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Reading assigments for each lecture should be completed before the day of the lecture, so you are more prepared to understand the material when I present it. This means spending around 15 minutes skimming the text, not necessarily understanding what you're reading, but at least making sure your eyes pass over the material so your brain can start processing it unconsciously. Because 15 minutes is so little of your time, I expect you to actually do it!
Easy problems: I tend to assign lots of easy problems at the beginning of a homework assignment, so please don't be intimidated by the number of them! This is so you learn how to do certain basic tasks quickly, which will be essential to finishing your exams on time.
Due dates: homeworks assigned on Monday and Tuesday will be due at the beginning of the following Thursday's lecture, and homeworks assigned on Wenesday, Thursday, and Friday will be due at the beginning of the following Monday's lecture: MonTue → Thurs , WedThurFri → Mon..
Topics in italics are topics that I will treat significantly differently from the text (in a way that past students have found helpful!). Topics in bold are the ones I think students have the most trouble with. Reading assignments in {braces} indicate optional reading: some concepts from that section will be introduced, but the whole section will not be covered yet.
| Lec | Date | Topics | Reading (before lecture) |
Homework assignments (complete after lecture) |
| 1 | Jun 22 | Review / introduction:
● R^2 ("the plane"), R^3 ("space"). ● Sets, set products, functions ● Graphs, level sets, ranges. ● Linear/translated linear functions. ● The linear approximation formula for functions (R^1) → R^n |
10.1, 12.1, {13.1}, {14.1} | 10.1: 1, 2, 3, 4, 23, 24, 25, 26, 27.
In 10.1, when a given function is trans. linear, underline it! (once each is enough) 12.1: 1, 2, 3, 4, 6, 10, 11, 13, 15, 18, 22, 23, 24, 26, 27, 31, 39. |
| 2 | Jun 23 | ● Parametric vs. implicit equations in general.
● Implicit curves in R^2 (level sets of functions (R^2) → R^1). ● Parametric curves in R^2 (ranges of functions (R^1) → R^2). |
10.1, {14.1} | 10.1: 6, 7, 9, 10, 11, 12, 15, 16, 28, 33, 38, 41, 46.
In 10.1, when a given function is trans. linear, underline it! (once each is enough) |
| 3 | Jun 24 | ● Polar coordinates.
● The "r,θ" half-plane. ● Areas |
10.3, 10.4 | 10.3: 1a-c, 3a-c, 8, 9, 11, 15, 16, 17, 18, 19, 20, 29, 34, 35, 47, 57, 58, 63, 64.
10.4: 5, 6, 7, 8, 9, 10, 11, 12. |
| 4 | Jun 25 | ● Vectors in R^1, R^2 and R^3.
● Points vs. vectors. ● The dot product (of vectors). ● The "norm" or "magnitude" of a vector. ● The "unitization" or "direction" of a vector. ● Vectors as displacements/velocities. ● Magnitudes as distances/speeds. ● Components and projections. ● The angle cosine between vectors. |
12.2, 12.3 | 12.2: 2, 3, 4, 5, 6, 7, 10, 13, 17, 22, 23, 24, 25, 26, 27, 32, 37.
12.3: 3, 4, 5, 6, 9, 10, 11, 13, 14, 15, 20, 23a-d, 25, 26, 35, 36, 41, 43, 50, 51, 53, 58. |
| 5 | Jun 26 | ● Ordered parallelotopes.
● Determinants and signed areas/volumes. ● The cross product (of vectors in R^3). ● The "scalar triple product" (cross product job #1). |
12.4 | 12.4: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13a-f, 20, 21, 22, 33, 34, 38, 45, 49a-c. |
| 6 | Jun 29 | ● Implicit and parametric equations of lines and planes.
● Implicit equations of cylinders and quadric surfaces. |
12.5, 12.6 | 12.5: 1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 14, 17, 23, 24, 27, 28, 43, 46, 49, 50, 62, 63.
12.6: 1, 3, 4, 7, 8, 21-28. |
| 7 | Jun 30 | ● Functions (R^1) → R^n as "vector functions" and "parametric curves".
● Derivatives and integrals of vector functions. ● The linear approximation formula for (R^1) → R^n. ● Tangents to parametric curves (LAF!) ● Arc lengths |
13.1, 13.2, {13.3}, |
13.1: 1, 2, 3, 5, 7, 15, 18, 19, 20, 25, 28, 41, 42.
13.2: 3, 5, 9, 11, 13, 15, 17, 19, 24, 25, 33, 38. 13.3 1, 2, 3, 4. 10.2: 3, 4, 7, 8, 17, 18, 41, 42. 10.4: 45, 46, 47, 48. |
| 8 | Jul 1 | ● Functions (R^n) → R^1 as "functions of several variables" and "scalar fields".
