October 20 Chapter 14.7: Understand how to think about global extrema on region. Notice that the idea (find the critical points on the interior, then test the boundary) is essentially the same as in Calculus I, but the execution is much slower and more annoying.
Chapter 14.8: Skim this chapter. Pay attention to the pictures and surrounding discussion, not the (extremely difficult) algebra, which we'll address next week.
Assignment 22
Monday, October 23 Chapter 14.8: Understand why the technique of Lagrange multipliers works when maximizing a function subject to an algebraic constraint. Be able to write down the necessary equations (and, in simple cases, solve them). Assignment 23
Wednesday, October 25 Chapter 14.8: Practice writing down and solving the equations involved in using Lagrange multipliers. The algebra here has a tendency to get really hard; you cannot expect to use one problem as an example of the algebraic steps to attack another. The only course is lots of practice, in which you manipulate the equations in many (algebraically correct) ways, even if you don't expect your manipulations to be profitable.
Two things to keep in mind, as useful general principles:
  1. You'll usually start with three equations in the variables x,y,λ. You don't actually care about the value of λ, and two of the equations can be easily solved for λ in terms of x and y. It is usually a good idea to do so, and then observe that the two expressions for λ are equal. This will (usually) reduce to the case where you have two equations in the two variables x and y.
  2. Remember that division by zero yields nonsense. If you are ever tempted to divide by a function h(x,y) (either because you need to do it to solve for λ or because you're trying to cancel a common factor), remember that you don't know whether or not h(x,y) is equal to zero. Thus, the desire to do this division is actually dividing the problem into two cases, each of which is likely to contain a critical point.
    The first case is where h(x,y)=0. Here you don't get to divide, but you do get to assume that h(x,y)=0, which will almost always be very helpful in solving the system.
    The second case is the one where you do get to divide. You don't get the power of a new equation, but this is what you were tempted to do all along, and the division is extremely likely to help.
Assignment 24
Friday, October 27 Chapter 14.8: Continue to practice the algebra involved in Lagrange multipliers. Try doing all the odd-numbered problems in the text, and checking with the back of the book. Also make sure you are clear on how to set up a problem with three variables and two constraints.
Chapters 14.5 through 14.7: Review the relationships among the chain rule ( dz=zxdx + zydy), tangent vectors (<dx,dy,dz>), and the gradient vector, and be sure you can solve questions involving these objects. Also make sure you know how to find and classify critical points.
Assignment 25
Monday, October 30 Chapter 15.1: Understand that volumes can be approximated by Riemann sums of rectangular prisms, just as areas were approximated by Riemann sums of rectangles. Understand that a double integral expresses the limit of such a Riemann sum, and so computes the exact volume. Be able to compute (definite) double integrals over a rectangular base using the fundamental theorem of calculus and/or Fubini's theorem. Be aware that Fubini's theorem is a useful tool for making hard problems easier. Assignment 26
Wednesday, November 1 Chapter 15.2: Be very good at computing double integrals when the limits of integration are given. Assignment 27
Friday, November 3 Study for the exam. This written assigment about the chain rule. (And solutions.)
Monday, November 6 Chapter 15.2: Master setting up double integrals based on a picture or a verbal description of the region. Assignment 28
Wednesday, November 8 Chapter 15.3: Understand that the mechanics for an iterated triple integral are essentially the same as for an iterated double integral, but we have to integrate three times instead of two. Be comfortable with Fubini's theorem about switching order, and recognize that in principle there are six orders to choose from. Make sure you're comfortable through example two. Assignment 29
Friday, November 10 Chapter 15.3: Be comfortable with the idea of setting up triple integrals over an arbitrary region, and practice with as many examples as you can. (Soon you'll want to feel like you could handle problems such as Example 5 or Example 6 without having to look anything up or seek help.) Assignment 30 This written assigment about Riemann sums and simple double integrals.
Monday, November 13 Chapter 12.7: Know what polar, cylindrical, and spherical coordinates are, and be comfortable converting both points and equations from polar to rectangular coordinates and back.
Chapter 15.3: Keep practicing.
Assignment 31
Wednesday, November 15 Chapter 15.3: Keep practicing, until you feel very comfortable setting these problems up. Assignment 32
Friday, November 17 Chapter 15.3: Keep practicing, until you have mastered these sorts of problems.
