Homework assignments for Jeff Mermin's Math 2163 section (problems on WebAssign; see here for usage directions.)

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, and be able to work with it. WebAssign 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). WebAssign 2
Monday, August 28Chapter 12.4: Memorize the formula for the cross product. (Use the definition in terms of a "symbolic determinant" (p. 654), or the i, j, k relations (p. 656), 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. WebAssign 3
Wednesday, August 30 Chapter 12.4: Understand 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 recognize this situation algebraically (i.e., without first computing the cross product). WebAssign 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.
WebAssign 5
Monday, September 4Labor day
Wednesday, September 6 This handout: Skim to understand what the questions are. Understand the key geometric value of the cross product: x × y is perpendicular to both x and y. Pick one page at random, and have a partner do the same. Then master your page, and carefully talk your partner through it. WebAssign 6
Friday, September 8This handout: With a partner, master the remaining pages. WebAssign 7
Monday, September 11Chapter 13.1: Know what vector-valued functions are, and be able to match up 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). WebAssign 8
Wednesday, September 13 Chapter 13.2: Understand that limits, derivatives and integrals of vector-valued functions can all be computed "component-wise". (In other words, calculus works out as nicely as you could hope for these things.) Then make sure you remember how to do those things. WebAssign 9
Friday, September 15 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 can almost never be evaluated algebraically, but be able to recognize some of the very rare cases (like the ones in the book) where it can. WebAssign 10
Monday, September 18Chapters 13.2 and 13.5: Understand how vector-valued Initial Value Problems can come up (13.2, Example 8 and 13.5, Examples 2 through 4), and be able to solve simple ones like these examples.
Skim Chapter 13.4 and the rest of Chapter 13.5, enough that you'll recognize the big ideas in the future.
WebAssign 11
Wednesday, September 20 Chapter 14.1: Understand what functions of two variables are, and be able to tell their graphs apart using traces. WebAssign 12
Friday, September 22Study for the exam.
Monday, September 25Chapter 14.1: Understand what level curves and contour maps are. Be able to sketch the contour graph of a function, to read off important features from a contour map, and to distinguish functions based on their contour maps. WebAssign 13
Wednesday, September 27 Chapter 14.2: Understand that the idea of continuity is exactly the same for multivariable functions as it was for single-variable functions, and that all the familiar laws of limits still hold. But also know that the definition of limit is considerably more complicated (that is, Example 7 is a thing that happens), and that in non-obvious cases they rarely exist. WebAssign 14
Friday, September 29Chapter 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 84). WebAssign 15
Monday, October 2Chapter 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 13). Understand what differentials are (dx, dy, and dz in the chain rule), and be able to use them for estimation and finding tangent planes. WebAssign 16
Wednesday, October 4 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. WebAssign 17
Friday, October 6Chapter 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.
Reread chapters 14.4 and 14.5, noticing how the chain rule ties them together.
WebAssign 18
Monday, October 9Chapter 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. WebAssign 19
Wednesday, October 11 Chapter 14.7: Memorize the second derivative test (which is more complicated than the one you know from Calculus I) for classifying critical points as local maxima, local minima, or saddle points. Recognize how much the complexity of the situation has increased. WebAssign 20
Friday, October 13Chapters 14.4 through 14.6: Reread these chapters, noticing how the chain rule addresses every issue. WebAssign 21
Monday, October 16Chapter 14.7: Skim the part of this chapter on global extrema. Notice that the idea (find the critical points on the inside, then check the boundary) is the same as it was in the one-variable case, but that the associated algebra is much longer and more annoying.
Chapter 14.8: Skim this chapter too. Pay attention to the figures (1,2,3,5,7,8) and surrounding discussion rather than the (extremely difficult) algebra. Your computer can solve these systems of equations for you, but only if you set them up properly.
WebAssign 22
Wednesday, October 18 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. WebAssign 23
Friday, October 20 (Fall Break)
Monday, October 23Chapter 15.2: Be able to compute double integrals when the limits of integration are given. Be good at setting up double integrals (that is, be able to find appropriate limits of integration) over an arbitrary planar region. WebAssign 24
Wednesday, October 25 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. WebAssign 25
Friday, October 27 (Study for the exam.)
Monday, October 30Chapter 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. (By the end of the week 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.) WebAssign 26
Wednesday, November 1 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 15.4: Be able to recognize "polar rectangles" and similar, and to know when it might make sense to integrate using polar (or spherical) coordinates instead of rectangular coordinates.
Be very clear that dxdy is not equal to drdθ, and that dxdydz is not equal to dρdθdφ. Memorize the appropriate conversion, or know exactly where to go to look it up.
WebAssign 27
Friday, November 3This handout: Know what differential forms are, understand that [dx][dy]=-[dy][dx], and be able to use them with the chain rule to derive the conversion factors for polar and spherical coordinates. WebAssign 28
Monday, November 6This handout: Understand how differential forms are used like u-substitution in multi-dimensional integrals, and have some feel for when a substitution makes sense. Know how to convert between [dx][dy] and [du][dv]. WebAssign 29
Wednesday, November 8 Chapter 15.6: Understand that this is about the same issues as in the handout. Be able to change coordinates in a double or triple integral, without messing up either the region of integration or the integrand. This means being comfortable with either the Jacobians in chapter 15.6 or the differential forms in the handout. WebAssign 30
Friday, November 10Chapters 15.4 and 15.6: Be comfortable with coordinate changes, and with polar and spherical coordinates as a special case of a coordinate change.
Chapter 15.5: Skim this chapter. Do your best to understand at least one of the applications.
WebAssign 31
Monday, November 13Chapter 15: Make sure your comfortable setting up and evaluating double and triple integrals, in all the situations we've encountered. WebAssign 32
Wednesday, November 15 Chapter 15: Master the setting up of integrals, in all the settings we've studied. WebAssign 33
Friday, November 17Study for the exam.
Monday, November 20Chapter 16.1: Know what vector fields are, and understand the relationship to arrow plots. WebAssign 34Written assignment one. (Answers.)

Extra credit one
Wednesday, November 22 (Thanksgiving)
Friday, November 24(Thanksgiving)
Monday, November 27Chapter 16.1: Be able to find a potential function, given a conservative vector field. Be able to recognize conservative vector fields based on the cross-partial condition. WebAssign 35
Wednesday, November 29Chapter 16.2: Be able to set up and evaluate vector and scalar line integrals, and know which is appropriate in a given situation. Understand that vector integrals are usually easier. WebAssign 36
Friday, December 1Chapter 16.3: Be able to recognize situations where a vector line integral is path-independent, and take advantage of the path independence. WebAssign 37
Monday, December 4Chapter 16.3: Continue to practice taking advantage of situations where vector integrals are path-independent. WebAssign 38
Wednesday, December 6Chapter 17.1: Know the statement of Green's Theorem, and be able to use it to evaluate path integrals by taking an appropriate double integral (as in Example 2). WebAssign 36
Friday, December 8Chapter 17.1: Understand that Green's Theorem can be useful in both directions. Be able to use it to compute area with a path integral (as in Example 3), and to manipulate difficult integrals into easier ones as in Example 4. (Skim the bit on the "vector form" of Green's theorem, but don't worry about it.) WebAssign 37