● Implicit surfaces in R^3 (level sets of functions (R^3) → R^1) ● Limits and continuity. |
14.1, 14.2 | 14.1: 6, 9, 18, 19, 21, 23, 31, 32, 35, 36, 39, 40, 49, 50.
14.2: 1, 2, 5, 7, 9, 13, 19, 25, 26, 29, 31, 37, 44. |
| 9 | Jul 2 | ● The definition of vector derivations (aka "directional derivatives").
● Partial derivatives and Clairaut's Theorem. |
{14.6}, 14.3 | 14.3: 5, 8, 9, 15, 16, 18, 19, 21, 22, 24, 25, 31, 37, 39, 42. |
| * | Jul 3 | NO CLASS | NO CLASS | |
| 10 | Jul 6 | ● Vector derivations and the gradient vector.
● The linear approximation formula for (R^n) → R^1 (the gradient). ● The gradient as the direction and magnitude of maxmimum increase. |
14.6 without tangent planes |
14.6: 4, 5, 6, 7, 8, 9, 11, 13, 20, 21, 22, 23, 24, 27, 28, 29, 35, 37. |
| 11 | Jul 7 | ● Differentiability = linear approximability.
● Tangent lines/planes as trans. linear approximations to graphs, parametrizations, and level sets. (LAF!) |
14.4, 14.6 | 14.4: 1, 3, 5, 6, 11, 12, 15, 16, 17, 18, 33, 34, 42, 45.
14.6: 39, 40, 41, 43, 47, 49, 50, 52, 53, 55, 56, 57. |
| 12 | Jul 8 | ● Composition of trans. linear functions.
● The chain rule: "total change = sum of responsible changes". ● Implicit differentiation (pretending you've isolated a variable). | .
14.5 | 14.5: 1, 3, 5, 7, 9, 10, 11, 12, 13, 14, 21, 22, 24, 27, 28, 31, 34, 48, 49, 50. |
| 13 | Jul 9 | ● Max/mins on an open set: critical points, the Hessian determinant and the second derivative test.
● (Topological) interior, exterior, and boundary points. ● Open and closed sets defined. ● Max/mins on a closed set. ● Max/mins on parametric curves: single-variable calculus! [[[[[Bring text; it's great for this section!]]]]] |
14.7 | 14.7: 1, 2, 5, 6, 9, 10, 13, 14, 19, 29, 30, 33, 34, 39, 42, 43, 44, 50, 51. |
| 14 | Jul 10 | ● Max/mins on a level set: Lagrange multipliers.
[[[[[Bring text; it's great for this section!]]]]] |
14.8 | 14.8: 3, 4, 9, 10, 13, 14, 17, 18, 19, 20, 40, 41, 42. |
| 15 | Jul 13 | REVIEW | REVIEW | |
| * | Jul 14 | MIDTERM | MIDTERM | |
| 16 | Jul 15 | ● Double integrals over rectangles.
● Iterated inegrals. |
{15.1}, 15.2 | 15.1: 1, 3, 11, 13, 17.
15.2: 1, 3, 5, 7, 8, 9, 11, 13, 15, 17, 19, 25, 27. |
| 17 | Jul 16 | ● Double integrals over general regions.
● "dxdy" and "dydx" type regions. ● Double integrals in polar coordinates. ● "drdθ" type regions. |
15.3, 15.4 | 15.3: 1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 31, 39, 45, 55.
15.4: 1, 3, 7, 9, 11, 15, 19, 23, 29, 36, 37. |
| 18 | Jul 17 | ● Triple integrals.
● "dzdydx" type regions, etc.. ● Visualization in terms of shadows. ● Linearly defined regions (polytopes) and the "corners" method. |
15.6 | 15.6: 3, 5, 7, 9, 11, 13, 17, 19, 20, 21, 22. |
| 19 | Jul 20 | ● Triple integrals in cylindrical coordinates.
● Triple integrals in spherical coordinates. |
15.7, 15.8 | 15.7: 1, 3, 5, 9, 15, 16, 17, 19, 21, 22, 23a, 27.
15.8: 1, 3, 5, 7, 11, 17, 19, 21, 23, 25, 27, 29a, 35. |
| 20 | Jul 21 | ● Visualization techniques for cylindrical and spherical coordinates.
● The "r,z" half-plane. ● The "φ,ρ" quarter-plane. |
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| 21 | Jul 22 | ● Functions (R^n) → R^n as "changes of variables" (parametrizations).