Chapter 15.4: Be able to recognize "polar rectangles" and similar, and to know when it might make sense to integrate using polar or spherical coordinates.
Be very clear that dxdy is not equal to drdθ, and that dxdydz is not equal to dρdθdφ. Memorize the appropriate conversions, or know exactly where to go to look them up.
Assignment 33
Monday, November 27 Get some rest, and enjoy the break!
To the extent that you have some time for math (and for sure in the weekend before we return to school), practice the sorts of problems that appear in chapters 15.2, 15.3, and 15.4. Make sure that you can handle the sort of problems that appear in the polar coordinates chapter (either by using polar or spherical coordinates, or by correctly writing the messier solutions using rectangular coordinates.
Assignment 34
Wednesday, November 29 Review Chapter 15, and bring your questions to class. Assignment 35
Friday, December 1 Study for the exam.


Ambitions at the start of the semester
Wednesday, August 23Chapters 12.1 and 12.2: Understand what vectors are and why it is useful to represent them using rectangular coordinates. Be able to add, subtract, and scale vectors, both conceptually and with coordinates. Understand the point-direction form for the equation of a line (called the "vector parametrization" on page 685), and be able to work with it. Assignment 1
Friday, August 25Chapter 12.3: Be able to compute the dot product of two vectors, and internalize its algebraic properties (Theorem 1). Understand the geometric interpretation of the dot product in terms of angles and projections (in particular, know what it means for the dot product of two vectors to be zero). Assignment 2
Monday, August 28Chapter 12.4: Memorize the formula for the cross product. (Use the definition in terms of a "symbolic determinant" (Definition, p. 701), or the i, j, k relations (Formulas 5 and 6, p. 703), or the formula (a,b,c) × (d,e,f) = (bf-ce, cd-af, ae-bd).) Practice on dozens of random vectors until it becomes automatic. Make sure that you're getting the correct sign in the second entry. Also internalize the key difference between the two products: The dot product of two vectors is a scalar, and the cross product is a vector. Assignment 3
Wednesday, August 30Chapter 12.4: Be comfortable with the algebraic (Theorem 2) and geometric (Theorem 1) properties of the cross product. Be able to interpret the cross product of two vectors in terms of parallelism, perpendicularity, and area. In particular, know what it means for the cross product of two vectors to be zero, and be able to save yourself some effort by recognizing this situation (i.e., avoid computing the cross product when the geometry tells you it will be zero).
Continue to practice computing both dot and cross products by hand, until you never make mistakes.
Assignment 4
Friday, September 1Chapter 12.5: Understand the equation of a plane, and be able to work with it. Understand that the normal vector is perpendicular to all vectors inside the plane. Be able to compute the equation of a line given two points, and of a plane given three points. Be able to find three points on a plane, or two points on a line, given its equation. Understand the ideas involved in the solutions to Examples 2 and 4. Assignment 5
Monday, September 4 Monday is Labor Day, so there's no class. But if you have the chance over the weekend, get started on this handout. Understand the nature of the questions about points, lines, and planes, and recognize the utility of the cross product in answering them: Given two vectors, we can generate a common perpendicular by taking the cross product.
Wednesday, September 6This handout: Skim the whole thing to understand what the questions are. Understand the key geometric value of the cross product: x × y is perpendicular to both x and y. Choose one of the seven sections, and work through it with at least one partner. Assignment 6
Friday, September 8This handout: Make sure you understand the suggested plans throughout. Work through at least one new section with a partner. Assignment 7
Monday, September 11This handout: Work through the remaining sections, and make sure you feel confident about this sort of problem. Assignment 8
Wednesday, September 13 Chapter 13.1: Know what vector-valued functions are, and be able to match graphs with parametrizations. Understand the distinction between a path (which has only one parametrization) and a curve (which is just a collection of points, and can have many parametrizations). Assignment 9Written Assignment 1
Friday, September 15Chapter 13.2: Understand that limits, derivatives, and integrals of vector-valued functions can all be computed "component-wise", so calculus works out as well as we might hope for these things. Then make sure you remember how to do them from Calculus I and II. Assignment 10
Monday, September 18 Chapter 13.3: Know the formula for the arc-length of a parametrized curve, and be able to set it up given any parametrization. Understand that the resulting integral is almost always impossible, but be able to recognize some of the cases, like those in the book, where it can be evaluated.
