## Math 212: Multivariable calculus II, Spring 2016

MTWF 12-12:50pm, Library 204
Office Hours: Mon 11-12, Wed 2-3, and by appointment in Library 313

Math
Center: SuMTWTh 7-9pm, Library 387

Textbook: *Vector Calculus*, 4th ed., by Susan Colley.

### Week 1: January 25-29

- Monday: Course overview. Hand out syllabus.
- Tuesday: Read 5.1. Area, volume, and iterated integrals.
- Wednesday: Read pp. 314 - middle of 321. Entry quiz. Double integrals over rectangles.
- Friday: Read pp. middle of 321 - middle of 328. HW due. Double integrals over general regions in the plane.

#### Homework due Friday

- Review the syllabus.
- 5.1: 2, 4, 8, 16
- 5.2: 2, 10

### Week 2: February 1-5

- Monday: Review pp.321-328. Entry quiz. More double integrals over general regions.
- Tuesday: Read the proof of Theorem 2.6 (starting on p. 328) and 5.3. HW due. Fubini's theorem and changing the order of integration.
- Wednesday: Read 5.4. Entry quiz. Triple integrals.
- Friday: HW due. Problem review. More triple integrals.

#### Homework due Tuesday

- 5.2: 28, 34, 40

#### Homework due Friday

- 5.2: 24, 25, 41 (counts as two problems)
- 5.3: 2[c], 8[c]
- 5.4: 22[c]

*Note*: Problems marked with a [c] are [computational] in
nature and do not require full sentence/paragraph-formatted answers.
You must still provide sufficient work, diagrams, and explanations so
that your work can be easily followed. These problems will be graded
on a 5-point scale. Problems without a [c] require full explanations
for full credit (5 points for mathematical insight, 3 points for
writing) *even if they appear computational in nature*.

### Week 3: February 8-12

- Monday: Review 5.4, especially the second half. Practice with triple integrals.
- Tuesday: Read these notes [pdf]. HW due. Integration in $\mathbb{R}^n$.
- Wednesday: Read pp.30 - top of 34 and bottom of 56 - 58. Entry quiz. Determinants.
- Friday: Read pp.349 - middle of 355. HW due. Determinants and coordinate transforms.

#### Homework due Tuesday

- 5.3: 12
- 5.4: 8, 26[c]

#### Homework due Friday

- 1.4: 2[c], 4[c]
- 1.6: 25, 26, 28, 29
- 5.5: 4

### Week 4: February 15-19

- Monday: Read pp.355-364. Change of variables for double integrals.
- Tuesday:
*No class.* - Wednesday: More change of variables. Review.
- Friday: First in-class exam.

### Week 5: February 22-26

- Monday: Read pp.364-371. Change of variables redux. Exam review.
- Tuesday: HW due. Common changes of variable: polar, cylindrical, and spherical coordinates.
- Wednesday: Optional reading: Shurman notes [pdf] §6.8. Entry quiz. Proof of COV: concepts and overview.
- Friday: Optional reading: Shurman notes[pdf] §6.9. HW due. Proof of COV: some details.

#### Homework due Tuesday

- 5.5: 8, 20, 25

#### Homework due Friday

- 5.5: 7, 30
- Shurman: 6.7.11, 6.7.12 (count (a-b), (c), and (d-e) as separate problems)

### Week 6: February 29 - March 4

- Monday: Read Shurman [pdf] §6.7. Entry quiz. Applications of COV.
- Tuesday: Read Colley pp.408-421. HW due. Scalar and vector line integrals.
- Wednesday: Entry quiz. More line integrals and physical applications.
- Friday: Read Colley §6.2. HW due. Green's theorem.

#### Homework due Tuesday

- Shurman: 6.7.13 and 6.7.14

#### Homework due Friday

- 5.6: 5, 22
- 6.1: 12, 22, 37, 39

### Week 7: March 7-11

- Monday: Entry quiz. Green's theorem - finish proof, examples, curl and divergence formulations.
- Tuesday: HW due. Exam review. Distribute take-home exam.
- Wednesday: Read Colley §6.3. Entry quiz. Conservative vector fields.
- Friday: Take-home exam due. Finding scalar potentials.

#### Homework due Tuesday

- 6.2: 4[c], 10, 17
- Invent a region $D$ to which you can apply the result from 17 above. Compute the area of $D$ by computing one of $\oint_{\partial D} x\,dy$ or $-\oint_{\partial D} y\,dx$.