● Matrices and linear maps (R^n) → R^n. ● The linear approximation formula for (R^n) → R^n (the Jacobian matrix). ● The Jacobian determinant as a signed stretch factor. ● Change of variables in integrals. |
15.9 | 15.9: #1-6 see handout; 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24. |
| 22 | Jul 23 | ● Visualizing changes of variables: transforming regions via points and functions.
● Visualizing linear changes of variables: "simultaneous axes" method. ● Density and mass, moments and center of mass. |
{15.5} | 15.5: 3, 5, 7, 9, 11, 13, 15.
15.6: 37, 39. 15.7: 23b 15.8: 29b. |
| 23 | Jul 24 | ● Monomial rectangles: a one sub method.
● REVIEW |
REVIEW | |
| * | Jul 27 | MIDTERM | MIDTERM | |
| 24 | Jul 28 | ● Points, vectors, covectors and scalars: what "dx" and "dy" mean on their own!
● Scalar fields. ● Functions (R^n) → R^n as "(co)vector fields". ● Curve ("line") integrals of scalar fields wrt curve length. |
16.1, {16.2} | 16.1: 1, 5, 7, 9, 11, 12, 13, 14, 21, 23.
16.2: 1, 3, 5, 7, 9, 11, 12, 13, 15, 19, 21, 33, 35, 46. |
| 25 | Jul 29 | ● Oriented curves.
● O.C. integrals of (co)vector fields. ● The Fundamental Theorem for O.C. Integrals (1-D integrals ↔ 0-D integrals, in R^n) ● Gradient fields = path independent fields = conservative fields. ● Scalar curl (R^2 only): testing for gradient fields. |
16.2, 16.3 | 16.3: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. |
| 26 | Jul 30 | ● 0-, 1-, and 2-connected regions.
● The de Rham sequence in R^2. (grad-scurl) ● The Poincaré lemma in R^2. ● Scurl as "circulation per area" at a point. ● Green's (Scurl) Theorem (2-D integrals ↔ 1-D integrals, in R^2). |
16.4 | 16.4: 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 19, 22, 29. |
| 27 | Jul 31 |
● Using scurl for O.C. integrals: (1) path independence, (2) solving ∇f=F, and (3) Green's Theorem.
● Why is scurl = circulation-per-area at a point? ● Why is circulation additive? |
<none> | |
| 28 | Aug 3 | ● The curl of a vector field:
(1) CurlF•(a×b) as approximate circulation about ordered parallelogram a,b at a point. (curl-Jacobian trick) (2) The axis vector of rotation at a point. ● The divergence of a vector field: "outflux per volume" at a point. ● The de Rham sequence in R^3 (grad-curl-div). ● The Poincaré lemma in R^3. |
16.5 | 16.5: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12=important!, 13, 14, 15, 16, 17, 19, 20, 21, 23, 33, 34. |
| 29 | Aug 4 | ● Differential boundaries in R^2: how not to confuse Green's Theorem.
● True/false practice with de Rham and Poincaré. | <none> | |
| 30 | Aug 5 | ● Parametric surfaces in R^3 (ranges of functions (R^2) → R^3). ● "Implied equations" describe supersets. ● The linear approximation formula for (R^2) → R^3 (the Jacobian matrix). ● Tangent planes to parametric surfaces. ● Surface integrals of scalar fields wrt surface area. ● Oriented surfaces. ● O.S./O.C. fluxes (integrals) of vector fields. |
16.6, 16.7 | 16.6: 1, 3, 5, 13, 14, 19, 20, 21, 23, 37, 39, 41, 43, 45.
16.7: 1, 3, 4, 5, 6, 7, 9, 10, 11, 13, 15, 17, 19, 20, 21, 23, 25, 27, 29, 40. |
| 31 | Aug 6 | ● Differential boundaries in R^3.
● Stokes' (Curl) Theorem (2-D integrals ↔ 1-D integrals, in R^3). ● Gauss' (Divergence) Theorem (3-D integrals ↔ 2-D integrals, in R^3). |
16.8, 16.9 | 16.8: 1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 14, 16.
16.9: 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 24. |
| 32 | Aug 7 | ● Using curl for O.C. integrals: (1) path independence, (2) solving ∇f=F, and
(3) Stokes' Theorem.
● Curl fields = "patch independent" fields = "bubble-conservative" fields. ● Using divergence for O.S. integrals: (1) patch independence and (2) Gauss' Theorem. (hard to solve curlF=G!) |
<none> | |
| 33 | Aug 10 | Summary of Multivariable Calculus | Handout | <none> |
| 34 | Aug 11 | Using multivariable calculus (1) | Handouts | |
| 35 | Aug 12 | Using multivariable calculus (2) | Handouts | |
| 36 | Aug 13 | PRE-FINAL | PRE-FINAL | |
| * | Aug 14 | FINAL EXAM | FINAL |