Know what the phrase "arc-length parametrization" means, and understand why I haven't assigned any problems on it.
Skim Chapters 13.4 and 13.5, enough that you'll recognize the big ideas in the future.
Assignment 11
Wednesday, September 20Review Chapters 12 and 13, and bring your questions to class. Assignment 12
Friday, September 22Study for the first exam.
Monday, September 25 Chapter 14.1: Understand what functions of two variables are, and be able to tell their graphs apart using traces, level curves, or contour maps. Assignment 13
Wednesday, September 27Chapter 14.1: Be able to sketch the contour graph of a function, and to read off important features from a contour map.
Chapter 14.2: Understand that the ideas of continuity and limits are exactly the same for multivariable functions as they were for single-variable functions. But also know that the definition of limit is considerably more complicated and that in the interesting cases limits very rarely exist. (In particular, understand that Example 7 is a thing that happens.)
Assignment 14
Friday, September 29 Chapter 14.3: Understand that a partial derivative is just a derivative with the other independant variables treated as constants, so that everything you learned in Calculus I still applies. Know the notation and definitions for higher-order partial derivatives, and internalize Clairaut's theorem (but read exercise 80). Assignment 15
Monday, October 2 Chapter 14.4: Know the chain rule: If z is a function of x and y, then dz=zxdx + zydy. (This isn't called the chain rule in the book, but it's the key to all of chapter 14.) Understand what differentials (dx, dy, dz) are, and be able to use them for estimation and finding tangent planes. Assignment 16
Wednesday, October 4 Chapter 14.4: Understand how to think of the differentials dx, dy, and dz as limits of Δx, Δy, and Δz, and continue to emphasize the chain rule dz=zxdx + zydy in your thinking about how they're related. Also observe how the chain rule shows up in the formula for a tangent plane. Assignment 17
Friday, October 6 Chapter 14.5: Know what the gradient vector is, and be able to use it to find "directional derivatives". Understand that directional derivatives are another way to do estimation, and contain essentially the same information as the tangent plane. Assignment 18
Monday, October 9 Chapter 14.6: Know what the book calls the chain rule, and understand its relationship to the real chain rule: If z is a function of x and y, then dz=zxdx + zydy. Be aware that this leads to a formula for implicit differentiation, but don't bother learning the formula. Assignment 19
Wednesday, October 11 Chapter 14.7: Understand what critical points are. Be able to recognize critical points, and master the algebraic procedure for finding them. Know the three different types of critical point, and know what their contour maps look like. Assignment 20
Friday, October 13 Chapter 14.7: Memorize the second derivative test (which is much more complicated than the one from Calculus I) for classifying critical points as local maxima, local minima, or saddle points. Understand how dramatically more complicated the situation has become.
Chapter 14.7: Skim the part about global extrema. Notice that the idea (find the critical points on the interior, then test the boundary) is essentially the same as in calculus I, but the execution is much slower and more annoying.
Chapter 14.8: Skim this too. Pay attention to the pictures and surrounding discussion, not the (extremely difficult) algebra. Your computer will be able to solve these systems of equations, but only if you input the right ones.
Assignment 21
Monday, October 16 Chapter 15.1: Understand that volumes can be approximated by Riemann sums of rectangular prisms, just as areas were approximated by Riemann sums of rectangles. Understand that a double integral expresses the limit of such a Riemann sum, and so computes the exact volume. Be able to compute (definite) double integrals over a rectangular base using the fundamental theorem of calculus and/or Fubini's theorem. Be aware that Fubini's theorem is a useful tool for making hard problems easier. Assignment 21
Wednesday, October 18 Chapter 15.2: Be very good at computing double integrals when the limits of integration are given. Assignment 22
Friday, October 20 Chapter 15.2: Master setting up double integrals based on a picture or a verbal description of the region. Assignment 23
Monday, October 23 Chapter 15.3: Understand that the mechanics for an iterated triple integral are essentially the same as for an iterated double integral, but we have to integrate three times instead of two. Be comfortable with Fubini's theorem about switching order, and recognize that in principle there are six orders to choose from. Make sure you're comfortable through example two. Assignment 24
Wednesday, October 25 Chapter 15.3: Be comfortable with the idea of setting up triple integrals over an arbitrary region, and practice with as many examples as you can. (Soon you'll want to feel like you could handle problems such as Example 5 or Example 6 without having to look anything up or seek help.) Assignment 25
Friday, October 27 Chapter 12.7: Know what polar, cylindrical, and spherical coordinates are, and be comfortable converting both points and equations from polar to rectangular coordinates and back.