#### Take-home exam due at the start of class Friday.

### Week 8: March 14-18

- Monday: Review §6.3. Entry quiz. Finding scalar potentials. Exam discussion.
- Tuesday: HW due. Proofs of Theorems 3.3 and 3.5.
- Wednesday: Entry quiz. Planimeter activity.
- Friday: HW due. Winding number revisited.

#### Homework due Tuesday

- 6.3: 18, 19, 20

#### Homework due Friday

- 6.3: 6[c], 8[c], 37
- p.453: 28, 30, 31, 32, 33

### Spring Break: March 21-25

### Week 9: March 28 - April 1

- Monday: Read Colley §7.1. Intro to $k$-surface integrals and parametrized surfaces.
- Tuesday: Read §7.2. HW due. Surface integrals.
- Wednesday: Entry quiz. More surface integrals.
- Friday: Read §7.3. HW due. Stokes's theorem.

#### Homework due Tuesday

- p.452: 14

#### Homework due Friday

- p.452: 15
- 7.1: 2, 4, 10, 24, 30

### Week 10: April 4-8

- Monday: Entry quiz. Stokes's theorem: examples and proof.
- Tuesday: HW due. Practice with Stokes's theorem. Intro to Gauss's theorem.
- Wednesday: Entry quiz. Proof of Gauss's theorem.
- Friday: HW due. Practice with Gauss's theorem.

#### Homework due Tuesday

- 7.2: 6, 8
- 7.3: 2

#### Homework due Friday

- 7.2: 17
- 7.3: 10, 11, 12, 23

### Week 11: April 11-15

- Monday: Read §8.1. Intro to differential forms.
- Tuesday: Algebra of forms.
- Wednesday: Exam review.
- Friday: In-class exam. You may bring one two-sided 8.5"x11" page of notes. Emphasis on material covered since Exam 2.

#### Homework due Tuesday

- 7.3: 18, 20
- A friend makes the following argument to you:
*If $S$ is a closed surface (so that $\partial S = \varnothing$) and $F$ is a $C^1$ vector field on $\mathbb{R}^3$, then by Stokes's theorem, $\iint_S (\nabla \times F)\cdot dS = \oint_\varnothing F\cdot ds = 0$.*Evaluate your friend's argument, and then justify their assertion by using Gauss's theorem.

#### In-class exam on Friday. No HW due.

### Week 12: April 18-22

- Monday: Read Shurman §§9.6-9.7. Entry quiz. Review of differential forms; algebra of forms.
- Tuesday: Read Shurman §9.8. HW due. Differentiation of forms.
- Wednesday: Read Shurman §9.9. Entry quiz. Pullback of forms.
- Friday: Read Shurman §9.10. HW due. Change of variables for forms.

#### Homework due Tuesday

- Shurman 9.3: 1, 2
- Shurman 9.5: 4
- Read through Shurman exercise 9.5.5 and convince yourself that you could execute all the steps. No need to turn anything in. (Note that Shurman uses $T'$ to denote $JT$.)

#### Homework due Friday

- Shurman 9.7: 1, 2, 4
- Shurman 9.8: 1, 2, 4, 5 (Note that Shurman uses $D_i$ to denote the $i$-th partial derivative $\partial/\partial x_i$. Further, $\langle x,y\rangle$ is used to denote the dot product $x\cdot y$.)

### Week 13: April 25-29

- Monday: Read Shurman 9.12-9.13. Entry quiz. Singular cubes, chains, manifolds, and boundaries.
- Tuesday: Read Shurman 9.14-9.15. HW due. Generalized Stokes's theorem. Classical theorems revisited.
- Wednesday: Review, synthesis, examples.
- Friday: Review, synthesis, examples. Distribute take-home final. Due May 6 by 5pm.

#### Homework due Tuesday

- 9.9.3(a)
- 9.10.1

#### Take-home final distributed in class on Friday. Due May 6 by 5pm.

### The LaTeX document preparation system

Poor handwriting? Love escape characters? Too much free time? Try LaTeX!

- LaTeX at Reed.
- A short guide [pdf] to writing mathematics with LaTeX.
- Sample LaTeX input / output: sample.tex / sample.pdf.

Kyle M. Ormsby