Chapter 15.3: Keep practicing.
Assignment 26
Monday, October 30 Chapter 15.4: Be able to recognize "polar rectangles" and similar, and to know when it might make sense to integrate using polar or spherical coordinates.
Be very clear that dxdy is not equal to drdθ, and that dxdydz is not equal to dρdθdφ. Memorize the appropriate conversions, or know exactly where to go to look them up.
Assignment 27
Wednesday, November 1 Review Chapters 13 and 14, and bring your questions to class.
Friday, November 3 Study for the exam.
Monday, November 6 Chapters 15.2 through 15.4: Practice as much as you can. Assignment 28
Wednesday, November 8 This handout: Read the text and work through the first six examples (stopping halfway down page 6). (Numerical solutions are here if you need them.) Make sure you're comfortable with what differential forms are, and in particular with their anticommutativity ([dx][dy]=-[dy][dx]). Be able to use them with the chain rule to derive the conversion factor for polar coordinates. Assignment 29
Friday, November 10 This handout (and answers): Work through the remaining examples. Understand that we use differential forms in two or more variables like we'd use u-substitution in one variable. Have some feeling for when it might make sense to use them. Make sure you can cleanly convert back and forth between [dx][dy] and [du][dv]. Assignment 30
Monday, November 13 Chapter 15.6: Understand that this chapter is addressing the same issues as the handout. Be good at changing coordinates in a double or triple integral, without messing up either the integrand or the region of integration. (This means mastering either Jacobians or differential forms.) Assignment 31
Wednesday, November 15 Chapters 15.4, 15.6, and the handout: Keep practicing. Make sure you're comfortable with coordinate changing, and understand that polar and spherical coordinates are a special kind of coordinate change. Assignment 32
Friday, November 17 Chapters 15.3 through 15.6, and the handout: Keep practicing. (For 15.5, just skim, making sure you have a cultural awareness of these applications.) Assignment 33Written assignment 2
Monday, November 20 Get some rest, after a brutal couple of months. If you want to work on math, continue to practice setting up integrals or start skimming Chapter 16.
Wednesday, November 22 Get some rest, after a brutal couple of months. If you want to work on math, continue to practice setting up integrals or start skimming Chapter 16.
Friday, November 24 Get some rest, after a brutal couple of months. If you want to work on math, continue to practice setting up integrals or start skimming Chapter 16.
Monday, November 27 Chapter 16.1: Know what vector fields are, and understand the relationship to arrow plots. Assignment 34
Wednesday, November 29 Review Chapter 15, and bring your questions to class.
Friday, December 1 Study for the exam.
Monday, December 4 Chapter 16.1: Be able to recognize conservative vector fields, based on the cross-partials condition. Given a conservative vector field, be able to find a potential function using integration. Assignment 35
Wednesday, December 6 Chapter 16.2: Be able to set up and evaluate vector line integrals. (Don't worry about scalar line integrals; we don't have time this semester.) Be able to recognize the situations where vector line integrals are appropriate.
Chapter 16.3: Be able to test vector fields for conservativeness based on the cross-partials condition, and to take advantage of conservative vector fields in vector integrals.
Assignment 36Written assignment 2
Friday, December 8 Skim Chapter 17.1, and be aware of Green's theorem, which helps us address non-path-independent integrals.
Review, and bring your questions to class.
Assignment 37
Friday, December 3 Review, and bring your questions to class. Assignment 38
Monday, December 11Study for the exam, which starts at 10:00 in our regular classroom.Extra credit AchieveExtra credit assignment on area